Introduction

From the point of view taken in these lectures, motivic cohomology with coefficients in an abelian group A is a family of contravariant functors

H^{p,q}(−, A) : Sm/k → Ab

from smooth schemes over a given field k to abelian groups, indexed by integers p and q. The idea of motivic cohomology goes back to P. Deligne, A. Beilinson and S. Lichtenbaum.

Most of the known and expected properties of motivic cohomology (predicted in ^{\underline{BMS}} [ABS87] and ^{\underline{Licht2}} [Lic84]) can be divided into two families. The first family concerns properties of motivic cohomology itself – there are theorems concerning homotopy invariance, Mayer-Vietoris and Gysin long exact sequences, projective bundles, etc. This family also contains conjectures such as the Beilinson-Soule vanishing conjecture (H^{p,q} = 0 for p < 0) and the Beilinson-Lichtenbaum conjecture, which can be interpreted as a partial étale descent property for motivic cohomology. The second family of properties relate motivic cohomology to other known invariants of algebraic varieties and rings. The power of motivic cohomology as a tool for proving results in algebra and algebraic geometry lies in the interaction of the results in these two families; specializing general theorems about motivic cohomology to the cases when they may be compared to classical invariants, one gets new results about these invariants.

The idea of these lectures was to define motivic cohomology and to give careful proofs for the elementary results in the second family. In this sense they are complimentary to the study of ^{\underline{orange}} [VSF00], where the emphasis is on the properties of motivic cohomology itself. In the process, the structure of the proofs force us to deal with the main properties of motivic cohomology as well (such as homotopy invariance). As a result, these lectures cover a considerable portion of the material of ^{\underline{orange}} [VSF00], but from a different point of view.

One can distinguish the following “elementary” comparison results for motivic cohomology:

1. H^{p,q}(X, A) = 0 for q < 0, and for a connected X one has

H^{p,0}(X, A) = \{ \begin{matrix}A & for p = 0 \\ 0 & for p ≠ 0 \end{matrix}

2. one has

H^{p,1}(X, ℤ) = \{ \begin{matrix}O^{∗}(X) & for p = 1 \\ P ic(X) & for p = 2 \\ 0 & for p ≠ 1, 2 \end{matrix}

3. for a field k, one has H^{p,p}(Spec(k), A) = K_{p}^{M}(k) ⊗ A where K_{p}^{M}(k) is the p-th Milnor K-group of k

4. for a strictly henselian local scheme S over k and an integer n prime to char(k), one has

H^{p,q}(S, ℤ/n) = \{ \begin{matrix}μ_{n}^{⊗q}(S) & for p = 0 \\ 0 & for p ≠ 0 \end{matrix}

where μ_{n}(S) is the groups of n-th roots of unity in S

5. one has H^{p,q}(X, A) = CH^{q}(X, 2q − p; A). Here CH^{i}(X, j; A) denotes the higher Chow groups of X introduced by S. Bloch in ^{\underline{acahkt}} [Blo86], ^{\underline{hmov}} [Blo94]. In particular,

H^{2q,q}(X, A) = CH^{q}(X) ⊗ A,

where CH^{q}(X) is the classical Chow group of cycles of codimension q modulo rational equivalence.

Deeper comparison results include the theorem of M. Levine comparing CH^{i}(X, j; ℚ) with the graded pieces of the gamma filtration in K_{∗}(X) ⊗ ℚ \underline{Levine2} [Lev94], and the construction of the spectral sequence relating motivic cohomology and algebraic K-theory for arbitrary coefficients in ^{\underline{BL}} [BL94] and ^{\underline{FS}} [FS00].

The lectures in this book may be divided into two terms. The fall term lectures contain the definition of motivic cohomology and the proofs for all of the comparison results listed above except the last one. The spring term lectures contain more advanced results in the theory of sheaves with transfers and the proof of the final comparison result (5).

The definition of motivic cohomology which is used here goes back to the work of Andrei Suslin in about 1985. As far as I understand, when he came up with his idea he was able to prove the first three of the comparison results stated above. In particular the proof of the comparison (3) between motivic cohomology and Milnor’s K-groups given in these lectures is exactly Suslin’s original proof. The proofs of the last two comparison results (4) and (5) are also based on results of Suslin. Suslin’s formulation of the Rigidity Theorem (Theorem ^{\underline{loccons}} 7.20) is a key result needed for the proof of (4), and Suslin’s moving lemma (to appear in the second part of the lectures) is a key result needed for the proof of (5).

Two main new ideas appeared since then and made it possible to finish the proofs of the comparisons (4) and (5). The first one, which originated in the context of the qfh-topology and was later transferred to sheaves with transfers (definition ^{\underline{pst}} 2.1), is that the sheaf of finite cycles ℤ_{tr}(X) is the free object generated by X. This idea led to a group of results which is represented here by lemma ^{\underline{etext}} 6.23. The second one which is the main result of ^{\underline{preth}} [CohTh] is represented here by theorem ^{\underline{theorem''}} 13.7. Taken together they allow one to efficiently do homotopy theory in the category of sheaves with transfers.

A considerable part of the first half of the lectures is occupied by the proof of (4). Instead of stating it in the form used above, I prove a more detailed theorem. For a given weight q, the motivic cohomology groups H^{p,q}(X, A) are defined as the hypercohomology (in the Zariski topology) of X with coefficients in a complex of sheaves A(q)|_{X_{Zar}}. This complex is the restriction to the small Zariski site of X (i.e., the category of open subsets of X) of a complex A(q) defined on the site of all smooth scheme over k with the Zariski and even the étale topology. Restricting A(q) to the small étale site of X, we may consider the étale or Lichtenbaum version of motivic cohomology:

H_{L}^{p,q}(X, A) := ℍ_{et}^{p}(X, A(q)|_{X_{et}})

Theorem ^{\underline{lichet}} 10.2 asserts that the étale motivic cohomology of any X with coefficients in ℤ/n(q) where n is prime to char(k) are isomorphic to H_{et}^{p}(X, μ_{n}^{⊗q}). This implies the comparison result (4), since the Zariski and the étale motivic cohomology of a strictly henselian local scheme X clearly agree. There should also be analog of (4) for the case of ℤ/p coefficients where p = char(k), but I do not know much about it.

This book is written by Carlo Mazza and Charles Weibel on the basis of the lectures on the motivic cohomology which I gave at the Institute for Advanced Study in Princeton in 1999/2000. At this moment only the first half of the text, corresponding to the fall term of 1999, is available.

Vladimir Voevodsky Institute for Advanced Study

May 2001.

Introduction to the second part.

The main goals of the second part are to introduce the triangulated category of motives, and to prove the final comparison theorem (5). Both require understanding the passage from a homotopy invariant presheaf with transfers to its associated sheaves for the Zariski and Nisnevich topologies.

The first few lectures (11, 12 and 13) investigate some basic properties of homotopy invariant presheaves with transfers, and their associated Nisnevich sheaves. A crucial role will be played by theorem ^{\underline{theorem''}} 13.7: if F is a homotopy invariant presheaf with transfers, and k is a perfect field, then the associated Nisnevich sheaf F_{Nis} is homotopy invariant, and so is its cohomology. For expositional purposes, the proof of this result is postponed and occupies lectures 20 to 23.

In lectures 14 and 15 we introduce the triangulated category of motives DM_{Nis}^{eff.−}(k, R) and study its basic properties. In particular we give a projective bundle theorem (^{\underline{motivepe}} 15.12) and show that the product on motivic cohomology (defined in ^{\underline{mult}} 3.11) is graded-commutative.

Lectures 16 to 19 deal with algebraic cycles, leading up to the proof of the final comparison theorem (5), that for any smooth separated scheme X over a perfect field k, we have:

H^{p,q}(X, ℤ) ≅ CH^{q}(X, 2q − p).

There are three main ingredients. First we show (in ^{\underline{thm2}} 16.7) that the motivic complex ℤ(i) is quasi-isomorphic to the Suslin-Friedlander chain complex ℤ^{SF} (i), which is built using equidimensinal cycles; our proof of this requires the field to be perfect. Then we show (in ^{\underline{hcgpst}} 17.20) that Bloch’s higher Chow groups are presheaves with transfers over any field. The final ingredient is a result of Suslin (^{\underline{equiqiso}} 18.3) comparing equidimensional cycles to higher Chow groups over any affine scheme.

The final lectures (20 to 23) are dedicated to the proof of ^{\underline{theorem''}} 13.7. Using technical results from lecture 20, we first prove (in ^{\underline{n0}} 21.3) that F_{Nis} is homotopy invariant. The proof that its cohomology is homotopy invariant occupies most of lecture 23. We conclude with a proof that the sheaf F_{Nis} admits a “Gersten” resolution.

Contents

1 The category of finite correspondences 7

1A The category Cor_{S} . . . . . . . . . . . . . . . . . . . . . . . . . 11 2 Presheaves with transfers 15 3 Motivic cohomology 22 4 Weight one motivic cohomology 26 5 Relation to Milnor K-Theory 30 6 Étale sheaves with transfers 36 7 Relative Picard group and ... 45 8 Derived tensor products 53

8A Tensor Triangulated Categories . . . . . . . . . . . . . . . . . . . 60 9 𝔸^{1}-weak equivalence 64 10 Étale motivic cohomology and ... 72 11 Standard triples 78 12 Nisnevich sheaves 84 13 Nisnevich sheaves with transfers 90 14 The category of motives 95 15 The complex ℤ(n) and ℙ^{n} 101 16 Equidimensional cycles 106 17 Higher Chow groups 110

17A Cycle maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 18 Higher Chow groups and ... 123

18A Generic Equidimensionality . . . . . . . . . . . . . . . . . . . . . 129 19 Motivic cohomology and ... 132 20 Covering morphisms of triples 138 21 Zariski sheaves with transfers 146 22 Contractions 153 23 Homotopy Invariance of Cohomology 158

Lecture 1

The category of finite correspondences

In this lecture we shall define the additive category Cor_{k} of finite correspondences over a field k. The objects of Cor_{k} will be the smooth separated schemes of finite type over k. The morphisms in Cor_{k} from X to Y will be the finite correspondences, which are special kinds of cycles in X × Y. Composition is defined so that Cor_{k} contains the category Sm/k of smooth separated schemes of finite type over k.

Our point of view will be that a cycle in a scheme T is a formal ℤ-linear combination of irreducible closed subsets of T. Each irreducible closed subset W is the support of its associated integral subscheme W̃ so W and W̃ determine each other. Thus we can ascribe some algebraic properties to W. We say that W is finite along a morphism T → S if the restriction W̃ → S is a finite morphism. A cycle ∑ n_{i}W_{i} is said to be finite along a morphism if each W_{i} is finite.

elcor Definition 1.1. If X is a connected scheme and Y any separated scheme, an elementary correspondence from X to Y is an irreducible closed subset W of X × Y whose associated integral subscheme is finite and surjective over X. By an elementary correspondence from a non connected scheme X to Y, we will mean an elementary correspondence from a connected component of X to Y.

The group Cor(X, Y) is the free abelian group generated by the elementary correspondences from X to Y. The elements of Cor(X, Y) will be called finite correspondences.

If X is not connected and X = ∪X_{i} is the decomposition into its irreducible components, our definition implies that Cor(X, Y) = ⊕_{i}Cor(X_{i}, Y).

embed Example 1.2. Let f : X → Y be a morphism in Sm/k. If X is connected, the graph Γ_{f} of f is an elementary correspondence from X to Y. If X is not connected, the sum of the components of Γ_{f} is a finite correspondence from X to Y. Indeed the projection Γ_{f} → X is an isomorphism, and Γ_{f} is closed because Y is separated over k.

Week 2 - October 13, 1999

For example the graph Γ_{1} of the identity on X is the support of the diagonal Δ(X) ⊂ X × X. We will write id_{X} for the finite correspondence Γ_{1} from X to itself. It will be the identity element of Cor(X, X) for the composition product. Note that id_{X} is an elementary correspondence when X is integral.

If X is connected and f : X → Y is finite and surjective, the transpose of Γ_{f} in Y × X is a finite correspondence from Y to X. This explains the name “finite correspondence”.

fincor Lemma 1.3. Every subscheme Z of X × Y which is finite and surjective over X determines a finite correspondence [Z] from X to Y.

Proof. If Z is integral then its support [Z] is by definition an elementary correspondence. In general we associate to Z the finite correspondence ∑ n_{i}W_{i}, where the W_{i} are the irreducible components of the support of Z which are surjective over X and n_{i} is the geometric multiplicity of W_{i} in Z, i.e., the length of the local ring of Z at W_{i} (See ^{\underline{serre}} [Ser65] or ^{\underline{fulton}} [Ful84]).

We must now define an associative and bilinear composition for finite correspondences. For this, it suffices to define the composition W ∘ V of elementary correspondences V ∈ Cor(X, Y) and W ∈ Cor(Y, Z). Our definition will use the push-forward of a finite cycle.

Let p : T → S be any morphism. If W is a irreducible closed subset of T finite along p, the image V = f(W) is a closed irreducible subset of S and d = [k(W) : k(V)] is finite. In this case we define the push-forward of the cycle W along p to be the cycle p_{∗}W = d · V; see ^{\underline{fulton}} [Ful84]. By additivity we may define the push-forward of any finite cycle along p.

finpush Lemma 1.4. Suppose that f : T → T′ is a morphism of separated schemes of finite type over a Noetherian base S. Let W be an irreducible closed subset of T which is finite over S. Then f(W) is closed and irreducible in T′ and finite over S. If W is finite and surjective over S, then so is f(W).

Proof. By Ex.II.4.4 of ^{\underline{hart}} [Har77], f(W) is closed in T′ and proper over S. Since f(W) has finite fibers over S, it is finite over S by ^{\underline{ega3}} [EGA III, 4.4.2]. If W → S is surjective, so is f(W) → S.

Let Ṽ and W̃ be the underlying integral subschemes associated to V and W respectively. Form the intersection T = (Ṽ × Z) ∩ (X × W̃) in X × Y × Z. The composition W ∘ V of V and W is defined to be the push-forward of the finite correspondence [T], along the projection p : X × Y × Z → X × Z; see ^{\underline{fulton}} [Ful84]. By lemma ^{\underline{inter}} 1.7 below, the cycle [T] is finite over X × Z. Thus the push-forward p_{∗}[T] is defined; it is a finite correspondence from X to Z by lemma ^{\underline{fincor}} 1.3.

We can easily check that id_{X} is the identity of Cor(X, X), and that the composition of finite correspondences is associative and bilinear (see ^{\underline{manin}} [Man68] and ^{\underline{fulton}} [Ful84, 16.1]).

Definition 1.5. Let Cor_{k} be the category whose objects are the smooth separated schemes of finite type over k and whose morphisms from X to Y are elements of Cor(X, Y). It follows from the above remarks that Cor_{k} is an additive category with ∅ as the zero object.

ega Lemma 1.6. Let Z be an integral scheme, finite and surjective over a normal scheme S. Then for every morphism T → S with T connected, every component of T ×_{S} Z is finite and surjective over T.

Proof. See ^{\underline{ega4}} [EGA IV, 14.4.4].

Recall that two subvarieties Z_{1} and Z_{2} are said to intersect properly if Z_{1} ∩ Z_{2} = ∅ or codim(Z_{1} ∩ Z_{2}) = codim Z_{1} + codim Z_{2}.

inter Lemma 1.7. Let Ṽ ⊂ X×Y and W̃ ⊂ Y ×Z be subschemes finite and surjective over X and Y respectively. Then Ṽ ×Z and X ×W̃ intersect properly, and each component of the push-forward of the cycle [T] of T = (Ṽ × Z) ∩ (X × W̃) is finite and surjective over X.

Proof. Without loss of generality we can suppose both X and Y connected. We form the pullback of Ṽ and W̃.

[figure: Ṽ ×_{Y} W̃; W̃; Z]

[figure: Ṽ; Y; f.surj.]

X {f.surj. \atop }

By ^{\underline{ega}} 1.6, each component of Ṽ ×_{Y} W̃ is finite and surjective over Ṽ and therefore over X too. The image T of the evident map Ṽ ×_{Y} W̃ → X × Y × Z is the intersection of Ṽ × Z and X̃ × W. Thus each irreducible component T_{i} of T is the image of an irreducible component of Ṽ ×_{Y} W̃. By ^{\underline{finpush}} 1.4, we know that each T_{i} is finite and surjective over X. Therefore dim T_{i} = dim X for all i, i.e., Ṽ ×Z and X × W̃ intersect properly.

Let p(T_{i}) denote the image of T_{i} under the map p : X × Y × Z → X × Z. By lemma ^{\underline{finpush}} 1.4, each p(T_{i}) is an irreducible closed subscheme of X × Z which is finite and surjective over X. Since the components of p_{∗}[T] are the supports of the p(T_{i}), we are done.

Remark 1.8. It is possible to reformulate finite correspondences in terms of the category Cor_{S}, where S is a Noetherian scheme; see ^{\underline{relchow}} [RelCy]. The objects of this category are the schemes of finite type over S. The morphisms are cycles satisfying certain technical conditions. An advantage of this approach is that we could work with singular schemes, even though we will use only smooth schemes during this series of lectures. Therefore we refer the reader to the appendix of this lecture for a general outline of this approach.

We can think of Cor_{k} as an enlargement of the category Sm/k of smooth schemes over k. Indeed, these categories have the same objects, and it is a routine computation (exercise!) to check that Γ_{g} ∘ Γ_{f} equals Γ_{g∘f}. We define a faithful functor Sm/k → Cor_{k} by:

X ↦ X (f : X → Y) ↦ Γ_{f}.

The category Cor_{k} possesses one more important structure: tensor product.

tensor Definition 1.9. If X and Y are two objects in Cor_{k}, then we define the tensor

product of X and Y as:

X ⊗ Y = X × Y.

If V and W are elementary correspondences from X to X′ and from Y to Y′, then V × W gives a finite correspondence from X ⊗ Y to X′ ⊗ Y′ by ^{\underline{fincor}} 1.3.

It is easy to verify that ⊗ makes Cor_{k} a symmetric monoidal category (see \underline{mac} [Mac71]).

rateq Exercise 1.10. If S = Spec k then Cor_{k}(S, X) is the group of zero-cycles in X. If W is a finite correspondence from 𝔸^{1} to X, and s, t : Spec k → 𝔸^{1} are k-points, show that the zero-cycles W ∘ Γ_{s} and W ∘ Γ_{t} are rationally equivalent (Cf. ^{\underline{fulton}} [Ful84, 1.6]).

galcor Exercise 1.11. Let x be a closed point on X, considered as a correspondence from S = Spec(k) to X. Show that the composition S → X → S is multiplication by the degree [k(x) : k], and that X → S → X is given by X × x ⊂ X × X.

In particular, if L/k is a finite Galois extension with Galois group G and T = Spec(L), show that Cor_{k}(T, T) ≅ ℤ[G] and that T → S → T is the norm ∑_{g∈G} Γ_{g}. Conclude that Cor_{k}(S, Y) ≅ Cor_{k}(T, Y)^{G} for every Y in Sm/k.

kfunct Exercise 1.12. If k ⊂ F is a field extension, there is an additive functor Cor_{k} → Cor_{F} sending X to X_{F}. If F is finite and separable over k, there is an additive functor Cor_{F} → Cor_{k} sending U to U. These are adjoint: if U is smooth over F and X is smooth over k, there is a canonical identification:

Cor_{F} (U, X_{F}) = Cor_{k}(U, X).

limcor Exercise 1.13. (a) Let F be a field extension of k and X and Y two smooth schemes over k. Writing X_{F} for X ×_{Spec k} Spec F and so on, show that Cor_{F} (X_{F} , Y_{F}) is the limit of the Cor_{E}(X_{E}, Y_{E}) as E ranges over all finitely generated field extensions of k contained in F.

(b) Let F be the function field of a smooth variety S over k and X → S be a smooth morphism. Show that Cor_{F} (X ×_{S} Spec F, Y ×_{k} Spec F) is the direct limit of the Cor_{k}(X ×_{S} U, Y) as U ranges over all open subschemes of S.

Appendix 1A - The category Cor_{S}

It is possible to generalize the notion of finite correspondence to construct a category Cor_{S}, associated to any Noetherian scheme S; see ^{\underline{relchow}} [RelCy]. The objects of this category are the schemes of finite type over S and the morphisms are the elements of an abelian group Cor_{S}(X, Y) whose elements are the “universally integral” cycles W ⊂ X ×_{S} Y, each component of which is finite and surjective over X. The pull-backs of universally integral cycles are always defined.

In order to compose an elementary correspondence V in Cor_{S}(X, Y) with a correspondence W in Cor_{S}(Y, Z) we must form the pullback W along V → Y to a cycle in W_{V} in V ×_{S} Z ⊂ X ×_{S} Y ×_{S} Z (See ^{\underline{a8}} 1A.9). This is why we need to restrict to universally integral cycles, because not every cycle has such a pullback.

Relabeling, we are reduced to the following basic setup for pulling back cycles. We are given a cycle W in X, a structure map X → S and a map V → S. The problem is to define a pullback cycle W_{V} in X ×_{S} V in a natural way. This is easy if V is flat over S (see ^{\underline{fulton}} [Ful84, 1.7]), but in general the problem is quite difficult even for V = Spec K.

One way to attack the problem is to use discrete valuation rings (DVR’s), introducing the notion of pullback along a fat point of S. This approach was introduced in ^{\underline{relchow}} [RelCy]. Recall that if K is a field, a K-point of S (or point) is a morphism Spec K → S.

Definition 1A.1. A fat point of S is a DVR D, a field K and morphisms

Spec K → Spec D → S,

so that the point of Spec K goes to the closed point of Spec D and the generic point of Spec D goes to a generic point of S. We say that the fat point lies over the underlying K-point Spec K → S.

Every point s in S has a fat point lying over it in the sense that there is a field extension k(s) ⊂ K and a fat point over Spec K → S. If s is a generic point of S, this is trivial (D = K). Otherwise, it is well known that for every two points s < s′ in a Noetherian scheme S there exists a DVR D and a map Spec D → S sending the two points of Spec D to s and s′; see ^{\underline{ega1}} [EGA I, 6.5.8].

1a.2 Theorem 1A.2. Let D be a DVR with field of fractions F. If X is a scheme of finite type over D and W_{F} is closed in the generic fiber X_{F} then there exists a unique closed subscheme W_{D} of W_{F} in X which is flat over Spec D.

Proof. Locally X has coordinate ring A, X_{F} has coordinate ring A ⊗_{D} F, and W_{F} has coordinate ring (A ⊗_{D} F)/(f_{1}, …, f_{n}), where f_{i} ∈ A for every i = 1, …, n. Let R_{0} be A/(f_{1}, …, f_{n}) and let R be R_{0}/I where I is the torsion submodule of the D-module R_{0}. Is is easy to see that R is independent of the choice of the f_{i}’s. Locally W_{D} is Spec R.

We are not yet able to define the pullback along a K-point, but using the previous theorem we can define a pullback along a fat point.

Given a fat point of S over a K-point s, and a closed subscheme W in X, we may form the flat pullback W_{F} along Spec F → S and the closed subscheme W_{D} as in ^{\underline{1a.2}} 1A.2. Then we define the pullback of W to be the cycle associated to the fiber W_{s} of the scheme W_{D} over the closed point Spec K of D. The pullback [W_{s}] is a cycle in X_{K} = X ×_{S} Spec K. Thus for every fat point over s we have a candidate for the pullback of W along s. However, two fat points over the same K-point may give two possibly distinct candidates.

Example 1A.3. Let S be the node over a field k and X its normalization. There are two fat points over the singular point s ∈ S, corresponding to the two k-points of X_{s} = {p_{0}, p_{1}}. The pullbacks of W = X along these fat points are [p_{0}] and [p_{1}], respectively.

If W is flat and equidimensional over S (or s is generic) the pullback just defined coincides with the classic pullback of a cycle along the K-point (see \underline{relchow} [RelCy, 3.2.4]), so it is independent of the choice of the fat point.

In order to have a useful object we need to get rid of the dependency of the pullback from the choice of the fat point.

Definition 1A.4. Let W = ∑ n_{i}W_{i} be a cycle on X. We say that W is dominant over S if each term W_{i} of W is dominant over a component of S. We say that a dominant cycle W is a relative cycle on X over S if its pullbacks coincide along all fat points over any common K-point; see ^{\underline{relchow}} [RelCy, 3.1.3].

As in ^{\underline{relchow}} [RelCy], we write Cycl(X/S, r) for the free abelian group of the relative cycles W on X over S such that each component has dimension r over S. It turns out that every effective relative cycle in Cycl(X/S, r) must be equidimensional over S; see ^{\underline{relchow}} [RelCy, 3.1.7]. If S is normal, this is also a sufficient condition; the following result is proven in ^{\underline{relchow}} [RelCy, 3.4.2].

a4 Theorem 1A.5. If S is normal or geometrically unibranch and W is a cycle on X which is dominant equidimensional over S, then W is a relative cycle.

We now have a good definition for the pullback of a relative cycle along a map Spec K → S supporting a fat point. We want to generalize this to pullbacks along any Zariski point s of S. However, there may be no fat points over s. For example, it may be that every fat point Spec K → S has K inseparable over k(s). To fix this, it turns out that we need to invert the characteristic p.

Example 1A.6. Let k be a purely inseparable extension of k_{0} with [k : k_{0}] = p and set X = Spec(k[t]). Let S = Spec A where A ⊂ k[t] is the ring of polynomials f(t) where f(0) ∈ k_{0}. If T_{0} = Spec k_{0} is the origin of S then the pullback of W = X to T = X ×_{S} T_{0} = Spec k is 1/p times [T], because every fat point of S must lie over a field extension of k.

Given f : V → S, the pullback W_{V} of a relative cycle W is a unique and well-defined relative cycle of X ×_{S} V over V, except that the coefficients may lie in ℤ[1/p] in characteristic p. It is characterized by the fact that for every point v of V and every fat point of V over v, the pullbacks of (W_{V})_{v} and W_{f(v)} agree. See ^{\underline{relchow}} [RelCy, 3.3.1].

See Example 3.5.10 in ^{\underline{relchow}} [RelCy] for a relative cycle for which the coefficient 1/p occurs in its pullbacks, yet both X and S are normal.

ui Definition 1A.7. A relative cycle W is called universally integral when its

pullbacks W_{V} always have integer coefficients; see ^{\underline{relchow}} [RelCy, 3.3.9].

We define c(X/S, 0) to be the free abelian group on the universally integral relative cycles of X which are finite and surjective over S. Finally we set Cor_{S}(X, Y) = c(X ×_{S} Y/X, 0). That is, Cor_{S}(X, Y) is the group of universally integral cycles on X ×_{S} Y whose support is finite over X (i.e., proper over X of relative dimension 0).

In ^{\underline{relchow}} [RelCy] the notation z(X/S, 0) was used for the subgroup of Cycl(X/S, 0) generated by universally integral cycles, and the notation c(X/S, 0) was introduced for the subgroup generated by the proper cycles in z(X/S, 0).

The following theorem was proved in ^{\underline{relchow}} [RelCy, 3.3.15] and ^{\underline{relchow}} [RelCy, 3.4.8].

a7 Theorem 1A.8. Any relative cycle of X over S is universally integral provided that either

1. S is regular, or

2. X is a smooth curve over S.

a8 Definition 1A.9. The composition of relative cycles V ∈ Cor_{S}(X, Y) and W ∈ Cor_{S}(Y, Z) is defined as follows. Form the pullback W_{V} of W with respect to the map V → Y. The composition W ∘ V is defined to be the push-forward of W_{V} along the projection p : X × Y × Z → X × Z. By ^{\underline{relchow}} [RelCy, 3.7.5], the composition will be a universally integral cycle which is finite and surjective over X.

corscork Example 1A.10. By definition, c(X/S, 0) = Cor_{S}(S, X). If S and X are smooth over a field k, then clearly Cor_{S}(S, X) ⊆ Cor_{k}(S, X) via the embedding of X in S × X. Hence, for every map S′ → S, there is a map c(X/S, 0) → c(X ×_{S} S′/S′, 0) induced by composition in Cor_{k}.

[figure: c(X/S, 0); ⊂; Cor_{k}(S, X); c(X ×_{S} S′/S′, 0); . . . . . . . . .; ⊂; Cor_{k}(S′, X)] corc Example 1A.11. If S = Spec k for a field k and X and Y are smooth over S,

then the group Cor_{S}(X, Y) = c(X × Y/X, 0) agrees with the group Cor_{k}(X, Y) of definition ^{\underline{elcor}} 1.1.

To see this, note that c(X × Y/X, 0) ⊆ Cor_{k}(X, Y) by definition. By ^{\underline{a4}} 1A.5 and ^{\underline{a7}} 1A.8, every cycle in X × Y which is finite and surjective over X is a universally integral relative cycle, so we have equality.

Since composition in Cor_{S} (as defined in ^{\underline{a8}} 1A.9) evidently agrees with composition in Cor_{k}, we see that Cor_{k} is just the restriction of Cor_{S} to Sm/k.

1a12 Example 1A.12. Suppose that V ⊂ S is a closed immersion of regular schemes and let W be an equidimensional cycle on a scheme X of finite type over S. It it shown in ^{\underline{relchow}} [RelCy, 3.5.8] that the pullback cycle W_{V} coincides with the image of W under the pull-back homomorphism for the map V ×_{S} X → X as defined in ^{\underline{serre}} [Ser65] and ^{\underline{fulton}} [Ful84], using an alternating sum of T or terms.

Presheaves with transfers

In order to define motivic cohomology we need to introduce the notion of a presheaf with transfers. In this lecture we develop the basic properties of presheaves with transfers.

pst Definition 2.1. A presheaf with transfers is a contravariant additive functor F : Cor_{k} → Ab. We will write P reSh(Cor_{k}), or PST(k) or PST if the field is understood, for the functor category whose objects are the presheaves with transfers and whose morphisms are natural transformations. By additivity, there is a pairing Cor_{k}(X, Y) ⊗ F (Y) → F (X).

Restricting to the subcategory Sm/k of Cor_{k}, we see that a presheaf with transfers F may be regarded as a presheaf of abelian groups on Sm/k which is equipped with extra “transfer” maps F (Y) → F (X) indexed by the finite correspondences from X to Y.

const Example 2.2. Every constant sheaf A on Sm/k may be regarded as a presheaf with transfers. If W is an elementary correspondence from X to Y (both connected), the transfer A → A is multiplication by the degree of W over X.

The following theorem is a special case of a well known result on functor categories, see ^{\underline{w}}[Wei94] 1.6.4 and Exercises 2.3.7 and 2.3.8.

injectives Theorem 2.3. The category PST(k) is abelian and has enough injectives and projectives.

schrier Example 2.4. The sheaf O^{∗} of global units and the sheaf O of global functions

are two examples of a presheaves with transfers.

Recall first that if X is normal and W → X is finite and surjective then there is a norm map N : O^{∗}(W) → O^{∗}(X) induced from the usual norm map on the function fields, k(W)^{∗} → k(X)^{∗}. Indeed if f ∈ O^{∗}(W) then Nf and Nf^{−1} are both in the integrally closed subring O(X) of k(X).

Similarly, there is a trace map Tr : O(W) → O(X) induced from the usual trace map on the function fields, k(W) → k(X). Indeed if f ∈ O(W) then Trf belongs to the integrally closed subring O(X) of k(X).

Week 3 - October, 20 1999

If W ⊂ X × Y is an elementary correspondence from X to Y, we define the transfer map O^{∗}(Y) → O^{∗}(X) associated to W to be the composition:

O^{∗}(Y) O^{∗}(W) ^{N} O^{∗}(X).

We define the transfer O(Y) → O(X) associated to W to be the composition

O(Y) O(W) ^{Tr} O(X).

We omit the verification that these transfers are compatible with the composition in Cor_{k}. It is clear from the transfer formula that the subsheaf μ_{n} of n^{th} roots of unity in O^{∗} is also a presheaf with transfers, and that the subsheaf k of O is just the constant sheaf with transfers described in ^{\underline{const}} 2.2.

chipst Example 2.5. The classical Chow groups CH^{i}(−) are presheaves with transfers. To see this, we need to construct a map ϕ_{W} : CH^{i}(Y) → CH^{i}(X) for each elementary finite correspondence W from a smooth scheme X to a smooth scheme Y, and check that this defines a functor from Cor_{k} to abelian groups.

The correspondence homomorphism ϕ_{W} is given by the formula ϕ_{W} (α) = q_{∗}(W · p^{∗}α), where α ∈ CH^{i}(Y). Here p^{∗} : CH^{i}(Y) → CH^{i}(X × Y) is the flat pullback along the projection X × Y → Y, the ‘·’ is the intersection product (see ^{\underline{intmult}} 17A.1), and q : X × Y → X is the projection. If Y were proper, this would be exactly the formula given in Chapter 16 of ^{\underline{fulton}} [Ful84]. For general Y, we need to observe that W · p^{∗}α has finite support over X, so that the push-forward q_{∗}(W · p^{∗}α) is defined in CH^{i}(X).

The verification that the definition of ϕ_{W} is compatible with the composition of correspondences is now a routine calculation using the projection formula; it is practically the same as the calculation in the proper case, which is given in \underline{fulton} [Ful84, 16.1.2].

311 Example 2.6. The presheaf K_{0} cannot be a presheaf with transfers under any transfer. To see this, it suffices to find a finite étale cover f : Y → X of degree 2 and an element x ∈ K_{0}(X) such that f^{∗}(x) = 0 but 2x ≠ 0. Indeed, if Φ ∈ Cor(X, Y) is the canonical transfer, then f ∘Φ = 2 in Cor(X, X) (cf. ^{\underline{galcor}} 1.11), so any presheaf with transfers F would have F (Φ)f^{∗}(x) = 2x for all x ∈ F (X).

Let L be a line bundle on a smooth variety X satisfying L^{2} ≅ O_{X} but [L ⊕ L] ≠ [O_{X} ⊕ O_{X}] in K_{0}(X). It is well-known that such L exists; see \underline{swanvbpm} [Swa62]. It is also well-known that there is an étale cover f : Y → X of degree 2 with Y = Spec(O_{X} ⊕ L); see ^{\underline{hart}} [Har77, IV Ex.2.7]. Since f^{∗}L ≅ O_{Y}, the element x = [L] − [O_{X}] of K_{0}(X) satisfies f^{∗}(x) = 0 but 2x ≠ 0, as required.

Representable functors provide another important class of presheaves with transfers. They provide an embedding of Sm/k into an abelian category:

[figure: Sm/k; Cor_{k}; PST(k).; X; X; ℤ_{tr}(X)] defztr Definition 2.7. If X is a smooth scheme over k we let ℤ_{tr}(X) denote the

presheaf with transfers represented by X, so that ℤ_{tr}(X)(U) = Cor(U, X). By the Yoneda lemma,

Hom_{PST}(ℤ_{tr}(X), F) ≅ F (X).

It follows that ℤ_{tr}(X) is a projective object in PST(k).

As a presheaf, the map ℤ_{tr}(X)(U) → ℤ_{tr}(X)(V) associated to V → U is just the pullback of cycles along V × X → U × X.

The notation ℤ_{tr}(X) was introduced in ^{\underline{bk}} [SV00] while the notation L(X) was used in ^{\underline{tri}} [TrCat] and c_{equi}(X/ Spec k, 0) in ^{\underline{relchow}} [RelCy] and ^{\underline{preth}} [CohTh].

We will write ℤ for the presheaf with transfers ℤ_{tr}(Spec k); it is just the constant presheaf ℤ on Sm/k, equipped with the transfer maps of ^{\underline{const}} 2.2. Thus the structure map X → Spec k induces a natural map ℤ_{tr}(X) → ℤ.

psttimes Exercise 2.8. If F is a presheaf with transfers and X is a smooth scheme, define F^{X}(U) = F (U × X). Show that F^{X} is a presheaf with transfers and that every morphism X → Y induces a morphism F^{Y} → F^{X} of presheaves with transfers.

trkl Exercise 2.9. If k ⊂ L is a separable field extension, every X in Sm/L is an inverse limit of schemes X_{α} in Sm/k. For every presheaf with transfers F over k, we set F (X) = \underrightarrow{lim} F (X_{α}). Show that this makes F a presheaf with transfers over L.

Given a pointed scheme (X, x), we define ℤ_{tr}(X, x) to be the cokernel of the map x_{∗} : ℤ → ℤ_{tr}(X) associated to the point x : Spec k → X. Since x_{∗} splits the structure map ℤ_{tr}(X) → ℤ, we have a natural splitting ℤ_{tr}(X) ≅ ℤ⊕ℤ_{tr}(X, x).

Of particular interest to us are the pointed scheme 𝔾_{m} = (𝔸^{1} − {0}, 1), and the presheaf with transfers ℤ_{tr}(𝔾_{m}) = ℤ_{tr}(𝔸^{1} − {0}, 1).

wedges Definition 2.10. If (X_{i}, x_{i}) are pointed schemes for i = 1, …, n we define ℤ_{tr}((X_{1}, x_{1}) ∧ … ∧ (X_{n}, x_{n})), or ℤ_{tr}(X_{1} ∧ … ∧ X_{n}), to be:

coker (⨁_{i} ℤ_{tr}(X_{1} × ⋯ X̂_{i} ⋯ × X_{n}) ^{id×···×x_{i}×···×id} ℤ_{tr}(X_{1} × ⋯ × X_{n})) .

By definition ℤ_{tr}((X, x)^{∧1}) = ℤ_{tr}(X, x) and ℤ_{tr}((X, x)^{∧q}) = ℤ_{tr}((X, x) ∧ … ∧ (X, x)) for q > 0. By convention ℤ_{tr}((X, x)^{∧0}) = ℤ and ℤ_{tr}((X, x)^{∧q}) = 0 when q < 0.

les Lemma 2.11. The presheaf ℤ_{tr}((X_{1}, x_{1}) ∧ … ∧ (X_{n}, x_{n})) is a direct summand of ℤ_{tr}(X_{1} × ⋯ × X_{n}). In particular, it is a projective object of PST.

Proof. This is a consequence of the stronger fact that the following sequence of presheaves with transfers is split exact (see ^{\underline{w}}[Wei94, 1.4.1]):

0 → ℤ →^{{x_{i}}} ⊕_{i}ℤ_{tr}(X_{i}) → ⊕_{i,j}ℤ_{tr}(X_{i} × X_{j}) → …

… → ⊕_{i,j}ℤ_{tr}(X_{1} × ⋯ X̂^{i} ⋯ X̂^{j} ⋯ × X_{n}) → ⊕_{i}ℤ_{tr}(X × ⋯ X̂^{i} ⋯ × X_{n}) → → ℤ_{tr}(X_{1} × ⋯ × X_{n}) → ℤ_{tr}(X_{1} ∧ ⋯ ∧ X_{n}) → 0.

Since ℤ_{tr}(X_{1}∧⋯∧X_{n}) is a summand of a projective object, it is projective.

To illustrate this lemma note ℤ_{tr}(X) ≅ ℤ ⊕ ℤ_{tr}(X, x) and that:

ℤ_{tr}(X_{1} × X_{2}) ≅ ℤ ⊕ ℤ_{tr}(X_{1}, x_{1}) ⊕ ℤ_{tr}(X_{2}, x_{2}) ⊕ ℤ_{tr}(X_{1} ∧ X_{2}).

We shall also need a functorial construction of a chain complex associated to a presheaf with transfers. For this we use the cosimplicial scheme Δ^{•} over k which is defined by:

Δ^{n} = Spec k[x_{0}, …, x_{n}]/ (∑_{i=0}^{n} x_{i} = 1) .

The j^{th} face map ∂_{j} : Δ^{n} → Δ^{n+1} is given by the equation x_{j} = 0. Although this construction is clearly taken from topology, the use of Δ^{•} in an algebraic setting originated with D. Rector in ^{\underline{rec}} [Rec71].

cstar Definition 2.12. If F is a presheaf of abelian groups on Sm/k, F (Δ^{•}) and F (U × Δ^{•}) are simplicial abelian groups. We will write C_{•}F for the simplicial presheaf defined as U ↦ F (U × Δ^{•}). That is, C_{n}(F)(U) = F (U × Δ^{n}).

As usual, we can take the alternating sum of the face maps to get a chain complex of presheaves with transfers which (using ∗ in place of •), we will call C_{∗}F. It sends U to the complex of abelian groups:

… → F (U × Δ^{2}) → F (U × Δ^{1}) → F (U) → 0.

Note that the Dold-Kan correspondence (see ^{\underline{w}}[Wei94, 8.4.1]), which describes an equivalence between simplicial presheaves and positive chain complexes of presheaves, associates to F (− × Δ^{•}) a normalized subcomplex C_{∗}^{DK}F of the complex C_{∗}F, which is quasi-isomorphic to the complex C_{∗}F.

If A is the constant presheaf A(U) = A then C_{∗}A is quasi-isomorphic to A. Indeed, C_{∗}^{DK}(A) is A regarded as a complex concentrated in degree zero.

hinv Definition 2.13. A presheaf F is homotopy invariant if for every X the map p^{∗} : F (X) → F (X × 𝔸^{1}) is an isomorphism. As p : X × 𝔸^{1} → X has a section, p^{∗} is always split injective. Thus homotopy invariance of F is equivalent to p^{∗} being onto.

The homotopy invariant presheaves of abelian groups form a Serre subcategory of presheaves, meaning that if 0 → F_{0} → F_{1} → F_{2} → 0 is an exact sequence of presheaves then F_{1} is homotopy invariant if and only if both F_{0} and F_{2} are. In particular if F and G are homotopy invariant presheaves with transfers then the kernel and the cokernel of every map f : F → G are homotopy invariant with transfers.

Let i_{α} : X ^{⊂} X × 𝔸^{1} be the inclusion x ↦ (x, α). We shall write i_{α}^{∗} for F (i_{α}) : F (X × 𝔸^{1}) → F (X).

swan Lemma 2.14. F is homotopy invariant if and only if i_{0}^{∗} = i_{1}^{∗} : F (X × 𝔸^{1}) → F (X) for all X.

Proof. (^{\underline{swan}} [Swa72, 4.1]) One direction is clear, so suppose that i_{0}^{∗} = i_{1}^{∗} for all X. Applying F to the multiplication map m : 𝔸^{1} × 𝔸^{1} → 𝔸^{1}, (x, y) ↦ xy, yields the diagram

F (X × 𝔸^{1})  \overline{i_{0}^{∗}} F (X)

[figure: 1_{X×𝔸^{1}}]

[figure: F (X × 𝔸^{1}); (i_{1} × 1_{𝔸^{1}})^{∗}; F (X × 𝔸^{1} × 𝔸^{1}); (1_{X} × m)^{∗}; (i_{0} × 1_{𝔸^{1}})^{∗}; F (X × 𝔸^{1}).; p^{∗}]

Hence p^{∗}i_{0}^{∗} = (1 × i_{0})^{∗}m^{∗} = (1 × i_{1})^{∗}m^{∗} = id. Since i_{0}^{∗}p^{∗} = id, p^{∗} is an isomorphism.

theta Definition 2.15. For i = 0, …, n we define θ_{i} : Δ^{n+1} → Δ^{n} × 𝔸^{1} to be the map that sends the vertex v_{j} to v_{j} × {0} for j ≤ i and to v_{j−1} × {1} otherwise. (See Figure ^{\underline{decomp}} 2.1.) This is the algebraic analogue of the standard simplicial decomposition of the polyhedron Δ^{n} × Δ^{1} (see Figure ^{\underline{decomp}} 2.1).

1 0 0

2 0 1 1

𝔸^{1}

θ_{0}

θ_{1}

Δ^{2}

[figure] [figure]

Δ^{1}

[figure]

Figure 2.1: Simplicial decomposition of Δ^{n} × 𝔸^{1}  decomp

2.11 Lemma 2.16. Let F be a presheaf. Then the maps i_{0}^{∗}, i_{1}^{∗} : C_{∗}F (X × 𝔸^{1}) → C_{∗}F (X) are chain homotopic.

Proof. The maps θ_{i} defined in ^{\underline{theta}} 2.15 induce maps h_{i} = F (1_{X} × θ_{i}) : C_{n}F (X × 𝔸^{1}) → C_{n+1}F (X). The h_{i} form a simplicial homotopy (^{\underline{w}}[Wei94, 8.3.11]) from i_{1}^{∗} = ∂_{0}h_{0} to i_{0}^{∗} = ∂_{n+1}h_{n}. By ^{\underline{w}}[Wei94, 8.3.13], the alternating sum s_{n} = ∑(−1)^{i}h_{i} is a chain homotopy from i_{1}^{∗} to i_{0}^{∗}.

Combining ^{\underline{swan}} 2.14 and ^{\underline{2.11}} 2.16, we obtain

aletter Corollary 2.17. If F is a presheaf then the homology presheaves H_{n}C_{∗}F : X ↦ H_{n}C_{∗}F (X) are homotopy invariant for all n.

Example 2.18. (^{\underline{swan}} [Swa72, 4.2]) The surjection F → H_{0}C_{∗}F is the universal morphism from F to a homotopy invariant presheaf. h0ztr Exercise 2.19. Set H_{0}^{sing}(X/k) = H_{0}C_{∗}ℤ_{tr}(X)(Spec k). Show that there is

a natural surjection from H_{0}^{sing}(X/k) to CH_{0}(X), the Chow group of zero cycles modulo rational equivalence (see exercise ^{\underline{rateq}} 1.10). If X is projective, H_{0}^{sing}(X/k) ≅ CH_{0}(X). If X = 𝔸^{1}, show that H_{0}^{sing}(𝔸^{1}/k) = ℤ. We will return to this point in ^{\underline{h0sing}} 7.1.

thm6 Lemma 2.20. Let F be a presheaf of abelian groups. Suppose that for every smooth scheme X there is a natural homomorphism h_{X} : F (X) → F (X × 𝔸^{1}) which fits into the diagram

F (X)
[figure: 0; id]
[figure: F (X); F (i_{0}); F (X × 𝔸^{1}); h_{X}; F (i_{1}); F (X)]

Then the complex C_{∗}F is chain contractible.

The assertion that h_{X} is natural means that for every map f : X → Y we have a commutative diagram

[figure: F (X); h_{X}; F (X × 𝔸^{1}); F (Y); h_{Y}; F (Y × 𝔸^{1})]

Proof. By naturality, h_{X} induces a map C_{∗}h : C_{∗}F (X) → C_{∗}F (X × 𝔸^{1}). By \underline{2.11} 2.16, the identity map id = i_{1}^{∗}(C_{∗}h) is chain homotopic to 0 = i_{0}^{∗}(C_{∗}h).

cstaro Example 2.21. The paradigm for lemma ^{\underline{thm6}} 2.20 is the sheaf of global functions. The complex C_{∗}O is chain contractible, because O(X × 𝔸^{1}) ≅ O(X)[t] and h_{X}(f) = tf satisfies the conditions of ^{\underline{thm6}} 2.20.

Here is a second application of ^{\underline{thm6}} 2.20. Note that the projection p : X×𝔸^{1} → X induces a map ℤ_{tr}(X × 𝔸^{1}) → ℤ_{tr}(X).

217half Corollary 2.22. C_{∗}ℤ_{tr}(X ×𝔸^{1}) → C_{∗}ℤ_{tr}(X) is a chain homotopy equivalence.

Proof. Let F denote the cokernel of ℤ_{tr}(i_{0}) : ℤ_{tr}(X) → ℤ_{tr}(X × 𝔸^{1}) induced by i_{0} : X → X × 𝔸^{1}. That is, each F (U) is the cokernel of Cor(U, X) → Cor(U, X × 𝔸^{1}). Let H_{U} denote the composition of the product with 𝔸^{1} and multiplication 𝔸^{1} × 𝔸^{1} → 𝔸^{1} :

Cor(U, X × 𝔸^{1}) → Cor(U × 𝔸^{1}, (X × 𝔸^{1}) × 𝔸^{1}) → Cor(U × 𝔸^{1}, X × 𝔸^{1}).

Since H_{U} sends Cor(U, X × {0}) to Cor(U × 𝔸^{1}, X × {0}), it induces a natural map h_{U} : F (U) → F (U × 𝔸^{1}). For U = X × 𝔸^{1} it is easy to see that the composition of H_{U} with i_{0}, i_{1} : U → U × 𝔸^{1} sends 1_{U} ∈ Cor(U, X × 𝔸^{1}) to the projection i_{0}p : U → X → X × 𝔸^{1} and 1_{U}, respectively. Therefore F (i_{0})h_{U}(1_{U}) = 0 and F (i_{1})h_{U}(1_{U}) = 1_{U} for U = X × 𝔸^{1}. For any other U, every element f̄ ∈ F (U) is the image of 1_{X×𝔸^{1}} under some correspondence f : U → X × 𝔸^{1}, so again F (i_{0})h_{U}(f̄) = 0 and F (i_{1})h_{U}(f̄) = f̄. Therefore \underline{thm6} 2.20 applies to show that C_{∗}F is chain contractible. Since C_{∗}ℤ_{tr}(X × 𝔸^{1}) ≅ C_{∗}ℤ_{tr}(X) ⊕ C_{∗}F, we are done.

An elementary 𝔸^{1}-homotopy between two morphisms f, g : X → Y is a map h : X × 𝔸^{1} → Y so that f and g are the restrictions of h along X × 0 and X × 1. This relation is not transitive (exercise!). To correct this, we pass to correspondences.

a1he Definition 2.23. We say that two finite correspondences from X to Y are 𝔸^{1}- homotopic if they are the restrictions along X × 0 and X × 1 of an element of Cor(X × 𝔸^{1}, Y). This is an equivalence relation on Cor(X, Y).

We say that f : X → Y is an 𝔸^{1}-homotopy equivalence if there exists a g : Y → X so that fg and gf are 𝔸^{1}-homotopic to the identity.

The projection p : X × 𝔸^{1} → X is the prototype of an 𝔸^{1}-homotopy equivalence, and the zero-section is an 𝔸^{1}-homotopy inverse.

cor1 Lemma 2.24. If f : X → Y is an 𝔸^{1}-homotopy equivalence with 𝔸^{1}-homotopy inverse g, then f_{∗} : C_{∗}ℤ_{tr}(X) → C_{∗}ℤ_{tr}(Y) is a chain homotopy equivalence with chain homotopy inverse g_{∗}.

Proof. Applying C_{∗}ℤ_{tr} to the data gives a diagram

C_{∗}ℤ_{tr}(X)
[figure: g_{∗}f_{∗}; 1_{X}]
[figure: C_{∗}ℤ_{tr}(X); (i_{0})_{∗} ; ≅; C_{∗}ℤ_{tr}(X × 𝔸^{1}); h_{∗}; (i_{1})_{∗} ; ≅; C_{∗}ℤ_{tr}(X)]

and similarly for Y. The horizontal maps are chain homotopy equivalences by ^{\underline{217half}} 2.22, and are homotopy inverses to p_{∗}. From the right triangle, h_{∗} ≃ p_{∗}. From the left triangle, we get g_{∗}f_{∗} ≃ 1_{X}. Similarly, the diagram for Y gives f_{∗}g_{∗} ≃ 1_{Y}. Hence f_{∗} : C_{∗}ℤ_{tr}(X) → C_{∗}ℤ_{tr}(Y) is a chain homotopy equivalence with inverse g_{∗}.

2.24 Exercise 2.25. Show that there is a natural identification for every X and Y:

H_{0}C_{∗}ℤ_{tr}(Y)(X) = Cor(X, Y)/𝔸^{1}-homotopy equivalence.

We will return to the subject of 𝔸^{1}-homotopy in lectures ^{\underline{relpicandrig}} 7, ^{\underline{leca1we}} 9, ^{\underline{lecnishxfer}} 13, and ^{\underline{lecmot}} 14;

see ^{\underline{h0singa1he}} 7.2, ^{\underline{a1hebis}} 9.8 and ^{\underline{a1henisbis}} 14.11.

The motive associated to X will be the class M(X) of C_{∗}ℤ_{tr}(X) in an appropriate triangulated category DM_{Nis}^{eff,−}(k, R) constructed in ^{\underline{DMnis}} 14.1 from the derived category of PST(k). By ^{\underline{217half}} 2.22, we have M(X) ≅ M(X × 𝔸^{1}) for all X. More generally, any 𝔸^{1}-homotopy equivalence X → Y induces an isomorphism M(X) ≅ M(Y) by ^{\underline{cor1}} 2.24.

Motivic cohomology

Using the tools developed in the last lecture, we will define motivic cohomology. It will be the hypercohomology of special complexes, called motivic complexes.

motcompl Definition 3.1. For every integer q ≥ 0 the motivic complex ℤ(q) is defined as the following complex of presheaves with transfers:

ℤ(q) = C_{∗}(ℤ_{tr}(𝔾_{m}^{∧q}))[−q].

We consider ℤ(q) to be a bounded above cochain complex; the shifting convention for [−q] implies that the terms ℤ(q)^{i} = C_{q−i}ℤ_{tr}(𝔾_{m}^{∧q}) vanish whenever i > q, and the term with i = q is ℤ_{tr}(𝔾_{m}^{∧q}).

If A is any other abelian group then A(q) = ℤ(q) ⊗ A is another complex of presheaves with transfers.

When q = 0, we have ℤ(0) = C_{∗}(ℤ). This is quasi-isomorphic to ℤ, regarded as a complex concentrated in degree 0.

When q = 1, we have ℤ(1) = C_{∗}ℤ_{tr}(𝔾_{m})[−1]. We will give another description of ℤ(1) in the next lecture.

By convention ℤ(q) = 0 if q < 0.

We now show that these complexes of presheaves are actually complexes of sheaves with respect to the Zariski topology.

ztrsh Lemma 3.2. For every smooth Y, ℤ_{tr}(Y) is a sheaf in the Zariski topology, and C_{∗}ℤ_{tr}(Y) is a chain complex of sheaves.

Proof. We have to prove that whenever U is covered by U_{1} and U_{2} the sequence

0 → Cor(U, Y) → Cor(U_{1}, Y) ⊕ Cor(U_{2}, Y) → Cor(U_{1} ∩ U_{2}, Y)

is exact. We may suppose that U is connected and therefore (everything being smooth) irreducible. As every correspondence from U to Y is dominant it is completely determined by the fiber at the generic point of U. Hence Cor(U, Y) injects into each Cor(U_{i}, Y). To see that the sequence is exact at the other spot, take Z_{1} ⊂ U_{1} × Y and Z_{2} ⊂ U_{2} × Y that coincide on (U_{1} ∩ U_{2}) × Y. Each term of Z_{1} and Z_{2} must coincide, so we may assume that they are elementary correspondences. But then the union Z = Z_{1} ∪ Z_{2} in U × Y is a correspondence from U to Y, and its restriction to both U_{i} × Y is Z_{i}, i.e., Z is a preimage of the pair.

Now whenever F is a sheaf, each presheaf U ↦ F (U × X) is also a sheaf for the Zariski topology. In particular each C_{n}F is a sheaf and C_{∗}F is a complex of sheaves. Thus C_{∗}ℤ_{tr}(Y) is a complex of Zariski sheaves.

Lemma 3.3. The restriction ℤ(q)_{X} of ℤ(q) to the Zariski site of X is a complex of sheaves in the Zariski topology.

Similarly, if A is any abelian group, A(q) is a complex of Zariski sheaves.

Proof. Set Y = (𝔸^{1} − \{0\})^{q}. By lemma ^{\underline{ztrsh}} 3.2 we know that C_{∗}(ℤ_{tr}(Y)) is a complex of sheaves. The complex ℤ(q)[q] is a direct summand of C_{∗}(ℤ_{tr}(Y)) by lemma ^{\underline{les}} 2.11, so it must be a complex of sheaves too.

We know that the ℤ_{tr}(X) are sheaves of free abelian groups. Hence each A ⊗ ℤ_{tr}(X) is a Zariski sheaf. We conclude that the A(n) are Zariski sheaves by the same argument we used for the ℤ(n).

Note that A(q) represents the derived tensor product ℤ(q) ⊗^{L} A since ℤ(q) is a flat complex.

motcoho Definition 3.4. The motivic cohomology groups H^{p,q}(X, ℤ) are defined to be the hypercohomology of the motivic complexes ℤ(q) with respect to the Zariski topology:

H^{p,q}(X, ℤ) = ℍ_{Zar}^{p}(X, ℤ(q)).

If A is any abelian group, we define:

H^{p,q}(X, A) = ℍ_{Zar}^{p}(X, A(q)).

Vanishing Theorem 3.5. For every smooth scheme X and any abelian group A, we have H^{p,q}(X, A) = 0 when p > q + dim X.

Proof. By definition, the complex ℤ(q) is zero in degrees bigger then q. Since H^{i}(X, F) vanishes for every sheaf F when i > dim X, the result is now an immediate consequence of the hypercohomology spectral sequence.

Remark 3.6. We will prove in ^{\underline{secondvanish}} 19.3 that, for every smooth variety X and any abelian group A, we have H^{p,q}(X, A) = 0 for p > 2q as well.

The groups H^{p,q}(X, ℤ) are contravariant functorial in X. To see this we need to check that for a morphism f : X → Y we can construct a natural map ℤ(q)_{Y} → f_{∗}ℤ(q)_{X} (which may not be an isomorphism). This can be done by going back to the definition and using a diagonal argument. For U ⊂ Y open the components of this map are summands of Cor(U ×Δ^{n}, T) → Cor(f^{−1}U ×Δ^{n}, T) for appropriate schemes T.

The groups H^{p,q}(X, A) are also covariantly functorial in k. That is, if i : k ⊂ F is a field extension, there is a natural map H^{∗,∗}(X, A) → H^{∗,∗}(X_{F} , A). It is induced by the sheaf map ℤ(q)_{X} → i_{∗}ℤ(q)_{X_{F}} assembled from the natural maps ℤ_{tr}(Y)(U) → i_{∗}ℤ_{tr}(Y_{F})(U) = ℤ_{tr}(Y_{F})(U_{F}) of exercise ^{\underline{kfunct}} 1.12.

Proposition 3.7. If k ⊂ F is a finite and separable field extension and X is smooth over F, then the motivic complexes ℤ(q)_{X} are independent of the choice of the ground field (k or F). Hence the motivic cohomology groups H^{p,q}(X, A) are independent of the choice of the ground field.

Proof. By exercise ^{\underline{kfunct}} 1.12 the groups C_{n}ℤ_{tr}(T_{F})(U) = Cor_{F} (U ×_{F} Δ_{F}^{n} , T_{F}) and C_{∗}ℤ_{tr}(T)(U) = Cor_{k}(U ×_{k} Δ_{k}^{n}, T) are isomorphic for every T smooth over k. That is, C_{∗}ℤ_{tr}(T_{F})(U) ≅ C_{∗}ℤ_{tr}(T)(U). Letting T be products of 𝔸_{k}^{1} − {0}, the result follows from lemma ^{\underline{les}} 2.11.

The following colimit lemmas are elementary consequences of exercise ^{\underline{limcor}} 1.13, but will be useful later on.

Lemma 3.8. (Colimits) Let k ⊂ F be a field extension and X smooth over k. Then:

H^{∗,∗}(X_{F} , A) = colim_{k⊂E⊂F E of finite type} H^{∗,∗}(X_{E}, A).

If f : X → S is a smooth morphism of smooth schemes over k such that S is connected and F = k(S), then:

H^{∗,∗}(X ×_{S} Spec F, A) = colim_{U⊂S ; nonempty} H^{∗,∗}(X ×_{S} U, A).

And now we want to introduce a multiplicative structure on the sheaves ℤ(n). We will need the following construction:

tecnical Construction 3.9. If (X_{s}, x_{s}) are pointed schemes for s = 1, …, j, then for every i < j we define a morphism of presheaves with transfers:

ℤ_{tr}(X_{1} ∧ … ∧ X_{i}) ⊗ ℤ_{tr}(X_{i+1} ∧ … ∧ X_{j}) → ℤ_{tr}(X_{1} ∧ … ∧ X_{j}).

Indeed definition ^{\underline{tensor}} 1.9 provides a map:

ℤ_{tr}(X_{1} × … × X_{i})(U) ⊗ ℤ_{tr}(X_{i+1} × … × X_{j})(U)
= Cor_{k}(U, X_{1} × … × X_{i}) ⊗ Cor_{k}(U, X_{i+1} × … × X_{j}) →
→ Cor_{k}(U × U, X_{1} × … × X_{j}) = ℤ_{tr}(X_{1} × … × X_{j})(U × U).

Composing with the diagonal U → U × U, we have:

ℤ_{tr}(X_{1} × … × X_{i})(U) ⊗ ℤ_{tr}(X_{i+1} × … × X_{j})(U) ^{Δ} ℤ_{tr}(X_{1} × … × X_{j})(U).

Now recall that by definition ℤ_{tr}(X_{1}∧…∧X_{n}) is a quotient of ℤ_{tr}(X_{1}×…×X_{n}). It is easy to check that the map Δ factors through the quotient, giving the required morphism.

42 Construction 3.10. For each m and n we construct a map

ℤ(m) ⊗ ℤ(n) → ℤ(m + n)

using the map ℤ_{tr}(G_{m}^{∧m}) ⊗ ℤ_{tr}(𝔾_{m}^{∧n}) → ℤ_{tr}(𝔾_{m}^{∧m+n}) of ^{\underline{tecnical}} 3.9, as follows. For any smooth scheme U we need to build a map:

ℤ(m)[m](U) ⊗ ℤ(n)[n](U) → ℤ(m + n)[m + n](U)

Recall that by definition ^{\underline{motcompl}} 3.1, ℤ(n)[n](U) is the chain complex C_{∗}ℤ_{tr}(𝔾_{m}^{∧n})(U). Let us write the underlying simplicial object as A_{•}^{n} = ℤ_{tr}(𝔾_{m}^{∧n})(U × Δ^{•}), and the associated unnormalized chain complex as A_{∗}^{n}. Therefore we want a map A_{∗}^{m} ⊗ A_{∗}^{n} → A_{∗}^{m+n}. Similarly, we write (A^{m} ⊗ A^{n})_{∗} for the chain complex associated to diag(A_{•}^{m} ⊗ A_{•}^{n}). The Eilenberg-Zilber theorem (^{\underline{w}}[Wei94, 8.5.1]) yields a quasi-isomorphism ∇ : A_{∗}^{m} ⊗ A_{∗}^{n} → (A^{m} ⊗ A^{n})_{∗}. Therefore if we find a simplicial map diag(A_{•}^{m}⊗A_{•}^{n}) → A_{•}^{m+n} we have also a map (A^{m}⊗A^{n})_{∗} → A_{∗}^{m+n} which, composed with the previous one, gives the multiplicative structure.

Unfolding the definitions again, we have:

A_{i}^{n} = ℤ_{tr}(𝔾_{m}^{∧n})(U × Δ^{i}).

Thus the component diag(A^{m} ⊗ A^{n})_{i} → A_{i}^{m+n} is just the map of ^{\underline{tecnical}} 3.9:

ℤ_{tr}(𝔾_{m}^{∧m})(U × Δ^{i}) ⊗ ℤ_{tr}(𝔾_{m}^{∧n})(U × Δ^{i}) → ℤ_{tr}(𝔾_{m}^{∧m+n})(U × Δ^{i}).

Setting U′ = U × Δ^{i}, we use the map of construction ^{\underline{tecnical}} 3.9 with (X_{s}, x_{s}) = 𝔾_{m} for all s = 1, …, m + n.

This construction and the morphism in ^{\underline{tecnical}} 3.9 are associative and the map ∇ in the Eilenberg-Zilber theorem is homotopy associative (^{\underline{w}}[Wei94, 8.5.4]). It follows that the pairing of construction ^{\underline{42}} 3.10 is homotopy associative.

mult Corollary 3.11. For each smooth X, there are pairings:

H^{p,q}(X, ℤ) ⊗ H^{p′,q′}(X, ℤ) → H^{p+p′,q+q′}(X, ℤ).

In ^{\underline{skew}} 15.9 we will show that this pairing is skew-commutative, so that H^{∗,∗}(X, ℤ) is a bigraded-commutative ring.

Lecture 4

Weight one motivic cohomology

zreg Theorem 4.1. There is a quasi-isomorphism of complexes of presheaves with

transfers:

ℤ(1) ^{≃} O^{∗}[−1].

smallpq Corollary 4.2. Let X be a smooth scheme over k. Then we have:

H^{p,q}(X, ℤ) = \begin{cases}0 & q ≤ 1 and (p, q) ≠ (0, 0), (1, 1), (2, 1) \\ ℤ(X) & (p, q) = (0, 0) \\ O^{∗}(X) & (p, q) = (1, 1) \\ Pic(X) & (p, q) = (2, 1) \end{cases}

[figure: q; p; H^{0,2}; H^{1,2}; 0; 0; 0; ℤ(X); 0; 0; 0; 0; 0; 0; 0; H^{2,2}; H^{−1,2}; H^{−2,2}; H^{3,2}; Pic(X); O^{∗}(X); 0; 0; 0; 0; 0]

Figure 4.1: Weight q motivic cohomology

Week 4 - October 27th 1999

This theorem will follow from lemmas 4.3 − 4.6 below. An alternative proof is given in ^{\underline{svin}} [SV96].

Consider the functor M^{∗}(ℙ^{1}; 0, ∞) : Sm/k → Ab which sends a scheme X to the group of rational functions on X ×ℙ^{1} which are regular in a neighborhood of X × {0, ∞} and equal 1 on X × {0, ∞}. Clearly M^{∗}(ℙ^{1}; 0, ∞) is a sheaf for the Zariski topology. Given a rational function f on X × ℙ^{1} let D(f) denote its divisor.

Lemma 4.3. For all f in M^{∗}(ℙ^{1}; 0, ∞)(X), the divisor D(f) belongs to the subgroup Cor(X, 𝔸^{1} \ {0}) of the group of cycles on X × ℙ^{1}.

From the lemma we get a morphism of sheaves:

M^{∗}(ℙ^{1}; 0, ∞) ^{⊂} ℤ_{tr}(𝔸^{1} \ {0}).

ses Lemma 4.4. For any connected X there is a short exact sequence in Ab:

0  M^{∗}(ℙ^{1}; 0, ∞)(X)  ℤ_{tr}(𝔸^{1} \ {0})(X) ^{λ} ℤ ⊕ O^{∗}(X)  0.

Proof. We know that Pic(X × ℙ^{1}) ≅ Pic(X) × ℤ, so for any Z in Cor(X, 𝔸^{1}) ⊂ Cor(X, ℙ^{1}) there is a unique rational function f on X × ℙ^{1} and an integer n so that D(f) = Z and f/t^{n} = 1 on X × {∞}. If Z lies in Cor(X, 𝔸^{1} \ {0}), then f(0) ∈ O^{∗}(X). We define λ : ℤ_{tr}(𝔸^{1} \{0}) → ℤ⊕O^{∗} by λ(Z) = (n, (−1)^{n}f(0)). If u ∈ O^{∗}(X) and Z_{u} = D(t − u) then λ(Z_{u}) = (1, u). Since λ(Z_{u} − Z_{1}) = (0, u) λ is onto. The kernel of λ consists of all Z whose f lies in M^{∗}(ℙ^{1}; 0, ∞)(X), so we are done.

Lemma 4.5. The map λ respects transfers. Hence M^{∗}(ℙ^{1}; 0, ∞) is a PST.

Proof. It is easy to see that the first component of λ is a morphism in PST because it is the map Cor_{k}(X, 𝔸^{1} \ {0}) → Cor_{k}(X, Spec k), induced by the structure map π : 𝔸^{1} \ {0} → Spec k. To check the second component of λ, we see from exercise ^{\underline{limcor}} 1.13 that it suffices to check that the following diagram commutes for every finite field extension F ⊂ E.

ℤ_{tr}(𝔸^{1} \ \{0\})(Spec E)  \overline{}E^{∗}

[figure: ℤ_{tr}(𝔸^{1} \ {0})(Spec F); F^{∗}; N_{E/F}]

This is a straightforward verification using exercise ^{\underline{rateq}} 1.10.

Write M^{∗} for M^{∗}(ℙ^{1}; 0, ∞). By ^{\underline{cstar}} 2.12, (C_{i}F)(U) = F (U × Δ^{i}), so ^{\underline{ses}} 4.4 gives us:

0 → C_{∗}(M^{∗}) → C_{∗}ℤ_{tr}(𝔸^{1} \ {0}) → C_{∗}(ℤ ⊕ O^{∗}) → 0.

Splitting off 0 → C_{∗}ℤ = C_{∗}ℤ → 0 we get an exact sequence:

0 → C_{∗}(M^{∗}) → ℤ(1)[1] → C_{∗}(O^{∗}) → 0.

But C_{∗}(O^{∗}) ≃ O^{∗} because O^{∗}(U × Δ^{n}) = O^{∗}(U). We will prove in lemma ^{\underline{acycl}} 4.6 that the first term C_{∗}(M^{∗}) is acyclic. Therefore ℤ(1)[1] is quasi-isomorphic to O^{∗}. This is the statement of the theorem ^{\underline{zreg}} 4.1, shifted once.

acycl Lemma 4.6. If X is a smooth scheme over k, then C_{∗}(M^{∗})(X) is an acyclic complex of abelian groups. Hence C_{∗}(M^{∗}) is an acyclic complex of sheaves.

Proof. Let f ∈ C_{i}^{DK}(M^{∗})(X) be a cycle, i.e., an element vanishing in C_{i−1}^{DK}(M^{∗})(X). Then f is a regular function on some neighborhood U of Z = X × Δ^{i} × {0, ∞} in X × Δ^{i} × ℙ^{1}, and f = 1 on each face X × Δ^{i−1} × ℙ^{1}, as well as on Z. Consider the regular function h_{X}(f) = 1 − t(1 − f) on the neighborhood 𝔸^{1} × U of 𝔸^{1} × Z in 𝔸^{1} × X × Δ^{i} × ℙ^{1}, where t denotes the coordinate function of 𝔸^{1}. Then h_{X}(f) is a cycle in C_{i}^{DK}(M^{∗})(𝔸^{1} × X), because it equals 1 where f equals 1. The restrictions along t = 0, 1, from C_{i}^{DK}(M^{∗})(𝔸^{1} × X) to C_{i}^{DK}(M^{∗})(X), send h_{X}(f) to 1 and f, respectively. Since these restrictions are chain homotopy equivalent by ^{\underline{2.11}} 2.16, f is a boundary.

This completes the proof of theorem ^{\underline{zreg}} 4.1.

Remark 4.7. We will revisit this in lecture ^{\underline{relpicandrig}} 7 in ^{\underline{7.10}} 7.11.

Lemma ^{\underline{acycl}} 4.6 works more generally to show that C_{∗}M^{∗}(Y; Z)(X) is acyclic for every affine X, where M^{∗}(Y; Z)(X) is the group of rational functions on X × Y which are regular in a neighborhood of X × Z and equal to 1 on X × Z.

Now let us consider the complex ℤ/l(1). By theorem ^{\underline{zreg}} 4.1 ℤ(1) is quasiisomorphic to O^{∗}[−1]. Tensoring with ℤ/l we have ℤ/l(1) ≃ O^{∗}[−1] ⊗^{L} ℤ/l, which is just the complex [O^{∗} ^{l} O^{∗}] in degrees 0 and 1. Then we have the universal coefficients sequence:

0  H^{p,q}(X, ℤ)/l  H^{p,q}(X, ℤ/l)  _{l}H^{p+1,q}(X, ℤ)  0.

ztret Corollary 4.8. There is a quasi-isomorphism of complexes of étale sheaves

ℤ/l(1)_{ét} ≃ μ_{l}.

Proof. Since sheafification is exact (^{\underline{milne}} [Mil80] p. 63), theorem ^{\underline{zreg}} 4.1 gives ℤ(1)_{ét} ≃ O_{ét}^{∗}[−1], and hence

ℤ/l(1)_{ét} ≃ O_{ét}^{∗}[−1] ⊗^{𝕃} ℤ/l ≃ μ_{l}.

mul Corollary 4.9. If 1/l ∈ k and X is smooth, then H^{p,1}(X, ℤ/l) = 0 for p ≠ 0, 1, 2 while:

H^{0,1}(X, ℤ/l) = μ_{l}(X), H^{1,1}(X, ℤ/l) = H_{ét}^{1}(X, μ_{l}),

H^{2,1}(X, ℤ/l) = Pic(X)/l Pic(X).

Proof. The calculation of H^{p,1} for p ≠ 1 follows from the universal coefficients sequence, since the only nonzero Zariski cohomology groups of O^{∗} on a smooth scheme are H^{0} and H^{1}(X, O^{∗}) = Pic(X). For p = 1 note that corollary ^{\underline{ztret}} 4.8 gives a natural map

ℍ_{Zar}^{∗}(X, ℤ/l(1)) → ℍ_{ét}^{∗}(X, ℤ/l(1)_{ét}) = H_{ét}^{1}(X, μ_{l})

fitting into the diagram:

[figure: H_{Zar}^{1}(X, ℤ(1))/l; ⊂; H_{Zar}^{1}(X, ℤ/l(1)); _{l}H_{Zar}^{2}(X, ℤ(1)); H_{ét}^{1}(X, ℤ(1))/l; ≅; ⊂; H_{ét}^{1}(X, ℤ/l(1)); _{l}H_{ét}^{2}(X, ℤ(1)).; ≅]

Since H_{ét}^{1}(X, O^{∗}) = H_{Zar}^{1}(X, O^{∗}) by Hilbert’s Theorem 90 (see ^{\underline{milne}} [Mil80, III 4.9]), the 5-lemma concludes the proof.

Remark 4.10. (Deligne) If char k = l then H^{1,1}(X, ℤ/l) ≅ H_{ffp}^{1}(X, μ_{l}). In fact, the proof of ^{\underline{mul}} 4.9 is valid in this setting.

Relation to Milnor K-Theory

The Milnor K-theory K_{∗}^{M}(F) of a field F is defined to be the quotient of the tensor algebra T (F^{∗}) over ℤ by the ideal generated by the elements of the form x ⊗ (1 − x) where x ∈ F^{∗}. In particular, K_{0}^{M}(F) = ℤ and K_{1}^{M}(F) = F^{∗}.

The goal of this lecture is to prove the following:

motmil Theorem 5.1. For any field F and any n we have:

H^{n,n}(Spec F, ℤ) ≅ K_{n}^{M}(F).

We have already seen that this holds for n = 0, 1, because by definition ^{\underline{motcoho}} 3.4 H^{0,0}(Spec F, ℤ) = H_{Zar}^{0}(Spec F, ℤ) = ℤ and by theorem ^{\underline{zreg}} 4.1:

H^{1,1}(Spec F, ℤ) = H_{Zar}^{1}(Spec F, O^{∗}[−1]) = H_{Zar}^{0}(Spec F, O^{∗}) = F^{∗}.

The proof of theorem ^{\underline{motmil}} 5.1 will follow ^{\underline{bk}} [SV00, 3.4] which is based on ^{\underline{homgenlin}} [NS89]. It will consist of three steps:

1. Construction of θ : H^{n,n}(Spec F, ℤ) → K_{n}^{M}(F). This will use lemma ^{\underline{45}} 5.5.

2. Construction of λ_{F} : K_{n}^{M}(F) → H^{n,n}(Spec F, ℤ). This will be done using lemmas ^{\underline{xix}} 5.9 and ^{\underline{46}} 5.6. The proof of lemma ^{\underline{xix}} 5.9 will need lemma ^{\underline{410}} 5.8.

\underline{46} 5.6 + (^{\underline{410}} 5.8 ⇒ ^{\underline{xix}} 5.9) ⇒ ∃ λ_{F}

3. Proof that these two maps are inverse to each other. For this we will need lemma ^{\underline{48}} 5.10 (proved using lemma ^{\underline{49}} 5.11).

Before starting the proof of the theorem we need some additional properties of motivic cohomology and Milnor K-theory.

Recall that ℤ_{tr}(𝔾_{m}^{∧n})(Spec F) is a quotient of ℤ_{tr}((𝔸^{1} − {0})^{n})(Spec F), which by ^{\underline{rateq}} 1.10 is the group of zero cycles of (𝔸^{1} \ \{0\})^{n}.

Week 5 - November, 3rd 1999

44 Lemma 5.2. We have H^{p,q}(Spec F, ℤ) = H_{q−p}(C_{∗}(ℤ_{tr}(𝔾_{m}^{∧q}))(Spec F)) for all p and q. In particular we have

H^{n,n}(Spec F, ℤ) =H_{0} (C_{∗}(ℤ_{tr}(𝔾_{m}^{∧n}))(Spec F))
= coker (ℤ_{tr}(𝔾_{m}^{∧n})(𝔸^{1}) ^{∂_{0}−∂_{1}} ℤ_{tr}(𝔾_{m}^{∧n})(Spec F)) .

Proof. Write A_{∗} for C_{∗}ℤ_{tr}(𝔾_{m}^{∧q})(Spec F) so the right side is H_{q−p}A_{∗} = H^{p−q}A_{∗}. By definition ^{\underline{motcompl}} 3.1, the restriction of ℤ(q) to Spec F is the chain complex A_{∗}[−q]. Since Zariski cohomology on Spec F is just ordinary cohomology, we have

H^{p,q}(Spec F, ℤ) = H^{p}(A_{∗}[−q]) = H^{p−q}(A_{∗}) = H_{q−p}(A_{∗}).

ffe Lemma 5.3. If F ⊂ E is a finite field extension, then the proper push-forward of cycles induces a map N_{E/F} : H^{∗,∗}(Spec E, ℤ) → H^{∗,∗}(Spec F, ℤ). Moreover, if x ∈ H^{∗,∗}(Spec E, ℤ) and y ∈ H^{∗,∗}(Spec F, ℤ) then:

1. N_{E/F} : H^{0,0}(Spec E, ℤ) = ℤ → ℤ = H^{0,0}(Spec F, ℤ) is multiplication by the degree of E/F.

2. N_{E/F} : H^{1,1}(Spec E, ℤ) = E^{∗} → F^{∗} = H^{1,1}(Spec F, ℤ) is the classical norm map E^{∗} → F^{∗}.

3. N_{E/F} (y_{E} · x) = y · N_{E/F} (x) and N_{E/F} (x · y_{E}) = N_{E/F} (x) · y.

4. If F ⊂ E ⊂ K, and K is normal over F, we have:

N_{E/F} (x)_{K} = [E : F]_{insep} ∑_{j:E ^{⊂}  K} j^{∗}(x)  in H^{∗,∗}(Spec K, ℤ).

5. If F ⊂ E′ ⊂ E then N_{E/F} (x) = N_{E′/F} (N_{E/E′}(x)).

Proof. All but property 2 follow immediately from the corresponding properties of proper push-forward. Property 2 follows from property 4 since this formula also holds for the classical norm map N_{E/F} : E^{∗} → F^{∗}.

If F ⊂ E is a finite field extension, there is a “norm map” N_{E/F} : K_{n}^{M}(E) → K_{n}^{M}(F) satisfying the analogue of lemma ^{\underline{ffe}} 5.3. In addition, it satisfies the following condition (see ^{\underline{mennicke}} [Sus82]).

weil Theorem 5.4 (Weil Reciprocity). Suppose that L is an algebraic function field over k. For each discrete valuation w on L there is a map

∂_{w} : K_{n+1}^{M}(L) → K_{n}^{M}(k(w)),

and for all x ∈ K_{n+1}^{M}(L):

∑_{w} N_{k(w)/k}∂_{w}(x) = 0.

45 Corollary 5.5. Let p : Z → 𝔸_{F}^{1} be a finite surjective morphism and suppose that Z is integral. Let f_{1}, …, f_{n} ∈ O^{∗}(Z) and:

p^{−1}({0}) = ∐ n_{i}^{0}z_{i}^{0} p^{−1}({1}) = ∐ n_{i}^{1}z_{i}^{1}

where n_{i}^{ϵ} are the multiplicities of the points z_{i}^{ϵ} = Spec E_{i}^{ϵ} (ϵ = 0, 1). Define:

φ_{0} = ∑ n_{i}^{0}N_{E_{i}^{0}/F} ({f_{1}, …, f_{n}}_{E_{i}^{0}}) φ_{1} = ∑ n_{i}^{1}N_{E_{i}^{1}/F} ({f_{1}, …, f_{n}}_{E_{i}^{1}})

then we have:

φ_{0} = φ_{1} ∈ K_{n}^{M}(F).

Proof. Let L be the function field of Z and consider x = {t/t − 1, f_{1}, …, f_{n}}. At every infinite place, t/t − 1 equals 1 and ∂_{w}(x) = 0. Similarly, ∂_{w}(x) = 0 at all finite places except those over 0 and 1. If w_{i} lies over t = 0 then ∂_{w_{i}}(x) = n_{i}^{0}{f_{1}, …, f_{n}} in K_{n}^{M}(E_{i}^{0}); if w_{i} lies over t = 1 then ∂_{w_{i}}(x) = −n_{i}^{1}{f_{1}, …, f_{n}} in K_{n}^{M}(E_{i}^{1}). By Weil Reciprocity ^{\underline{weil}} 5.4, ∑ N∂_{w_{i}}(x) = φ_{0}−φ_{1} vanishes in K_{n}^{M}(F).

We are now ready to define the map θ. By ^{\underline{44}} 5.2 it is enough to find a map f from ℤ_{tr}(𝔾_{m}^{∧n})(Spec F) to K_{n}^{M}(F) which composed with the difference of the face operators is zero. Such a map must induce a unique map θ on the cokernel:

[figure: ℤ_{tr}(𝔾_{m}^{∧n})(𝔸^{1}); ∂_{0} − ∂_{1}; ℤ_{tr}(𝔾_{m}^{∧n})(Spec F); H^{n,n}(Spec F, ℤ)]

[figure: f]

K_{n}^{M}(F). \begin{matrix}θ \\  \end{matrix}

But now ℤ_{tr}(𝔾_{m}^{∧n})(Spec F) is a quotient of the free abelian group generated by the closed points of (𝔸_{F}^{1} \ \{0\})^{n} (by exercise ^{\underline{rateq}} 1.10), modulo the subgroup generated by all points of the form (x_{1}, …, 1, …, x_{n}) where the 1’s can be in any position. If x is a closed point of (𝔸_{F}^{1} \ {0})^{n} with residue field E then x is defined by a canonical sequence (x_{1}, …, x_{n}) of nonzero elements of E. Now E is a finite field extension of F, and {x_{1}, …, x_{n}} ∈ K_{n}^{M}(E). Using the norm map for Milnor K-theory N_{E/F} : K_{n}^{M}(E) → K_{n}^{M}(F), we define

f(x) = N_{E/F} ({x_{1}, …, x_{n}}).

Since {x_{1}, …, 1, …, x_{n}} = 0 in K_{∗}^{M}(E), this induces a well-defined map f : ℤ_{tr}(𝔾_{m}^{∧n})(Spec F) → K_{n}^{M}(F). By ^{\underline{45}} 5.5 the composition of f with the face operators is zero. We define θ to be the map induced on the cokernel.

If x is an F -point of (𝔸_{F}^{1} − {0})^{n} then its coordinates x_{1}, …, x_{n} are nonzero elements of F. We shall write [x_{1} : ⋯: x_{n}] for the class of x in H^{n,n}(Spec F, ℤ). The map θ is obviously surjective since θ([x_{1} : ⋯: x_{n}]) = {x_{1}, …, x_{n}} for x_{1}, …, x_{n} in F.

Now let us build the opposite map, λ_{F}. For this, we will use the multiplicative structure (^{\underline{mult}} 3.11) on H^{∗,∗}(X, ℤ). The following lemma is immediate from the construction ^{\underline{42}} 3.10 and lemma ^{\underline{44}} 5.2.

46 Lemma 5.6. For a_{1}, …, a_{n} ∈ F we have [a_{1} : ⋯: a_{n}] = [a_{1}] ⋯ [a_{n}].

By definition K_{∗}^{M}(F) = T (F^{∗})/(x ⊗ (1 − x)). Therefore we define a map:

T (F^{∗}) → ⊕_{n}H^{n,n}(Spec F, ℤ), a_{1} ⊗ … ⊗ a_{n} ↦ [a_{1}] ⋯ [a_{n}].

We will prove that this maps factors through K_{n}^{M}(F). By ^{\underline{46}} 5.6, it is enough to prove that [a : 1−a] is zero, which is the statement of the proposition ^{\underline{xix}} 5.9 below.

Example 5.7. We can use a special cycle to show that [a : −a] = 0. Consider the correspondence Z from 𝔸^{1} (parametrized by t) to X = 𝔸^{1} − {0} (parametrized by x) defined by

x^{2} − t(a + b)x − (1 − t)(1 + ab)x + ab = 0.

Restricting along t = 0, 1 yields correspondences [ab] + [1] and [a] + [b] in Cor(Spec F, X). Setting these equal recovers the identity [ab] = [a] + [b] in H^{1,1}(Spec F, ℤ) ≅ F^{∗}, because [1] = 0.

Let Y denote the composition of Z with the diagonal embedding X ^{⊂} X^{2}. Since [1 : 1] = [1][1] = 0, equating the restrictions along t = 0, 1 yields the identity [ab : ab] = [ab : ab] + [1 : 1] = [a : a] + [b : b] in H^{2,2}(Spec F, ℤ). Bilinearity (^{\underline{46}} 5.6) yields skew-commutativity: [a : b] + [b : a] = 0. In particular, 2[a : a] = 0.

Passing to E = F (\sqrt{a}), we see that 0 = 2[\sqrt{a} : \sqrt{a}] = [a : \sqrt{a}] in H^{2,2}(Spec E, ℤ). By ^{\underline{ffe}} 5.3, applying N_{E/F} yields 0 = [a : −a] in H^{2,2}(Spec F, ℤ).

410 Lemma 5.8. Suppose ∃n > 0 so that n[x : 1−x] = 0 for all finite extensions of F and x ≠ 0, 1 in F. Then [x : 1 − x] = 0 in H^{2,2}(Spec F, ℤ) for every x ≠ 0, 1.

Proof. Suppose n = m · p where p is a prime; we want to prove m[x : 1 − x] = 0. Let us consider y = \sqrt[p]{x} and E = F (y). Then 0 = mp[y : 1 − y] = m[x : 1 − y], and 1 − x = N_{E/F} (1 − y). Hence

0 = N_{E/F} (m[x : 1 − x]) = m · [x : N_{E/F} (1 − y)] = m[x : 1 − x].

The formula [x : 1 − x] = 0 follows by induction on n.

xix Proposition 5.9. The element [x : 1−x] in H^{2,2}(Spec F, ℤ) is the zero element.

Proof. Let Z be the finite correspondence from 𝔸^{1} (parametrized by t) to X = 𝔸^{1} − {0} (parametrized by x) defined by:

x^{3} − t(a^{3} + 1)x^{2} + t(a^{3} + 1)x − a^{3} = 0.

Let ω be a root of x^{2} + x + 1, so ω^{3} = 1, and E = F (ω). The fiber over t = 0 consists of a, ωa, and ω^{2}a and the fiber over t = 1 consists of a^{3} and two sixth roots of 1. Using the embedding x ↦ (x, 1 − x) of 𝔸^{1} − {0, 1} into X^{2}, Z yields a correspondence Z′ from 𝔸^{1} to X^{2}. Then in H^{2,2}(Spec E, ℤ)

∂_{0}(Z′) = [a : 1 − a] + [ωa : 1 − ωa] + [ω^{2}a : 1 − ω^{2}a] =

[a : 1 − a^{3}] + [ω : (1 − ωa)(1 − ω^{2}a)^{2}] is equal to

∂_{1}(Z′) = [a^{3} : 1 − a^{3}] + [−ω : 1 + ω] + [−ω^{2} : 1 + ω^{2}].

Multiplying by 3 eliminates terms [ω : b], noting that [−1 : 1+ω]+[−1 : 1+ω^{2}] = 0 as (1+ω)(1+ω^{2}) = 1. Therefore 0 = 2[a^{3} : 1−a^{3}] over E. Applying the norm yields 0 = 4[a^{3} : 1 − a^{3}] over F. Passing to the extension F (\sqrt[3]{a}) and norming yields 0 = 12[a : 1 − a] over F. Applying lemma ^{\underline{410}} 5.8 with n = 12, we see that 0 = [a : 1 − a] as well.

Proposition ^{\underline{xix}} 5.9 shows that the algebra map of lemma ^{\underline{46}} 5.6 induces a map on the quotient λ_{F} : K_{n}^{M}(F) → H^{n,n}(Spec F, ℤ). Now we need to check that λ_{F} and θ are inverse to each other. Since θ ∘ λ_{F} is the identity by construction, it is enough to prove that λ_{F} is surjective.

48 Lemma 5.10. The map λ_{F} is surjective.

Proof. By ^{\underline{44}} 5.2, it suffices to show that if x is a closed point of X = (𝔸_{F}^{1} \ \{0\})^{n} then [x] ∈ H^{n.n}(Spec F, ℤ) belongs to the image of λ_{F}. Set E = k(x), and choose a lift x̃ ∈ X_{E} of x. Since x is the proper push-forward of x̃, the definition of the norm map (see ^{\underline{ffe}} 5.3) implies that:

[x] = N_{E/F} ([x̃])  x̃ = (a_{1}, …, a_{n}) ∈ (𝔸^{1} \ {0})^{n}(E).

Since x̃ is a rational point of X_{E}, [x̃] is the image under λ_{E} of its coordinates. So [x] = N_{E/F} λ_{E}{a_{1}, …, a_{n}}. The lemma now follows from the assertion, proven in ^{\underline{49}} 5.11 below, that the diagram (^{\underline{norm}} 5.10.1) commutes.

K_{n}^{M}(E) \overline{\begin{matrix}λ_{E} \\  \end{matrix}}H^{n,n}(Spec E, ℤ)

[figure: K_{n}^{M}(F); N_{E/F}; λ_{F}; H^{n,n}(Spec F, ℤ).; N_{E/F}]

(5.10.1) norm

49 Lemma 5.11. If F ⊂ E is any finite field extension, then the diagram (^{\underline{norm}} 5.10.1) commutes.

Proof. By ^{\underline{ffe}} 5.3 (3) we may assume that [E : F] = l for some prime number l. Assume first that F has no extensions of degree prime to l and [E : F] = l. The Bass-Tate lemma (5.3) in ^{\underline{milnring}} [BT73] states that in this case K_{n}^{M}(E) is generated by the symbols a = {a_{1}, …, a_{n−1}, b} where a_{i} ∈ F and b ∈ E. The properties of the norm on K_{∗}^{M} and ^{\underline{46}} 5.6 yield:

λ_{F} N{a_{1}, …, a_{n−1}, b} = λ_{F} {a_{1}, …, a_{n−1}, N(b)} = [a_{1} : ⋯: a_{n−1}] · [Nb].

But using the assertions of lemma ^{\underline{ffe}} 5.3 we have:

Nλ_{E}(a) = N[a_{1} : ⋯: a_{n−1} : b] =^{(2)} [a_{1} : ⋯: a_{n−1}]·N[b] =^{(4)} [a_{1} : ⋯: a_{n−1}]·[Nb].

This concludes the proof in this case.

Now we use a standard reduction. For simplicity, we will write H^{p,q}(F) for H^{p,q}(Spec F, ℤ). If F′ is the union of all prime-to-l extensions of F then the kernel of H^{n,n}(F) → H^{n,n}(F′) is a torsion group of exponent prime to l by (1) and (3) of ^{\underline{ffe}} 5.3. Fix a ∈ K_{n}^{M}(E). By the above case, t = Nλ_{E}(a) − λ_{F} N(a) is a torsion element of H^{n,n}(F), of exponent prime to l.

Since the kernel of H^{n,n}(F) → H^{n.n}(E) has exponent l, t_{E} ≠ 0 if and only if t = 0. If E is an inseparable extension of F then by ^{\underline{ffe}} 5.3(4) we have t_{E} = lλ_{E}(a) − λ_{E}(la) = 0. If E is separable over F then E ⊗_{F} E is a finite product of fields E_{i} with [E_{i} : E] < l. Moreover, Weil Reciprocity implies that the diagrams

[figure: K_{n}^{M}(E); diag; ⊕K_{n}^{M}(E_{i}); H^{n,n}(E); diag; ⊕H^{n,n}(E_{i})]

[figure: K_{n}^{M}(F); N_{E/F}; K_{n}^{M}(E); ⊕N_{E_{i}/E}; H^{n,n}(F); N_{E/F}; H^{n,n}(E); ⊕N_{E_{i}/E}]

commute (see p.387 of ^{\underline{milnring}} [BT73]). By induction on l, we have

t_{E} = ⊕N_{E_{i}/E}λ_{E_{i}}(a_{E_{i}}) − ⊕λ_{E}N_{E_{i}/E}(a_{E_{i}}) = 0.

Since t_{E} = 0 we also have t = 0.

This completes the proof of theorem ^{\underline{motmil}} 5.1.

Étale sheaves with transfers

The goal of this lecture will be to study the relations between presheaves with transfers and étale sheaves. The main result (^{\underline{ettr}} 6.17) will be that sheafification preserves transfers.

Definition 6.1. A presheaf F of abelian groups on Sm/k is an étale sheaf if it restricts to an étale sheaf on each X in Sm/k. That is, if:

1. the sequence 0 → F (X) → F (U) → F (U ×_{X} U) is exact for every surjective étale morphism of smooth schemes U → X;

2. F (X ∐ Y) = F (X) ⊕ F (Y) for all X and Y.

We will write Sh_{ét}(Sm/k) for the category of étale sheaves, which is a full subcategory of the category of presheaves of abelian groups.

A presheaf with transfers F is an étale sheaf with transfers if its underlying presheaf is an étale sheaf on Sm/k. We will write Sh_{ét}(Cor_{k}) for the full subcategory of PST(k) whose objects are the étale sheaves with transfers.

For example, we saw in lecture 2 that the étale sheaves ℤ and O^{∗} have transfers, so they are étale sheaves with transfers. Lemma ^{\underline{etsh}} 6.2 shows that ℤ_{tr}(T) is an étale sheaf with transfers.

etsh Lemma 6.2. For any smooth scheme T, ℤ_{tr}(T) is an étale sheaf.

Proof. Since PST(k) is an additive category, we have the required decomposition of ℤ_{tr}(T)(X ∐ Y) = Hom_{PST}(X ∐ Y, T). To check the sheaf axiom for surjective étale maps U → X, we proceed as in the proof of ^{\underline{ztrsh}} 3.2.

As U × T → X × T is flat, the pullback of cycles is well-defined and is an injection. Hence the subgroup ℤ_{tr}(T)(X) = Cor_{k}(X, T) of cycles on X × T injects into the subgroup ℤ_{tr}(T)(U) = Cor_{k}(U, T) of cycles on U × T.

Week 6 - November 10th, 1999

To see that the sequence is exact at ℤ_{tr}(T)(U), take Z_{U} in Cor_{k}(U, T) whose images in Cor_{k}(U ×_{X} U, T) coincide. We may assume that X and U are integral; let F and L be their respective generic points. Then Z_{L} ∈ Cor_{F} (L, T_{F}) comes from a cycle Z_{F} in Cor_{F} (F, T_{F}) by ^{\underline{galcor}} 1.11, because if L lies in a Galois extension L′ and G = Gal(L′/F), then Z′_{L} lies in Cor_{F} (L′, T_{F})^{G} = Cor_{F} (L, T_{F}). Thus by ^{\underline{limcor}} 1.13 there is a Zariski open V ⊂ X and a cycle Z_{V} in Cor_{k}(V, T) agreeing with Z_{U} in Cor(U ×_{X} V, T). But U is finite and flat over X, so each term in Z_{U} is also finite over X. Hence each term in Z_{V} extends to a cycle in X × T finite over X, i.e., to a finite correspondence in Cor_{k}(X, T).

homshpst Corollary 6.3. Let F be an étale sheaf with transfers. Then

Hom_{Sh_{ét}(Cor_{k})}(ℤ_{tr}(X), F) = Hom_{PST}(ℤ_{tr}(X), F) = F (X).

muqiso Corollary 6.4. For any abelian group A, the A(n) are complexes of étale sheaves. If 1/n ∈ k, the motivic complex of étale sheaves ℤ/n(1) is quasiisomorphic to the étale sheaf μ_{n}.

Proof. The ℤ(n) are étale sheaves with transfers by lemmas ^{\underline{les}} 2.11 and ^{\underline{etsh}} 6.2. We know that the ℤ_{tr}(T) are sheaves of free abelian groups. Hence A ⊗ ℤ_{tr}(T) are étale sheaves. We conclude that the A(n) are étale sheaves by the same argument we used for the ℤ(n). The last assertion is just a restatement of corollary ^{\underline{ztret}} 4.8 using ^{\underline{etsh}} 6.2.

Exercise 6.5. Let π : X → S be a finite étale map, and π_{t} the induced finite correspondence from S to X. If F is any étale sheaf with transfers, show that π_{t}^{∗} : F (X) → F (S) is the étale trace map of ^{\underline{milne}} [Mil80, V.1.12]. Hint: If Y → S is Galois with group G, and factors through X, then Cor(S, X) = Cor(Y, X)^{G} by 6.2. Show that the image of π in Cor(Y, X) is the sum ∑ f of all S-maps from f : Y → X, and hence determines π_{t} ∈ Cor(S, X).

Locally constant étale sheaves form a second important class of étale sheaves with transfers.

Definition 6.6. The full subcategory Et/k of Sm/k consists of all the schemes of finite type over k which are smooth of dimension zero. Every S in Et/k is a finite disjoint union of spectra of separable field extensions of k.

It is well known (see ^{\underline{milne}} [Mil80] and ^{\underline{sga4}} [SGA4, VIII 2.2]) that the category of étale sheaves on Et/k is equivalent to the category of discrete modules over the profinite group Gal(k_{sep}/k). If F corresponds to the Galois module M and S = Spec(ℓ) then F (S) = M^{H}, where H = Gal(k_{sep}/ℓ).

We have the following functors:

Sh_{ét}(Et/k) [figure: π_{∗}; π^{∗}] Sh_{ét}(Sm/k),

where the restriction π_{∗} is the right adjoint of π^{∗}; they are both exact functors.

lcs Definition 6.7. An étale sheaf is locally constant if π^{∗}π_{∗}F → F is an isomorphism. We will write Sh_{ét}^{lc} for the full subcategory of Sh_{ét}(Sm/k) consisting of all locally constant sheaves. ex64 Exercise 6.8. Let F be the locally constant sheaf π^{∗}M corresponding to the G-

module M. If X is connected, and l is the separable closure of k in H^{0}(X, O_{X}), show that F (X) = M^{H} where H = Gal(k_{sep}/l). Conclude that π_{∗}F is the Galois module M. Note that F (X) = M^{H} is also defined if X is normal.

galois Lemma 6.9. The functors π^{∗} and π_{∗} induce an equivalence between the category Sh_{ét}^{lc} and the category of discrete modules over the profinite group Gal(k_{sep}/k).

Proof. If M is in Sh_{ét}(Et/k), then M → π_{∗}π^{∗}M is an isomorphism by Ex. ^{\underline{ex64}} 6.8. Thus π^{∗} is faithful. By category theory, π^{∗}π_{∗}π^{∗} ≅ π^{∗}, so for F locally constant we have a natural isomorphism π^{∗}π_{∗}F ≅ F.

galext Exercise 6.10. Let L be a Galois extension of k, and let G = Gal(L/k). Show that ℤ_{tr}(L) is the locally constant étale sheaf corresponding to the module ℤ[G].

81 Lemma 6.11. Any locally constant étale sheaf has a unique underlying étale sheaf with transfers.

Proof. Let Z′ ⊂ X × Y be an elementary correspondence and let Z be the normalization of Z′ in a normal field extension L of F = k(X) containing K = k(Z′). If G = Gal(L/F) then we also have G = Aut_{X}(Z), and it is well known that the set Hom_{X}(Z, Z′) of maps q : Z → Z′ over X is in one-one correspondence with the set of field maps Hom_{F} (K, L). The cardinality of this set is the separable degree of K over F.

Let M be a Galois module, considered as a locally constant étale sheaf. It is easy to check using exercise ^{\underline{ex64}} 6.8 that M(X) is isomorphic to M(Z′)^{G}.

Write i for the inseparable degree of K over F. Then the transfer map M(Y) → M(X) is defined to be the composite of M(Y) → M(Z′), multiplication by i, and the sum over all maps q : Z → Z′ over X of q^{∗} : M(Z′) → M(Z).

The verification that this gives M the structure of a presheaf with transfers is now straightforward, and we refer the reader to 5.17 in ^{\underline{svin}} [SV96] for details.

It is clear that the locally constant étale sheaves form an abelian subcategory of Sh_{ét}(Cor_{k}), i.e., the inclusion is an exact functor.

In order to describe the relation between presheaves and étale sheaves with transfers (see ^{\underline{exact}} 6.18), we need two preliminary results.

If p : U → X is an étale cover, we define ℤ_{tr}(Ǔ) to be the Čech complex

… ^{p_{0}−p_{1}+p_{2}} ℤ_{tr}(U ×_{X} U) ^{p_{0}−p_{1}} ℤ_{tr}(U) 0.

ztruu Proposition 6.12. Let p : U → X be an étale covering. Then ℤ_{tr}(Ǔ) is an étale resolution of the sheaf ℤ_{tr}(X), i.e., the following complex is exact as a complex of étale sheaves.

… ^{p_{0}−p_{1}+p_{2}} ℤ_{tr}(U ×_{X} U) ^{p_{0}−p_{1}} ℤ_{tr}(U) ^{p} ℤ_{tr}(X) → 0

Proof. As this is a complex of sheaves it suffices to verify the exactness of the sequence at every étale point. Since points in the étale topology are strictly Hensel local schemes, it is enough to prove that, for every Hensel local scheme S over k, the following sequence of abelian groups is exact.

⋯ → ℤ_{tr}(U)(S) → ℤ_{tr}(X)(S) → 0.  (6.12.1) ztrs

Here S is an inverse limit of smooth schemes S_{i}, and by abuse of notation ℤ_{tr}(T)(S) denotes lim ℤ_{tr}(T)(S_{i}).

To prove that (^{\underline{ztrs}} 6.12.1) is exact we need another reduction step. Let Z be a closed subscheme of X × S which is quasi-finite over S. We write L(Z/S) for the free abelian group generated by the irreducible connected components of Z which are finite and surjective over S. L(Z/S) is covariantly functorial on Z with respect to morphisms of quasi-finite schemes over S. Clearly, the sequence (^{\underline{ztrs}} 6.12.1) is the colimit of complexes of the form:

⋯ → L(Z_{U} ×_{Z} Z_{U}/S) → L(Z_{U}/S) → L(Z/S) → 0  (6.12.2) lemmino

where Z_{U} = Z ×_{X} U and the limit is taken over all Z closed subschemes of X × S which are finite and surjective over S. Therefore the proof of ^{\underline{ztruu}} 6.12 will be completed once we show that the sequence (^{\underline{lemmino}} 6.12.2) is exact for every subscheme Z of X × S which is finite and surjective over S.

Since S is Hensel local and Z is finite over S, Z is also Hensel. Therefore the covering Z_{U} → Z splits. Let s_{1} : Z → Z_{U} be a splitting. We set (Z_{U})_{Z}^{k} = Z_{U} ×_{Z} … ×_{Z} Z_{U}. It is enough to check that the maps s_{k} : L((Z_{U})_{Z}^{k}/S) → L((Z_{U})_{Z}^{k+1}/S) are contracting homotopies where s_{k} = L (s_{1} ×_{Z} id_{(Z_{U})_{Z}^{k}} ).

This is the end of the proof of ^{\underline{ztruu}} 6.12.

The proof shows that ℤ_{tr}(Ǔ) is also a Nisnevich resolution of ℤ_{tr}(X), i.e., the sequence of ^{\underline{ztruu}} 6.12 is also exact as a complex of Nisnevich sheaves. We can pinpoint why this proof holds in the étale topology and in the Nisnevich topology, but does not hold in the Zariski topology. This is because:

• If S is strictly Hensel local (i.e. a point in the étale topology) and Z is finite over S then Z is strictly Hensel.

• If S is Hensel local (i.e. a point in the Nisnevich topology) and Z is finite over S then Z is Hensel.

• If S is local (i.e. a point in the Zariski topology) and Z is finite over S then Z need not be local but will be semilocal.

6.12 Example 6.13. Let X be a semilocal scheme finite over a local scheme S. X is covered by its local subschemes U_{i}. If X → S does not split, its graph defines an element of ℤ_{tr}(X)(S) that cannot come from ⊕ℤ_{tr}(U_{i})(S). Hence ⊕ℤ_{tr}(U_{i}) → ℤ_{tr}(X) is not a surjection of Zariski sheaves.

We will see in ^{\underline{cor2}} 13.13 that Tot(C_{∗}ℤ_{tr}(Ǔ)) is a Zariski resolution of C_{∗}ℤ_{tr}(X). If U = {U_{i} → X} is a Zariski covering, we can replace the infinite complex

ℤ_{tr}(Ǔ) of ^{\underline{ztruu}} 6.12 by the bounded complex

ℤ_{tr}(Ǔ) : 0 → ℤ_{tr}(U_{1} ∩ … ∩ U_{n}) → … → ⊕_{i}ℤ_{tr}(U_{i}) → 0.

13.5 Proposition 6.14. Let U = {U_{i} → X} be a Zariski open covering of X. Then ℤ_{tr}(Ǔ) is an étale resolution of ℤ_{tr}(X), i.e., the following sequence is exact as a complex of étale sheaves:

0 → ℤ_{tr}(U_{1} ∩ … ∩ U_{n}) → … → ⊕_{i}ℤ_{tr}(U_{i}) → ℤ_{tr}(X) → 0.

Proof. If n = 2, we apply ^{\underline{ztruu}} 6.12 to U = U_{1} ∐ U_{2}. Since U ×_{X} U = U_{1} ∐(U_{1} ∩ U_{2}) ∐ U_{2}, we see that the image of ℤ_{tr}(U^{×3}) in ℤ_{tr}(U ×_{X} U) is ℤ_{tr}(U_{1})⊕ℤ_{tr}(U_{2}) in the exact complex of ^{\underline{ztruu}} 6.12. It follows that ℤ_{tr}(Ǔ) → ℤ_{tr}(X) is exact for n = 2. For n > 2, the exactness follows by induction on n.

6.14 Example 6.15. If U is the cover of ℙ^{1} by 𝔸^{1} = Spec k[t] and Spec k[t^{−1}], and we mod out by the basepoint t = 1, we obtain the exact sequence

0 → ℤ_{tr}(𝔾_{m}) → 2ℤ_{tr}(𝔸^{1}, 1) → ℤ_{tr}(ℙ^{1}, 1) → 0.

Applying C_{∗}, and recalling that C_{∗}ℤ_{tr}(𝔸^{1}, 1) ≃ 0, we obtain quasi-isomorphisms of étale complexes (or even Nisnevich complexes)

C_{∗}ℤ_{tr}(ℙ^{1}, 1) ≃ C_{∗}ℤ_{tr}(𝔾_{m})[1] = ℤ(1).

etcov Lemma 6.16. Let p : U → Y be an étale covering and f : X → Y a finite correspondence. Then there is an étale covering p′ : V → X and a finite correspondence f′ : V → U so that the following diagram commutes in Cor_{k}.

[figure: V; f′; U; X; p′; f; Y; p]

Proof. We may suppose that f is defined by the elementary correspondence Z ⊂ X × Y. Form the pullback Z_{U} = Z ×_{Y} U inside X × U. Since the projection Z_{U} → Z is étale and Z → X is finite, there is an étale cover V → X so that V ×_{X} Z_{U} → V ×_{X} Z has a section s. But then s(V ×_{X} Z_{U}) ⊂ V × U over V and defines the required finite correspondence V → U.

[figure: V ×_{X} Z_{U}; Z_{U}; ⊂; X × U; U]

[figure: V ×_{X} Z; Z; ⊂; X × Y; Y]

[figure]

[figure: V; X]

As in ^{\underline{milne}} [Mil80] pp. 61-65, the inclusion i : Sh_{ét}(Sm/k) → P reSh(Sm/k) has a left adjoint a_{ét}, and i ∘ a_{ét} is left exact. Hence the category of étale sheaves on Sm/k is abelian, and the functor a_{ét} is exact.

If F is a presheaf with transfers, the following theorem shows that its étale sheafification admits transfers. The same holds in the Nisnevich topology but not in the Zariski topology. However, we will prove later (in ^{\underline{azartr}} 21.15) that if F is a homotopy invariant presheaf with transfers, its Zariski sheafification admits transfers.

Recall that there is a forgetful functor φ : PST(k) → P reSh(Sm/k).

ettr Proposition 6.17. Let F be a presheaf with transfers, and write F_{ét} for a_{ét}φF. Then F_{ét} has a unique structure of presheaf with transfers such that F → F_{ét} is a morphism of presheaves with transfers.

exact Corollary 6.18. The inclusion functor Sh_{ét}(Cor_{k}) ^{⊂} ^{i} PST(k) has a left adjoint a_{ét}. The category Sh_{ét}(Cor_{k}) is abelian, a_{ét} is exact and commutes with the forgetful functor φ to (pre)sheaves on Sm/k.

The connections between these abelian categories, given by ^{\underline{ettr}} 6.17 and ^{\underline{exact}} 6.18, are described by the following diagram, where the φ are forgetful functors and both functors a_{ét} are exact.

[figure: P reSh(Sm/k); φ; PST(k); Sh_{ét}(Sm/k); i_{∪}; a_{ét}; φ; Sh_{ét}(Cor_{k}); i_{∪}; a_{ét}]

Proof of ^{\underline{ettr}} 6.17. Uniqueness. Suppose that two étale sheaves with transfers F_{1} and F_{2} satisfy the conditions of the theorem. We already know that F_{1}(X) = F_{2}(X) = F_{ét}(X) for all X and we just need to check that F_{1}(f) = F_{2}(f) holds when f : X → Y is a morphism in Cor_{k}. This is given if f comes from Sm/k.

Let y ∈ F_{1}(Y) = F_{2}(Y) = F_{ét}(Y). Choose an étale covering p : U → Y so that y|_{U} ∈ F_{ét}(U) is the image of some u ∈ F (U). Lemma ^{\underline{etcov}} 6.16 yields the following diagram.

[figure: V; f′; U; X; p′; f; Y; p]

Because y|_{U} comes from F (U), we have F_{1}(f′)(y|_{U}) = F_{2}(f′)(y|_{U}).

F_{1}(p′)F_{1}(f)(y) = F_{1}(f′)F_{1}(p)(y)  as the diagram commutes,
= F_{1}(f′)(y|_{U})  as p comes from Sm/k,
= F_{2}(f′)(y|_{U})  as y|_{U} comes from F (U),
= F_{2}(p′)F_{2}(f)(y)  as the diagram commutes,
= F_{1}(p′)F_{2}(f)(y)  as p′ comes from Sm/k.

This implies that F_{1}(f) = F_{2}(f) as p′ is a covering and F_{1} is an étale sheaf.

Existence. We need to define a morphism F_{ét}(Y) → F_{ét}(X) for each finite correspondence from X to Y. We first produce a map

F_{ét}(Y) → Hom_{Sh}(ℤ_{tr}(Y), F_{ét})

natural in Cor_{k}.

For all y ∈ F_{ét}(Y) there is an étale covering p : U → Y and an element u ∈ F (U) so that y and u agree in F_{ét}(U). By representability (see ^{\underline{defztr}} 2.7), u determines a morphism ℤ_{tr}(U) → F of presheaves with transfers. By shrinking U, we may arrange that the difference map sends u to zero in F (U ×_{X} U). A chase in the commutative diagram below (where U_{X}^{2} denotes U ×_{X} U) will produce the map of sheaves [y] : ℤ_{tr}(Y) → F_{ét}. The top row is exact by ^{\underline{ztruu}} 6.12.

[figure: 0 → Hom_{Sh}(ℤ_{tr}(Y), F_{ét}); Hom_{Sh}(ℤ_{tr}(U), F_{ét}); Hom_{Sh}(ℤ_{tr}(U_{X}^{2}), F_{ét}); Hom_{PST}(ℤ_{tr}(U), F); Hom_{PST}(ℤ_{tr}(U_{X}^{2}), F)]

We can now define a pairing Cor(X, Y) ⊗ F_{ét}(Y) → F_{ét}(X). Let f be a correspondence from X to Y and y ∈ F_{ét}(Y). By the map just described, y induces a morphisms of sheaves [y] : ℤ_{tr}(Y) → F_{ét}. Consider the composition:

ℤ_{tr}(X) ^{f} ℤ_{tr}(Y) ^{[y]} F_{ét}.

Hence there is a map ℤ_{tr}(X)(X) → F_{ét}(X). The image of the identity map will be the pairing of f and y.

We conclude with an application of these ideas to homological algebra. etinj Proposition 6.19. The abelian category Sh_{ét}(Cor_{k}) has enough injectives.

Proof. The category S = Sh_{ét}(Cor_{k}) has products and filtered direct limits are exact, because this is separately true for presheaves with transfers and for étale sheaves. That is, S satisfies axioms AB5 and AB3^{∗}. By ^{\underline{homshpst}} 6.3, the family of sheaves ℤ_{tr}(X) is a family of generators of S. It is well known (see ^{\underline{tohoku}} [Gro57, 1.10.1]) that this implies that S has enough injectives.

6.18 Example 6.20. Let F be an étale sheaf with transfers. We claim that the terms E^{n}(F) in its canonical flasque resolution (as an étale sheaf, see ^{\underline{milne}} [Mil80] p. 90) are actually étale sheaves with transfers. For this it suffices to consider E = E^{0}(F). Fix an algebraic closure k̄ of k. For every X we define:

E(X) = ∏_{x̄∈X(k̄)} Fx̄,

where X(k̄) is the set of k̄-points of X, and Fx̄ denotes the fiber of F at x̄. If U → X is étale, E(U) is the product ∏ Fx̄ over x̄ ∈ U(k̄). From this it follows that E is an étale sheaf, not only on X but on the big étale site of Sm/k. It is also easy to see that F (X) → E(X) is an injection.

In addition, E is a presheaf with transfers and F → E is a morphism in PST. For if Z ⊂ X × Y is an elementary correspondence from X to Y, we define the transfer E(Y) → E(X)

E(Y) = ∏_{ȳ∈Y (k̄)} Fȳ → ∏_{x̄∈X(k̄)} Fx̄ = E(X)

by stating that the component for x̄ ∈ X(k̄) is the sum over all ȳ ∈ Y (k̄) such that z = (x̄, ȳ) ∈ Z(k̄) of the localized transfers Fȳ → Fx̄. To see that F → E is a morphism in PST, we may take X to be strictly Hensel local, so F (X) = E(X). Since this forces Y to also be strictly Hensel semilocal, so F (Y) = E(Y), this is a tautology.

The same construction works in the Nisnevich topology, letting E(X) be the product over all closed points x ∈ X of F (Spec O_{X,x}^{h}) (see ^{\underline{nisres}} 13.3). However, example ^{\underline{6.12}} 6.13 shows that it does not work in the Zariski topology, because the transfer E(X) → E(S) need not factor through the sum of the E(U_{i}).

6.20 Lemma 6.21. If F is any étale sheaf with transfers, then its cohomology presheaves H_{ét}^{n}(−, F) are presheaves with transfers.

Proof. The canonical flasque resolution F → E^{∗}(F) of ^{\underline{6.18}} 6.20 is a resolution of sheaves with transfers. Since the forgetful functor from PST(k) to presheaves is exact, and H^{n}(−, F) is the cohomology E^{∗}(F) as a presheaf, we see that H^{n}(−, F) is also the cohomology of E^{∗}(F) in the abelian category PST(k).

Example 6.22. By ^{\underline{schrier}} 2.4, F = 𝔾_{m} is an étale sheaf with transfers. By ^{\underline{6.20}} 6.21, both the Picard group Pic(X) = H_{ét}^{1}(X, 𝔾_{m}) and the cohomological Brauer group Br′(X) = H_{ét}^{2}(X, 𝔾_{m})_{tors} are presheaves with transfers. etext Lemma 6.23. For any F ∈ Sh_{ét}(Cor_{k}) and any smooth X and i ∈ ℤ we have:

Ext_{Sh_{ét}(Cor_{k})}^{i}(ℤ_{tr}(X), F) = H_{ét}^{i}(X, F).

Proof. The case i = 0 is Hom(ℤ_{tr}(X), F) = F (X); this is ^{\underline{homshpst}} 6.3. For i > 0 it suffices to show that if F is an injective étale sheaf with transfers then H^{i}(X, F) is zero. Consider the canonical flasque resolution E^{∗}(F) of example ^{\underline{6.18}} 6.20. Since F is injective, the canonical inclusion F → E^{0} must split, i.e., F is a direct factor of E^{0} in Sh_{ét}(Cor_{k}). Since H_{ét}^{i}(X, F) is a direct summand of H^{i}(X, E^{0}), it must vanish for i > 0.

If we restrict to the category Sh_{ét}(Cor_{k}, R) of étale sheaves of R-modules with transfer, E^{0}(F) is a flasque sheaf of R-modules with transfer by ^{\underline{6.18}} 6.20. The proof of ^{\underline{etext}} 6.23 goes through word for word to prove the following variation.

etextr Porism 6.24. For any F ∈ Sh_{ét}(Cor_{k}, R) and any smooth X and i ∈ ℤ:

Ext_{Sh_{ét}(Cor_{k},R)}^{i}(R_{tr}(X), F) = H_{ét}^{i}(X, F).

The same proof also shows that lemmas ^{\underline{etext}} 6.23 and ^{\underline{etextr}} 6.24 hold for the Nisnevich topology (see ^{\underline{nistr}} 13.4). See ^{\underline{tri}} [TrCat, 3.1.8] for an alternative proof.

hexthcoho Exercise 6.25. Still assuming that cd_{R}(k) < ∞, let K be any complex of étale sheaves of R-modules with transfer. Show that its hyperext and hypercohomology agree in the sense that for any smooth X and i ∈ ℤ:

Ext^{i}(R_{tr}(X), K) ≅ ℍ_{ét}^{n}(X, K).

If one is willing to extend the constructions of this paper to possibly singular schemes, the cdh topology would play a major role. The constructions of this section can be carried out in the cdh topology as well.

Relative Picard group and Suslin’s Rigidity Theorem

relpicandrig

In this lecture we introduce the relative Picard group Pic(X̄, X_{∞}). When X̄ is a good compactification of X over S, its elements determine maps F (X) → F (S) for every homotopy invariant F. This pairing will be used to prove Suslin’s Rigidity Theorem ^{\underline{loccons}} 7.20.

Recall from ^{\underline{ui}} 1A.7 and ^{\underline{a7}} 1A.8 that if S is a smooth connected scheme and p : X → S a smooth morphism then we write c(X/S, 0) for the free abelian group generated by the irreducible closed subsets of X which are finite and surjective over S. In this lecture we will write C_{0}(X/S) for c(X/S, 0).

By ^{\underline{corscork}} 1A.10, given a map S′ → S, there is a map C_{0}(X/S) → C_{0}(X ×_{S} S′/S′), induced from

C_{0}(X/S) ^{⊂} ℤ_{tr}(X)(S) = Cor_{k}(S, X).

h0sing Definition 7.1. We define H_{0}^{sing}(X/S) to be the cokernel of the map

C_{0}(X × 𝔸^{1}/S × 𝔸^{1}) ^{∂_{0}−∂_{1}} C_{0}(X/S)

where ∂_{i} is induced by “t = i” : Spec k → 𝔸_{k}^{1}.

h0singa1he Example 7.2. If X = Y ×_{k} S then C_{0}(X/S) = Cor(S, Y) = ℤ_{tr}(Y)(S). In addition, X × 𝔸^{1} = Y ×_{k} (S × 𝔸^{1}) and the following diagram commutes:

[figure: C_{0}(X × 𝔸^{1}/S × 𝔸^{1}); C_{0}(X/S); ℤ_{tr}(Y)(S × 𝔸^{1}); =; ℤ_{tr}(Y)(S).; =]

Taking cokernels, we conclude (using ^{\underline{2.24}} 2.25) that:

H_{0}^{sing}(Y × S/S) = H_{0}C_{∗}ℤ_{tr}(Y)(S) = Cor(S, Y)/𝔸^{1}-homotopy.

In particular, this implies that two elements of Cor(S, Y) are 𝔸^{1}-homotopic exactly when they agree in H_{0}^{sing}(Y × S/S).

Week 7 - November, 17 1999

If S = Spec k then H_{0}^{sing}(X/S) is the cokernel H_{0}^{sing}(X/k) of ℤ_{tr}(X)(𝔸^{1}) → ℤ_{tr}(X)(S) discussed in exercise ^{\underline{h0ztr}} 2.19, because C_{0}(X/S) = ℤ_{tr}(X)(Spec k). Also by ^{\underline{h0ztr}} 2.19, there is a natural surjection H_{0}^{sing}(X/S) → CH_{0}(X). If X is projective, this surjection is an isomorphism.

hsingline Example 7.3. If S = Spec k, then ^{\underline{relpic}} 7.16 below shows that H_{0}^{sing}(ℙ^{1}/S) = H_{0}^{sing}(𝔸^{1}/S) = ℤ but H_{0}^{sing}(𝔸^{1} − {0}/S) = ℤ ⊕ k^{∗}.

hsing Remark 7.4. In ^{\underline{svin}} [SV96] the groups H_{∗}^{sing}(X/S) are defined to be the homology of the evident chain complex C_{∗}(X/S) with

C_{n}(X/S) = C_{0}(X × Δ^{n}/S × Δ^{n}).

We will consider the singular homology H_{∗}^{sing}(X/S) in lecture ^{\underline{algsinghom}} 10 below when S = Spec k, and C_{∗}(X/S) = C_{∗}ℤ_{tr}(X)(S).

Let F be a PST. The map Tr : C_{0}(X/S) ⊗ F (X) → F (S) is defined to be the inclusion C_{0}(X/S) ⊂ Cor_{k}(S, X) (see ^{\underline{corscork}} 1A.10) followed by evaluation on F (X).

C_{0}(X/S) ⊗ F (X) \overline{Tr} F (S)

[figure: evaluate]

Cor_{k}(S, X) ⊗ F (X)^{∩}

pairing Lemma 7.5. If F is homotopy invariant presheaf with transfers then the map Tr factors through H_{0}^{sing}(X/S) ⊗ F (X) → F (S).

Proof. Since F (X) = F (X × 𝔸^{1}), we have a diagram

[figure: C_{0}(X × 𝔸^{1}/S × 𝔸^{1}) ⊗ F (X); Tr; F (S × 𝔸^{1}); C_{0}(X/S) ⊗ F (X); ∂_{0} − ∂_{1}; Tr; F (S).; i_{0} − i_{1} = 0]

expair Example 7.6. If σ : S → X is a section of p, regarded as an element of H_{0}^{sing}(X/S), then T r(σ, −) is the usual map σ^{∗} : F (X) → F (S).

Remark 7.7. The pairing H_{0}^{sing}(X/S) ⊗ F (X) → F (S) is fundamental. It can be defined more generally for homotopy invariant presheaves equipped only with transfer maps Tr_{D} : F (X) → F (S) for any relative smooth curve X/S and any effective divisor D ⊂ X which is finite and surjective over S, such that the transfer maps form a “pseudo pretheory”. This construction applies to the K-theory presheaves K_{n}(X), equipped with the transfer maps of exercise ^{\underline{311}} 2.6, even though these are not presheaves with transfers.

In order to compute H_{0}^{sing}(X/S), it is useful to embed X in a slightly larger scheme X̄.

defgc Definition 7.8. A smooth curve p : X → S admits a good compactification X̄ if it factors as:

X ^{_{⊂} j} X̄

[figure: p]

Sp̄ ;

where j is an open embedding, X̄ is a proper normal but not necessarily smooth curve over S and Y = X̄ − X has an affine open neighborhood in X̄.

If S is affine, for example, then X = 𝔸^{1} × S admits ℙ^{1} × S as a good compactification. Similarly, if C is any smooth affine curve over k then C × S → S admits C̄ × S as a good compactification. The following result implies that every point x of every X has an open neighborhood U which has a good compactification over a generic projection X → 𝔸^{l−1}.

goodcomp Lemma 7.9. Let p : X → 𝔸^{l} be an étale map. If k is infinite, there exists a linear projection 𝔸^{l} → 𝔸^{l−1} so that the composition X → 𝔸^{l−1} is a curve with a good compactification.

Proof. There is an open U ⊂ 𝔸^{l} so that X is quasi-finite and surjective over U. Choose a linear projection 𝔸^{l} → 𝔸^{l−1} so that the restriction to 𝔸^{l} − U is finite; 𝔸^{l} has good compactification Y = ℙ^{1} × 𝔸^{l−1}. By Zariski’s Main Theorem (as formulated in ^{\underline{ega4}} [EGA IV, 8.12.6]), the map X → Y may be factored as an open immersion X ^{⊂} X̄ followed by a finite map p̄: X̄ → Y. Replacing X̄ by its normalization, we may assume that X̄ is normal. Note that p̄ is an affine map. Since Y is a good compactification of U, X̄ is a good compactification of X.

relpicdef Definition 7.10. If Y ^{⊂} ^{i} X̄ is closed we set 𝔾_{X̄,Y} = Ker(O∗ ; X̄ → i_{∗}O_{Y}^{∗}). The relative Picard group is defined to be:

Pic(X̄, Y) = H_{Zar}^{1}(X̄, 𝔾_{X̄,Y}).

By ^{\underline{milne}} [Mil80] p. 124, we also have Pic(X̄, Y) = H_{ét}^{1}(X̄, 𝔾_{X̄,Y}).

By ^{\underline{svin}} [SV96, 2.1], the elements of Pic(X̄, Y) are the isomorphism classes (L, t) of line bundles L on X̄ with a trivialization t on Y. The group operation is ⊗, i.e., (L, t) ⊗ (L′, t′) = (L ⊗ L′, t ⊗ t′).

7.10 Remark 7.11. For X̄ = S × ℙ^{1} and Y = S × {0, ∞}, the “stalk” (i^{∗}𝔾_{X̄,Y})(Y) of 𝔾_{X̄,Y} at Y is the group M^{∗}(ℙ^{1}; 0, ∞)(S) of lecture 4.

The cohomology of O^{∗} → i_{∗}O_{Y}^{∗} yields the exact sequence

O^{∗}(X̄) → O^{∗}(Y) → Pic(X̄, Y) → Pic(X̄) → Pic(Y).

Comparing this exact sequence for X̄ and X̄ × 𝔸^{1} yields: pichi Corollary 7.12. If X̄ is a normal scheme and Y is reduced, we have:

Pic(X̄, Y) = Pic(X̄ × 𝔸^{1}, Y × 𝔸^{1}).

Let us write j for the open embedding of X = X̄ − Y into X̄.

injec Lemma 7.13. If 1/n ∈ k, there is a natural injection

Pic(X̄, Y)/n ^{⊂} H_{ét}^{2}(X̄, j_{!}μ_{n}).

Proof. By Kummer Theory we have an exact sequence of étale sheaves:

0  j_{!}μ_{n}  𝔾_{X̄,Y} ^{n} 𝔾_{X̄,Y}  0.  (7.13.1) 7.11.1

Applying étale cohomology yields:

[figure: H^{1}(X̄, j_{!}μ_{n}); H^{1}(X̄, 𝔾_{X̄,Y}); n; H^{1}(X̄, 𝔾_{X̄,Y}); H_{ét}^{2}(X̄, j_{!}μ_{n}).]

But the middle groups are both Pic(X̄, Y).

picn Example 7.14. Suppose that S = Spec k and k is algebraically closed. If X̄ is a smooth connected curve, then Pic(X̄, Y) is an extension of Pic(X̄) by a finite product of |Y | − 1 copies of k^{∗}. Hence Pic(X̄, Y)/n ≅ H_{ét}^{2}(X̄, μ_{n}) ≅ ℤ/n.

Recall that C_{0}(X/S) is generated by closed subsets Z of X which are finite and surjective over S. Because X is smooth, each such subset is an effective Cartier divisor on X̄, and has an associated line bundle L equipped with a canonical map O → L. This map gives a trivialization of L on X̄ − Z, which is a neighborhood of Y. Thus a good compactification X̄ induces a homomorphism

C_{0}(X/S) → Pic(X̄, Y).

When Y lies in an affine open neighborhood, this map is onto because every trivialization on Y extends to a neighborhood of Y.

C0lift Exercise 7.15. In this exercise we make the lifting to C_{0}(X/S) explicit. Suppose that L is a line bundle on X̄ with a fixed trivialization t on an open neighborhood U of Y. Show that t gives a canonical isomorphism of L with a Cartier divisor L(D), i.e., an invertible subsheaf of the sheaf K of total quotient rings of O. (See ^{\underline{hart}} [Har77, II.6].) Show that L(D) comes from a Weil divisor D = ∑ n_{i}Z_{i} on X̄ with the Z_{i} supported on X̄ − U. Then show that the map C_{0}(X/S) → Pic(X̄, Y) sends ∑ n_{i}Z_{i} to (L, t).

Because C_{1}(X/S) → Pic(X̄, Y) factors through Pic(X̄ × 𝔸^{1}, Y × 𝔸^{1}), ^{\underline{pichi}} 7.12 shows that C_{0}(X/S) → P ic(X̄, Y) induces a homomorphism

H_{0}^{sing}(X/S) → Pic(X̄, Y). relpic Theorem 7.16. Let S be a smooth scheme. If p : X → S is a smooth quasi-

affine curve with a good compactification (X̄, Y), then:

H_{0}^{sing}(X/S) ^{≅} Pic(X̄, Y).

Proof. The kernel of C_{0}(X/S) → Pic(X̄, Y) consists of f ∈ K(X̄) which are defined and equal to 1 on Y. Since X is quasi-affine over S, Y contains at least one point in every irreducible component of every fiber of X̄ over S. Therefore the divisor D of tf+(1−t) defines an element of C_{0}(X×𝔸^{1}/S×𝔸^{1}) with ∂_{0}D = 0 and ∂_{1}D = (f). Hence (f) represents 0 in H_{0}^{sing}(X/S). This proves that the map H_{0}^{sing}(X/S) → Pic(X̄, Y) is an injection, hence an isomorphism.

Theorem ^{\underline{relpic}} 7.16 also holds when X is not quasi-affine over S, but the proof is more involved.

Corollary 7.17. If F is a homotopy invariant presheaf with transfers, there is a pairing

Pic(X̄, Y) ⊗ F (X) → F (S).

Example 7.18. If X is a smooth curve over k and 1/n ∈ k, then any two geometric points x, x′ : Spec k̄ → X induce the same map F (X) → F (Spec k̄). Here F is any homotopy invariant presheaf with transfers satisfying nF = 0. Indeed, [x] = [x′] in Pic(X̄, Y)/n by example ^{\underline{picn}} 7.14. This phenomenon is known as “rigidity,” and is a simple case of Theorem ^{\underline{loccons}} 7.20 below.

inj Corollary 7.19. Let p : X → S be a smooth curve with a good compactification. Assume that S is Hensel local and let X_{0} → S_{0} be the closed fiber of p. Then for every n prime to char k the following map is injective:

H_{0}^{sing}(X/S)/n → H_{0}^{sing}(X_{0}/S_{0})/n.

Proof. Kummer Theory yields the exact sequence ^{\underline{7.11.1}} 7.13.1 of étale sheaves, and similarly for (X̄_{0}, Y_{0}). Applying étale cohomology yields:

[figure: H^{1}(X̄, j_{!}μ_{n}); H^{1}(X̄, 𝔾_{X̄,Y}); n; H^{1}(X̄, 𝔾_{X̄,Y}); H_{ét}^{2}(X̄, j_{!}μ_{n}); H^{1}(X̄_{0}, j_{!}μ_{n}); H^{1}(X̄_{0}, 𝔾_{X̄,Y}); n; H^{1}(X̄_{0}, 𝔾_{X̄,Y}); H_{ét}^{2}(X̄_{0}, j_{!}μ_{n}).; ≅]

Since H^{2}(X̄, j_{!}μ_{n}) = H_{c}^{2}(X, μ_{n}), the right vertical map is an isomorphism by proper base change with compact supports (see ^{\underline{milne}} [Mil80, VI.3.2]). We have a diagram:

[figure: Pic(X̄, Y)/n; ⊂; H_{ét}^{2}(X̄, j_{!}μ_{n}); Pic(X_{0}, Y_{0})/n; ⊂; H_{ét}^{2}(X̄_{0}, j_{!}μ_{n}).; ≅] Corollary ^{\underline{inj}} 7.19 now follows from theorem ^{\underline{relpic}} 7.16.

It follows from ^{\underline{ex64}} 6.8 that every locally constant étale sheaf F is homotopy invariant, because H^{0}(X × 𝔸^{1}, O) ≅ H^{0}(X, O) ⊗_{k} k[t]. The following result shows that the converse is true for torsion sheaves. (Cf. ^{\underline{svin}} [SV96, 4.5].)

loccons Theorem 7.20. (Suslin’s “Rigidity Theorem”) Let F be a homotopy invariant presheaf with transfers, such that the groups F (X) are torsion of exponent prime to char k. Then F_{ét} is locally constant.

Proof. Let F_{0} = π_{∗}π^{∗}(F) be the locally constant sheaf for the group M = F (k_{sep}). We want to show that the adjunction F_{0} → F is an isomorphism of étale sheaves. It suffices to check on stalks. Since O_{X,x}^{sh} contains a separable closure of k, we may assume that k is separably closed. In this case ^{\underline{loccons}} 7.20 asserts that F_{ét} is the constant sheaf for the group M = F (Spec k). Since X is smooth at x, O_{X,x}^{sh} is isomorphic to the Henselization of 𝔸^{l} at {0}. Thus the Rigidity Theorem is a consequence of proposition ^{\underline{68}} 7.21 below.

68 Proposition 7.21. Let S_{l} be the Henselization at {0} in 𝔸^{l} over a separably closed field k. Assume that F is as is ^{\underline{loccons}} 7.20. Then F (S_{l}) = F (Spec k).

Proof. The hypothesis on F is inherited by F (X)_{n} = {x ∈ F (X) : nx = 0}. Therefore we may assume that F has exponent n for some prime n.

We use the following sequence of inclusions:

Spec k = S_{0} ⊂ … ⊂ S_{l−1} ⊂^{i} S_{l}.

By induction on l, it is enough to prove that the map F (i) : F (S_{l}) → F (S_{l−1}) is an isomorphism. For this it suffices to prove that F (i) is an injection, because it is split by the projection π

S_{l−1} [figure: π; i] S_{l}.

But F (S_{l}) = colim_{(X,x_{0})→(𝔸^{l},0)} F (X) where the colimit is taken over all diagrams:

S_{l} ^{π} X ^{p} 𝔸_{l}.

It suffices to show for every X that if φ ∈ F (X) has i_{l}^{∗}π^{∗}φ = 0 then π^{∗}φ = 0. By lemma ^{\underline{goodcomp}} 7.9 there is a curve X → 𝔸_{l−1} with a good compactification. Let X′ be the pullback in the following diagram:

S_{l−1}  \overline{i_{l}} S_{l}

[figure: s_{1}; π; Id]

X′  \overline{q} X

[figure: S_{l}; π_{l}; S_{l−1}; 𝔸^{l−1}]

The maps π and πi_{l}π_{l} : S_{l} → X induce two sections s_{1}, s_{2} : S_{l} → X′ of X′ → S_{l} which agree on the closed fiber X_{0} = X ×_{S} S_{0}. Given φ ∈ F (X) we need to show that π_{l}^{∗}i_{l}^{∗}π^{∗}φ = π^{∗}φ. But π^{∗}φ = s_{1}^{∗}q^{∗}(φ) and π_{l}^{∗}i_{l}^{∗}π^{∗}φ = s_{2}^{∗}q^{∗}(φ). The s_{i} coincide on the closed point of S_{l} by construction. So we are left to prove that s^{∗}(ψ) = (s′)^{∗}(ψ) for all ψ ∈ F (X′) and any s, s′ : S_{l} → X′ with s_{0} = s′_{0}. Consider the following diagram:

(Γ_{s} − Γ_{s′}) ⊗ ψ  s^{∗}(ψ) − s′^{∗}(ψ)

[figure: C_{0}(X′/S_{l}) ⊗ F (X′); H_{0}(X′/S_{l}) ⊗ F (X′); Tr; F (S_{l})]

[figure: H_{0}(X′_{0}/S_{0}) ⊗ F (X′); ∩; Tr; F (S_{0})]

By assumption, the element (Γ_{s} − Γ_{s′}) ⊗ ψ in the top left group goes to zero in H_{0}(X′_{0}/S_{0})⊗F (X′). Hence it vanishes in H_{0}(X′/S_{l})⊗F (X′) by the immersion of H_{0}(X′/S)/n in H_{0}(X′_{0}/S_{0})/n of ^{\underline{inj}} 7.19. Therefore s^{∗}(ψ) − s′^{∗}(ψ) vanishes in F (S_{l}).

We conclude this lecture with a description of the behavior of the relative Picard group for finite morphisms. We will need this description in the proof of

\underline{storto} 20.9.

covf Definition 7.22. Let (Ȳ , Y_{∞}) and (X̄, X_{∞}) be two good compactifications, say of Y and X, respectively. Any finite map f : Ȳ → X̄ which restricts to a map f : Y → X, yields a map f_{∗} : O^{∗}(Y_{∞}) → O^{∗}(X_{∞}) constructed as follows.

Consider α ∈ O^{∗}(Y_{∞}). We may extend α to α̃ ∈ O^{∗}(U) where U is an affine open neighborhood of Y_{∞}. Since f is finite, we may assume that U = f^{−1}(V), where V is an open neighborhood of X_{∞}. Since V is normal, there is a norm map N : O^{∗}(U) → O^{∗}(V) (see ^{\underline{schrier}} 2.4). We define f_{∗}(α) = N(α̃)|_{X_{∞}}. By ^{\underline{normszz}} 7.23 below, f_{∗}(α) is independent of the choice of the extension α̃. normszz Exercise 7.23. Let f : U → V be a finite morphism of normal schemes and let

Z ⊂ V be a reduced closed subscheme. If α ∈ O^{∗}(U) and α = 1 on the reduced closed subscheme f^{−1}(Z), show that N(α) = 1 on Z.

invrelpic Lemma 7.24. Let (Ȳ , Y_{∞}) and (X̄, X_{∞}) be good compactifications of Y and X, respectively. Let f be a finite map f : Ȳ → X̄ which restricts to a map f : Y → X. Then the following diagram is commutative:

[figure: O^{∗}(Y_{∞}); Pic(Ȳ , Y_{∞}); ≅; H_{0}(Y/S); O^{∗}(X_{∞}); f_{∗}; Pic(X̄, X_{∞}); ≅; H_{0}(X/S),]

where f_{∗} was defined in ^{\underline{covf}} 7.22 and the right vertical map is induced by the pushforward of cycles.

Proof. Choose α ∈ O^{∗}(Y_{∞}) and extend it to a rational function t on Ȳ which is regular in a neighborhood of the form f^{−1}(V). By definition, f_{∗}(α) extends to the regular function N(t) on V. The horizontal maps send α and f_{∗}(α) to (OȲ , α) and (OX̄, f_{∗}α). Let D and D′ be the Weil divisors on Ȳ and X̄ associated to t and N(t), respectively. We may regard D and D′ as classes in C_{0}(Y/S) and C_{0}(X/S). By ^{\underline{C0lift}} 7.15, D and D′ represent the images of (OȲ , α) and (OX̄, f_{∗}α) in H_{0}(Y/S) and H_{0}(X/S), respectively. The right vertical map send D to D′ because D′ = div(Nt) is the push-forward of D = div(t) (see ^{\underline{fulton}} [Ful84, 1.4]).

Derived tensor products

The goal of this lecture is to define a tensor product on the derived category of étale sheaves with transfer, starting with the tensor product X ⊗ Y = X × Y on Cor_{k} defined in ^{\underline{tensor}} 1.9. For this we first need to build a total tensor product on the category PST(k), and this construction makes sense in somewhat greater generality.

Let A be a small additive category. We define ℤ(A) to be the category of all additive presheaves on A, i.e., all contravariant additive functors F : A → Ab. It is an abelian category. The Yoneda embedding h : A → ℤ(A) allows us to define the additive category A^{⊕} as the closure of A under infinite direct sums in ℤ(A). If X_{i} are in A, we will consider X = ⊕X_{i} to be the object of A^{⊕} corresponding to the presheaf h_{X} = ⊕h_{X_{i}} in ℤ(A).

More generally, if R is a ring, we define R(A) to be the (abelian) category of all additive functors F : A → R-mod. By abuse of notation, we will write h_{X} for the functor A ↦ R ⊗_{ℤ} Hom_{A}(A, X) and call it “representable”.

projectives Lemma 8.1. Every representable presheaf h_{X} is a projective object of R(A), every F in R(A) has a projective resolution by representable functors, and every projective object of R(A) is a direct summand of a representable functor.

Proof. Since Hom_{R(A)}(h_{X}, F) ≅ F (X), each h_{X} is a projective object in R(A). Moreover every F in R(A) is a quotient of some h_{X}, X ∈ A^{⊕}, because of the natural surjection

⨁_{X in A} ⨁_{\begin{matrix}x∈F (X) \\ x≠0 \end{matrix}} h_{X} ^{x} F

Now suppose that A has an additive symmetric monoidal structure ⊗, such as A = Cor_{k}. (By this, we mean that ⊗ commutes with finite direct sums; see \underline{asmc} 8A.3.) We may extend ⊗ to a tensor product on A^{⊕} in the obvious way, and this extends to tensor product of projectives. We now extend ⊗ to a tensor product on all of R(A).

Week 8 - December 1 1999

If F and G are in R(A), we can form the presheaf tensor product (F ⊗_{R} G)(X) = F (X) ⊗_{R} G(X). However, it does not belong to R(A), since F ⊗_{R} G is not additive. In order to get a tensor product on R(A), we need a more complicated construction.

Our construction of ⊗ is dictated by the requirement that if X and Y are in A, then the tensor product h_{X} ⊗ h_{Y} of their representable presheaves should be represented by X ⊗ Y. As a first step, note that we can extend ⊗ to a tensor product ⊗ : A^{⊕} × A^{⊕} → A^{⊕} commuting with ⊕. Thus if L_{1} and L_{2} are in the category Ch^{−}(A^{⊕}) of bounded above cochain complexes (⋯ → F^{n} → 0 → ⋯), the chain complex L_{1} ⊗L_{2} is defined as the total complex of the double complex L_{1}^{∗} ⊗ L_{2}^{∗}.

73 Definition 8.2. If F and G are objects of R(A), choose projective resolutions P_{∗} → F and Q_{∗} → G and define F ⊗^{𝕃} G to be P ⊗ Q, i.e., Tot(P_{∗} ⊗ Q_{∗}). We define the tensor product and Hom presheaves to be:

F ⊗ G = H_{0}(F ⊗^{𝕃} G)
\underline{Hom}(F, G) : X ↦ Hom_{R(A)}(F ⊗ h_{X}, G)

Since any two projective resolutions of F are chain homotopy equivalent, the chain complex F ⊗^{𝕃} G is well-defined up to chain homotopy equivalence, and similarly for \underline{Hom}(F, G). In particular, since h_{X} and h_{Y} are projective, we have h_{X} ⊗^{𝕃} h_{Y} = h_{X} ⊗ h_{Y} = h_{X⊗Y} for all X and Y in A^{⊕}.

The following result implies that R(A) is an additive symmetric monoidal category (see ^{\underline{asmc}} 8A.3).

8.3 Lemma 8.3. The functor \underline{Hom}(F, −) is right adjoint to F ⊗ −. In particular, \underline{Hom}(F, −) is left exact and F ⊗ − is right exact.

Proof. Because R(A) has enough projectives, it suffices to observe that

Hom_{R(A)}(h_{X}, \underline{Hom}(h_{Y} , G)) = G(X ⊗ Y) = Hom_{R(A)}(h_{X} ⊗ h_{Y} , G).

ex84 Example 8.4. If A is the category of free R-modules over a commutative ring R, R(A) is equivalent to the category of all R-modules; the presheaf associated to M is M⊗_{R}, and \underline{Hom} and ⊗ are the familiar Hom_{R} and ⊗_{R}.

ex74 Exercise 8.5. If F_{i} and G_{i} are in R(A), show that there is a natural map

\underline{Hom}(F_{1}, G_{1}) ⊗ \underline{Hom}(F_{2}, G_{2}) → \underline{Hom}(F_{1} ⊗ F_{2}, G_{1} ⊗ G_{2}),

compatible with the monoidal pairing Hom_{A}(U×A_{1}, X_{1})⊗Hom_{A}(U×A_{2}, X_{2}) → Hom_{A}(U × U × A_{1} × A_{2}, X_{1} × X_{2}) → Hom_{A}(U × A_{1} × A_{2}, X_{1} × X_{2}).

Remark 8.6. If the (projective) objects h_{X} are flat, i.e., h_{X} ⊗ − is an exact functor, then ⊗ is called a balanced functor (^{\underline{w}}[Wei94, 2.7.7]). In this case F ⊗^{𝕃} G agrees (up to chain equivalence) with the usual left derived functor 𝕃(F ⊗ −)G. But we do not know when the h_{X} are flat. It is true in example ^{\underline{ex84}} 8.4, but probably not true in PST = ℤ(Cor_{k}).

We can now extend ⊗^{𝕃} to a total tensor product on the category Ch^{−}R(A) of bounded above cochain complexes (⋯ → F^{n} → 0 → ⋯). This would be the usual derived functor if ⊗ were balanced (see ^{\underline{w}}[Wei94, 10.6]), and our construction is parallel. If C is a complex in Ch^{−}R(A), there is a quasi-isomorphism P ^{≃} C with P a complex of projective objects. Any such complex P is called a projective resolution of C, and any other projective resolution of C is chain homotopic to P; see ^{\underline{w}}[Wei94, 5.7]. If D is any other complex in Ch^{−}R(A), and Q ^{≃} D is a projective resolution, we define

C ⊗^{𝕃} D = P ⊗ Q.

Because P and Q are bounded above, each (P ⊗ Q)^{n} = ⊕_{i+j=n}P^{i} ⊗ Q^{j} is a finite sum, and C ⊗^{𝕃} D is bounded above. Because P and Q are defined up to chain homotopy, the complex C ⊗^{𝕃} D is independent (up to chain homotopy equivalence) of the choice of P and Q. There is a natural map C ⊗^{𝕃} D → C ⊗D, which extends the map F ⊗^{𝕃} G → F ⊗ G of definition ^{\underline{73}} 8.2.

lem7.5 Lemma 8.7. Let C, C′ and D be bounded above complexes of presheaves.

1. If C and D are complexes over A^{⊕}, or complexes of projectives, then C ⊗^{𝕃} D ^{≃} C ⊗ D is a chain homotopy equivalence.

2. If f : C ^{≃} C′ is a quasi-isomorphism of complexes, then C ⊗^{𝕃} D → C′ ⊗^{𝕃} D is a chain homotopy equivalence.

Proof. If C is a complex over A^{⊕}, it is a complex of projectives. We may take P = C in the definition of ⊗^{𝕃}: C ⊗^{𝕃} D = C ⊗ Q. If D is also a complex of projectives, we may take Q = D as well. Part 1 is now immediate. In part 2, we may take P to be a projective resolution of both C and C′, so that C ⊗^{𝕃} D = C′ ⊗^{𝕃} D = P ⊗ Q.

dcTTC Proposition 8.8. The derived category D^{−}R(A), equipped with ⊗^{𝕃}, is a tensor triangulated category.

Proof. The category P of projective objects in R(A) is additive symmetric monoidal, and D^{−}R(A) is equivalent to the chain homotopy category K^{−}(P) by ^{\underline{w}}[Wei94, 10.4.8]. By ^{\underline{KTTC}} 8A.4, this is a tensor triangulated category under ⊗. The result now follows from the natural isomorphism ⊗ ≅ ⊗^{𝕃} in P of ^{\underline{lem7.5}} 8.7.

tpmap Definition 8.9. If C and D are bounded above complexes of presheaves, there is a canonical map from the presheaf tensor product C ⊗_{R} D to the tensor product C ⊗D. By right exactness of ⊗_{R} and ⊗ (see ^{\underline{8.3}} 8.3), it suffices to construct a natural map of presheaves h_{X} ⊗_{R} h_{Y} → h_{X⊗Y}. For U in A, this is just the monoidal product in A, followed by the diagonal Δ : U → U ⊗ U:

h_{X}(U) ⊗_{R} h_{Y} (U) = Hom_{A}(U, X) ⊗_{R} Hom_{A}(U, Y) ^{⊗}

Hom_{A}(U ⊗ U, X ⊗ Y) ^{Δ^{∗}} Hom_{A}(U, X ⊗ Y) = h_{X⊗Y} (U).

Having disposed with these generalities, we now specialize to the case where A is Cor_{k} and ⊗ is the tensor product X ⊗ Y = X × Y of ^{\underline{tensor}} 1.9. We have the Yoneda embedding

Cor_{k} ⊂ Cor_{k}^{⊕} ⊂ PST(k).

We will write ⊗^{tr} for the tensor product on PST = ℤ(Cor_{k}), or on PST(k, R) = R(Cor_{k}), and ⊗_{L}^{tr} for ⊗^{𝕃}. Thus there are natural maps C ⊗_{L}^{tr} D → C ⊗^{tr} D.

wedgetens Example 8.10. By lemma ^{\underline{projectives}} 8.1, h_{X} = R_{tr}(X) is projective and

R_{tr}(X) ⊗^{tr} R_{tr}(Y) = R_{tr}(X × Y).

Similarly if (X_{i}, x_{i}) are pointed schemes then the R_{tr}(X_{i}, x_{i}) are projective and from ^{\underline{les}} 2.11 we see that

R_{tr}(X_{1}, x_{1}) ⊗^{tr} ⋯ ⊗^{tr} R_{tr}(X_{n}, x_{n}) = R_{tr}((X_{1}, x_{1}) ∧ … ∧ (X_{n}, x_{n})).

In particular, R_{tr}(𝔾_{m})^{⊗^{tr}n} = R_{tr}(𝔾_{m}^{∧n}).

The next example, in which R = ℤ, shows that ⊗^{tr} does not behave well on locally constant sheaves.

Example 8.11. The complex ℤ ^{n} ℤ is a projective resolution of ℤ/n, so we have ℤ/n ⊗^{tr} ℤ_{tr}(X) = ℤ/n ⊗_{ℤ} ℤ_{tr}(X) = (ℤ/n)_{tr}(X) by ^{\underline{lem7.5}} 8.7.

If \sqrt{−1} ∉ k and l = k(\sqrt{−1}), let ℤ_{ϵ} = ℤ_{tr}(l)/ℤ denote the locally constant sheaf corresponding to the sign representation of G = Gal(l/k). We see from \underline{lem7.5} 8.7 that ℤ/n ⊗_{L}^{tr} ℤ_{ϵ} is quasi-isomorphic to the complex (ℤ/n) ⊗^{𝕃} (ℤ → ℤ_{tr}(l)), i.e.,

0 → ℤ/n → (ℤ/n)_{tr}(l) → 0.

Hence the presheaf (ℤ/n) ⊗^{tr} ℤ_{ϵ} sends Spec k to 0 and Spec l to ℤ/n. If n = 4, this is not an étale sheaf because (ℤ_{ϵ}/4ℤ_{ϵ})^{G} ≠ 0. It is easy to see, however, that its sheafification is the locally constant étale sheaf:

((ℤ/4) ⊗^{tr} ℤ_{ϵ})_{ét} ≅ μ_{4}.

The étale sheaf μ_{4} is the tensor product (ℤ/4)⊗_{ét} ℤ_{ϵ} of the two underlying étale sheaves.

Definition 8.12. If F and G are presheaves of R-modules with transfer, we write F ⊗_{ét}^{tr} G for (F ⊗^{tr} G)_{ét}, the étale sheaf associated to F ⊗^{tr} G. If C and D are bounded above complexes of presheaves with transfer, we shall write C ⊗_{ét}^{tr} D for (C ⊗^{tr} D)_{ét}, and C ⊗_{L,ét}^{tr} D for (C ⊗_{L}^{tr} D)_{ét} ≃ P ⊗_{ét}^{tr} Q, where P and Q are complexes of representable sheaves with transfers, and P ≃ C and Q ≃ D. There is a natural map C ⊗_{L,ét}^{tr} D → C ⊗_{ét}^{tr} D, induced by C⊗_{L}^{tr} → C ⊗^{tr} D.

83 Lemma 8.13. If F, F′ are étale sheaves of R-modules with transfer, and F is locally constant, then the map of ^{\underline{tpmap}} 8.9 induces an isomorphism

F ⊗_{ét} F′ ^{≅} F ⊗_{ét}^{tr} F′.

Proof. Let F correspond to the discrete Galois module M. As M = ∪M^{H} and ⊗^{tr} commutes with colimits, we may assume that M = M^{H} for some open normal H of Gal = Gal(k_{sep}/k). Thus M is a G-module. Choose a presentation over R[G]:

⊕R[G]^{α} → ⊕R[G]^{β} → M → 0.

As ⊗_{ét} and ⊗_{ét}^{tr} are both right exact, we may assume M = R[G] and F′ = R_{tr}(X). If L = (k_{sep})^{H} and T = Spec(L) then F = R_{tr}(T) by exercise ^{\underline{galext}} 6.10. But then F ⊗^{tr} F′ = R_{tr}(T × X), so it suffices to observe that R_{tr}(T) ⊗_{ét} R_{tr}(X) → R_{tr}(T × X) is an isomorphism. Since T × Y → Y is an étale cover, it suffices to observe that for Y over T

R_{tr}(T) ⊗_{ét} R_{tr}(X)(Y) ≅ R[G] ⊗ Cor(Y, X) ≅ ≅ R ⊗_{ℤ} Cor(Y, T × X) = R_{tr}(T × X)(Y).

We are now going to show (in ^{\underline{etqiso}} 8.16) that the tensor product ⊗_{L,ét}^{tr} induces a tensor triangulated structure on the derived category of étale sheaves of Rmodules with transfer. Using proposition ^{\underline{dcTTC}} 8.8, we have C ⊗_{L,ét}^{tr} D ≅ D ⊗_{L,ét}^{tr} C, and it suffices to show that ⊗_{L,ét}^{tr} preserves quasi-isomorphisms.

As a first step, fix Y and consider the right exact functor Φ(F) = R_{tr}(Y)⊗_{ét}^{tr} F, from the category PST(k, R) of presheaves of R-modules with transfer to the category of étale sheaves of R-modules with transfer. Its left derived functors L_{p} Φ(F) are the homology sheaves of the total left derived functor R_{tr}(Y) ⊗_{L,ét}^{tr} F. If C is a chain complex (bounded below in homological notation), the hyperhomology spectral sequence (see ^{\underline{w}}[Wei94, 5.7.6]) is

E_{p,q}^{2} = L_{p} Φ(H_{q}C) ⇒ 𝕃_{p+q} Φ(C).

8.x Example 8.14. If U → X is an étale cover, consider the augmented Čech

complex

Č : … → R_{tr}(U ×_{X} U) → R_{tr}(U) → R_{tr}(X) → 0.

Since Č_{ét} is exact by ^{\underline{ztruu}} 6.12, each homology presheaf H_{q}(U/X) = H_{q}(Č) satisfies H_{q}(U/X)_{ét} = 0. By definition, R_{tr}(Y) ⊗^{tr} Č is the augmented Čech complex

… → R_{tr}(U ×_{X} U × Y) → R_{tr}(U × Y) → R_{tr}(X × Y) → 0

for the étale cover U × Y → X × Y, so R_{tr}(Y) ⊗_{ét}^{tr} Č is again exact by ^{\underline{ztruu}} 6.12. Thus 𝕃_{n} Φ(Č) = 0 for all n. In particular, the 0^{th} homology presheaf H_{0}(U/X) satisfies

Φ H_{0}(U/X) = R_{tr}(Y) ⊗_{ét}^{tr} H_{0}(U/X) = H_{0} (R_{tr}(Y) ⊗_{ét}^{tr} Č) = 0.

The following lemma shows that in fact every derived functor L_{n} Φ vanishes on H_{0}(U/X).

8.4 Lemma 8.15. Fix Y and set Φ = R_{tr}(Y)⊗_{ét}^{tr}. If F is a presheaf of R-modules with transfer such that F_{ét} = 0, then L_{n} Φ(F) = 0 for all n.

Proof. Suppose that F_{ét} = 0. Each map R_{tr}(X) → F is defined by an x ∈ F (X), and there is an étale cover U_{x} → X such that x vanishes in F (U_{x}). Thus the composition R_{tr}(U_{x}) → R_{tr}(X) → F is zero, i.e., the given map factors through the cokernel H_{0}(U_{x}/X) of R_{tr}(U_{x}) → R_{tr}(X). It follows that the canonical surjection ⊕_{X,x}R_{tr}(X) → F factors through a surjection ⊕_{X,x}H_{0}(U_{x}/X) → F. If K denotes the kernel of this surjection then K_{ét} = 0.

We now proceed by induction on n, noting that L_{n} Φ = 0 for n < 0. For n = 0, we know that Φ H_{0}(U_{x}/X) = 0 by example ^{\underline{8.x}} 8.14. Since Φ is right exact, this yields Φ(F) = 0. For n > 0, we may assume that the lemma holds for L_{p} Φ when p < n. From the exact sequence

⊕_{X,x}(L_{n} Φ)H_{0}(U_{x}/X) → L_{n} Φ(F) → L_{n−1} Φ(K)

we see that it suffices to prove that (L_{n} Φ)H_{0}(U/X) = 0. We saw in ^{\underline{8.x}} 8.14 that H_{q}(U/X)_{ét} = 0, so L_{p} Φ H_{q}(U/X) = 0 by the inductive assumption. Hence the hypercohomology sequence for the complex Č collapses to yield

𝕃_{n} Φ(Č) ≅ (L_{n} Φ)H_{0}(Č) = (L_{n} Φ)H_{0}(U/X).

But we saw in example ^{\underline{8.x}} 8.14 that L_{n} Φ(Č) = 0, whence the result.

Now we prove that ⊗_{L,ét}^{tr} preserves quasi-isomorphisms.

etqiso Proposition 8.16. Let f : C → C′ be a morphism of bounded above complexes of presheaves of R-modules with transfer. If f induces a quasi-isomorphism C_{ét} → C′_{ét} between the associated complexes of étale sheaves, then C ⊗_{L,ét}^{tr} D → C′ ⊗_{L,ét}^{tr} D is a quasi-isomorphism for every D.

Proof. If P ^{≃} C is a projective resolution of presheaves, then P_{ét} → C_{ét} is a quasi-isomorphism of complexes of étale sheaves. Thus we may assume that C, C′ and D are complexes of representable presheaves. If A denotes the mapping cone of C → C′, it suffices to show that A ⊗_{L,ét}^{tr} D = A ⊗_{ét}^{tr} D is acyclic. As each row of the double complex underlying A ⊗_{ét}^{tr} D is a sum of terms A ⊗_{ét}^{tr} R_{tr}(Y), it suffices to show that A ⊗_{ét}^{tr} R_{tr}(Y) is acyclic. As in the proof of ^{\underline{8.4}} 8.15, its homology sheaves are the hyper-derived functors 𝕃_{n} Φ(A), Φ = ⊗_{ét}^{tr}R_{tr}(Y). In the hypercohomology spectral sequence

E_{p,q}^{2} = L_{p} Φ(H_{q}A) ⇒ 𝕃_{p+q} Φ(A)

the presheaves H_{q}A have (H_{q}A)_{ét} = 0 because A_{ét} is acyclic. By lemma ^{\underline{8.4}} 8.15 we have L_{q} Φ(H_{q}A) = 0 for all p and q. Hence the spectral sequence collapses to yield 𝕃_{n} Φ(A) = 0 for all n, i.e., 𝕃 Φ(A) ≃ R_{tr}(Y) ⊗_{ét}^{tr} A is acyclic.

D-TTC Corollary 8.17. The derived category of bounded above complexes of étale sheaves of R-modules with transfer is a tensor triangulated category.

Proof. Combine ^{\underline{etqiso}} 8.16 with ^{\underline{dcTTC}} 8.8 and ^{\underline{locTTC}} 8A.7.

82 Lemma 8.18. Let F be a locally constant étale sheaf of flat R-modules. Then the map E ⊗_{L,ét}^{tr} F → E ⊗_{ét}^{tr} F is a quasi-isomorphism for every étale sheaf with transfers E.

Proof. Suppose first that E = R_{tr}(Y). Choose a resolution C → F in the category of locally constant sheaves in which each C_{n} is a sum of representables R_{tr}(L_{n,α}) for finite Galois field extensions L_{n,α} of k. (This is equivalent to resolving the Galois module M corresponding to F by Galois modules R[G_{n,α}], and the existence of such a resolution of M is well known.) By proposition ^{\underline{etqiso}} 8.16, E ⊗_{ét}^{tr} C = E ⊗_{L,ét}^{tr} C is quasi-isomorphic to E ⊗_{L,ét}^{tr} F. By lemma ^{\underline{83}} 8.13,

E ⊗_{ét}^{tr} C = E ⊗_{ét} C ^{≃} E ⊗_{ét} F ^{≃} E ⊗_{ét}^{tr} F.

Hence the result is true for E = R_{tr}(Y).

In the general case, choose a projective resolution P → E in the category of presheaves of R-modules with transfer. Then we have quasi-isomorphisms

E ⊗_{L,ét}^{tr} F = P ⊗_{L,ét}^{tr} F ^{≃} P ⊗_{ét}^{tr} F ^{≃} P ⊗_{ét} F.

Because sheafification is exact, P → E is also a resolution in the category of étale sheaves of R-modules. Since F is flat in this category, we have the final quasi-isomorphism:

P ⊗_{ét} F ^{≃} E ⊗_{ét} F ^{≃} E ⊗_{ét}^{tr} F.

It is clear that ^{\underline{82}} 8.18 also holds if E is a bounded above complex of étale sheaves with transfers.

Tateinv Corollary 8.19. In the derived category of étale sheaves of ℤ/m-modules with transfer, the operation M ↦ M(1) = M ⊗_{L,ét}^{tr} ℤ/m(1) is invertible.

Proof. Indeed, if μ_{m}^{∗} is the Pontrjagin dual of μ_{m}, then combining ^{\underline{82}} 8.18, ^{\underline{83}} 8.13, and ^{\underline{ztret}} 4.8 yields:

μ_{m}^{∗} ⊗_{L,ét}^{tr} ℤ/m(1) ≃^{\underline{82} 8.18} μ_{m}^{∗} ⊗_{ét}^{tr} ℤ/m(1) ≅^{\underline{83} 8.13} μ_{m}^{∗} ⊗_{ét} ℤ/m(1) ≅^{\underline{ztret} 4.8} μ_{m}^{∗} ⊗_{ét} μ_{m} ≅ ℤ/m.

lctens Exercise 8.20. If E and F are bounded above complexes of locally constant étale sheaves of R-modules, show that E ⊗_{L,ét}^{tr} F is quasi-isomorphic to E ⊗_{R}^{𝕃} F, their total tensor product as complexes of étale sheaves of R-modules. (Hint: Use ^{\underline{83}} 8.13, ^{\underline{etqiso}} 8.16, and ^{\underline{82}} 8.18.)

Appendix 8A - Tensor Triangulated Categories

The notion of a tensor triangulated category is a generalization of the tensor product structure on the derived category of modules over a scheme, which played a central role in the development of the subject.

TTC Definition 8A.1. A tensor triangulated category is an additive category with two structures: that of a triangulated category and that of a symmetric monoidal category. In addition, we are given natural isomorphisms r and l of the form

C[1] ⊗ D ≅ ; l_{C,D} (C ⊗ D)[1] ≅ ; r_{C,D} C ⊗ D[1],

which commute in the obvious sense with the associativity, commutativity and unity isomorphisms. There are two additional axioms:

(TTC1) For any distinguished triangle C_{0}  C_{1}  C_{2} ^{∂} C_{0}[1] and any

D, the following triangles are distinguished:

[figure: C_{0} ⊗ D; C_{1} ⊗ D; C_{2} ⊗ D; l(∂ ⊗ D); (C_{0} ⊗ D)[1]; D ⊗ C_{0}; D ⊗ C_{1}; D ⊗ C_{2}; r(D ⊗ ∂); (D ⊗ C_{0})[1].]

(TTC2) For any C and D, the following diagram commutes up to multiplication by −1, i.e., rl = −lr:

C[1] ⊗ D[1] \overline{\begin{matrix}r \\  \end{matrix}}(C[1] ⊗ D)[1]

−1

[figure: (C ⊗ D[1])[1]; l; r; (C ⊗ D)[2].; l]

week8app.tex

This description is not minimal. For example the commutativity isomorphism τ : C ⊗ D ≅ D ⊗ C allows us to recover r from l and vice versa using the formula τlτ = r. In addition, l_{C,D} can be recovered from l_{1,D} : 1[1] ⊗ D ≅ D[1]. Moreover, if either of the two triangles in (TTC1) is distinguished, then both are distinguished.

twist Exercise 8A.2. Show that the canonical isomorphisms l^{i}r^{j}, r^{j}l^{i} : C[i]⊗D[j] ≅ (C ⊗ D)[i + j] differ by (−1)^{ij}, and are interchanged by the twist isomorphism τ on C ⊗ D and C[i] ⊗ D[j].

asmc Definition 8A.3. Let A be an additive category with a symmetric monoidal structure ⊗. We say that A is an additive symmetric monoidal category if (∐ A_{i}) ⊗ B ≅ ∐(A_{i} ⊗ B) for every finite direct sum ∐ A_{i} in A.

If C and D are bounded above complexes in A, the tensor product C ⊗ D has (C ⊗ D)^{n} = ⊕_{p+q=n}C^{p} ⊗ D^{q} and differential d ⊗ 1 + (−1)^{p} ⊗ d on C^{p} ⊗ D^{q}. It is associative.

We define the twist isomorphism τ : C ⊗ D → D ⊗ C componentwise, as (−1)^{pq} times the natural isomorphism C^{p} ⊗ D^{q} → D^{q} ⊗ C^{p} in A. It is a straightforward exercise to verify that the category Ch^{−}(A) is an additive symmetric monoidal category.

The degree n part of each of C ⊗ D[1], (C ⊗ D)[1], and C[1] ⊗ D are the same, and we define l_{C,D} to be the canonical isomorphism. The map r_{C,D} is multiplication by (−1)^{p} on the summand C^{p} ⊗D^{q}. A routine calculation verifies the following.

KTTC Proposition 8A.4. Let A be an additive symmetric monoidal category. Then the chain homotopy category K^{−}(A) of bounded above cochain complexes is a tensor triangulated category.

Example 8A.5. (See ^{\underline{verdier}} [Ver77].) Let A be the category of modules over a commutative ring, or more generally over a scheme. Then not only is K^{−}(A) a tensor triangulated category, but the total tensor product ⊗^{𝕃} makes the derived category D^{−}(A) into a tensor triangulated category. In effect, D^{−}(A) is equivalent to the tensor triangulated subcategory of flat complexes in K^{−}(A).

Example 8A.6. The smash product of based topological spaces leads to another example. If A → X → X/A → SA is a cofibration sequence, there is a natural homeomorphism (X/A) ∧ Y ≅ (X ∧ Y)/(A ∧ Y); see ^{\underline{whitehead}} [Whi78, III.2.3]. The suspension SX = S^{1} ∧ X has homeomorphisms

X ∧ (SY) ≅ ; r S(X ∧ Y) ≅ ; l (SX) ∧ Y

satisfying (TTC1) and (TTC2) up to homotopy. It follows easily that the stable homotopy category, which is triangulated by ^{\underline{w}}[Wei94, 10.9.18] and a symmetric monoidal category by ^{\underline{adams}} [Ada74, III.4], is a tensor triangulated category.

If W is a saturated multiplicative system of morphisms in a triangulated category D, closed under ⊕, translations, and cones, Verdier proved in ^{\underline{verdier}} [Ver77] that the localization D[W^{−1}] is also a triangulated category. locTTC Proposition 8A.7. Let D be a tensor triangulated category. Suppose that if

C → C′ is in W then C ⊗ D → C′ ⊗ D is in W for every D in D. Then the localization D[W^{−1}] is also a tensor triangulated category.

Proof. Because each ⊗D : D → D preserves W, ⊗ induces a symmetric monoidal pairing D[W^{−1}] × D[W^{−1}] → D[W^{−1}] by the universal property of localization (applied to W × W). Similarly, the natural isomorphisms r and l descend to D[W^{−1}]. Axiom (TTC2) is automatic, and axiom (TTC1) may be routinely verified for Verdier’s description of distinguished triangles in D[W^{−1}].

cyclic Exercise 8A.8. Let T be an invertible object in a symmetric monoidal category C, i.e., an object such that T ⊗ U ≅ 1 for some U. It is well known that endomorphisms of 1 commute; show that the same must be true for endomorphisms of T. Then show that the cyclic permutation of T ⊗ (T ⊗ T) must equal the identity morphism.

Let T be an object in a symmetric monoidal category (C, ⊗, 1). Let C[T^{−1}] denote the category whose objects are pairs (X, m) with X in C and m ∈ ℤ; morphisms (X, m) → (Y, n) in C[T^{−1}] are just elements of the direct limit lim_{i→∞} Hom(X ⊗ T^{⊗m+i}, Y ⊗ T^{⊗n+i}), where the bonding maps are given by the functor ⊗T : C → C. Composition is defined in the obvious way, and it’s easy to check that C[T^{−1}] is a category. There is a universal functor C → C[T^{−1}] sending X to (X, 0). Note that (X, m) ≅ X ⊗ T^{⊗m} in C[T^{−1}] for m ≥ 0.

tinvtrcat Exercise 8A.9. Let T be an object in a tensor triangulated category C. Show that C[T^{−1}] is a triangulated category, and that C → C[T^{−1}] is triangulated.

In order for the formula (X, m) ⊗ (Y, n) = (X ⊗ Y, m + n) to extend to a bifunctor on C[T^{−1}], we need to define the tensor f ⊗g of two C[T^{−1}]-morphisms in a natural way. In general, C need not be symmetric monoidal, as exercise \underline{cyclic} 8A.8 above shows.

tinvprop Proposition 8A.10. Let T be an object in a symmetric monoidal category (C, ⊗, 1) such that the cyclic permutation on T^{⊗3} is the identity in C[T^{−1}]. Then (C[T^{−1}], ⊗, 1) is also a symmetric monoidal category.

Proof. The hypothesis implies that permutations on T^{⊗n} commute with each other for n ≥ 3. The many ways to define f ⊗ g on X ⊗ T^{m+i} ⊗ Y ⊗ T^{n+j} are indexed by the (i, j)-shuffles, and differ only by a permutation, so f ⊗ g is independent of this choice. Therefore the tensor product is a bifunctor on C[T^{−1}]. The symmetric monoidal axioms may now be routinely verified as in \underline{adams} [Ada74, III.4]. The hexagonal axiom, that the two isomorphisms from X ⊗(Y ⊗ Z) to (Z ⊗ X) ⊗ Y) agree, follows because the cyclic permutation on T^{⊗3} is the identity.

TateTTC Corollary 8A.11. Let T be an object in a tensor triangulated category C such that the cyclic permutation on T^{⊗3} is the identity in C[T^{−1}]. Then C[T^{−1}] is a tensor triangulated category.

Proof. By ^{\underline{tinvtrcat}} 8A.9 and ^{\underline{tinvprop}} 8A.10, C[T^{−1}] is both triangulated and symmetric monoidal. The verification of the remaining axioms is straightforward.

invTTC Exercise 8A.12. Let T be an object in a tensor triangulated category D such that Hom(X, Y) → Hom(X ⊗ T, Y ⊗ T) is an isomorphism for every X and Y in D. Show that D[T^{−1}] is a tensor triangulated category.

𝔸^{1}-weak equivalence

leca1we

In this section we define the notion of 𝔸^{1}-weak equivalence between bounded above cochain complexes of étale sheaves with transfers, and 𝔸^{1}-local complexes. The category DM_{ét}^{eff,−} is obtained by inverting 𝔸^{1}-weak equivalences. The main result in this lecture (^{\underline{locconseff}} 9.32) is that when we restrict to sheaves of ℤ/n-modules the category DM_{ét}^{eff,−} is equivalent to the derived category of discrete Galois modules for the group Gal(k_{sep}/k). We will use these ideas in the next lecture to identify étale motivic cohomology with ordinary étale cohomology.

Since quasi-isomorphic complexes will be 𝔸^{1}-weak equivalent, it is appropriate to define the notion in the derived category D^{−} = D^{−}(Sh_{ét}(Cor_{k}, R)) of étale sheaves of R-modules with transfer. In D^{−}, we have the usual shift, and

A ^{f} B  cone(f)  A[1]

is a distinguished triangle for each map f. We refer the reader to ^{\underline{gm}} [GM88] or \underline{w}[Wei94] for basic facts about derived categories. We will also need the notion of a thick subcategory, which was introduced by Verdier in ^{\underline{verdier}} [Ver77]. We will use Rickard’s definition (see ^{\underline{rickard}} [Ric89]); this is slightly different from, but equivalent to, Verdier’s definition.

9.1 Definition 9.1. A full additive subcategory E of D^{−} is thick if:

1. Let A → B → C → A[1] be a distinguished triangle. Then if two out of A, B, C are in E then so is the third.

2. if A ⊕ B is in E then both A and B are in E.

Week 9 - December 8, 1999

If E is a thick subcategory of D^{−}, we can form a quotient triangulated category D^{−}/E as follows (see ^{\underline{verdier}} [Ver77]). Let W_{E} be the set of maps whose cone is in E; W_{E} is a saturated multiplicative system of morphisms. Then D^{−}/E is the localization D^{−}[W_{E}^{−1}], which may be constructed using calculus of fractions; see \underline{w}[Wei94, 10.3.7]. In particular, a morphism f : C → C′ becomes an isomorphism in D^{−}[W_{E}^{−1}] if and only if f is in W_{E}. defa1we Definition 9.2. A morphism f in D^{−} is called an 𝔸^{1}-weak equivalence if f

is in W_{𝔸} = W_{E_{𝔸}}, where E_{𝔸} is the smallest thick subcategory so that:

1. R_{tr}(X × 𝔸^{1}) → R_{tr}(X) are in E_{𝔸} for every smooth scheme X;

2. E_{𝔸} is closed under any direct sum that exists in D^{−}.

We set DM_{ét}^{eff,−}(k, R) = D^{−}[W_{𝔸}^{−1}].

It is clear that the notion of 𝔸^{1}-weak equivalence in D^{−} = D^{−}(Sh(Cor_{k}, R)) makes sense for other topologies. For the Nisnevich topology, the localization DM_{Nis}^{eff,−}(k, R) of D^{−} is the triangulated category of motivic complexes introduced and studied in ^{\underline{tri}} [TrCat].

l12et Lemma 9.3. The smallest class in D^{−} which contains all the R_{tr}(X) and is closed under quasi-isomorphisms, direct sums, shifts, and cones is all of D^{−}.

Proof. First we show that for any complex D_{∗}, if all D_{n} are in the class, then so is D_{∗}. If β_{n}D is the brutal truncation 0 → D_{n} → D_{n−1} → ⋯ of D_{∗}, then D_{∗} is the union of the β_{n}D. Each β_{n}D is a finite complex, belonging to the class, as an inductive argument shows. Since there is an exact sequence

0  ⊕ β_{n}D  ⊕ β_{n}D  D_{∗}  0,

it follows that D_{∗} is in the class.

Thus it suffices to show that each sheaf F is in the class. Now there is a resolution L_{∗} → F by sums of the representable sheaves R_{tr}(X), given by lemma ^{\underline{projectives}} 8.1. Since each L_{n} is in this class, so is L_{∗} and hence F.

86 Lemma 9.4. If f : C → C′ is an 𝔸^{1}-weak equivalence, then for every D the map f ⊗ Id : C ⊗_{L,ét}^{tr} D → C′ ⊗_{L,ét}^{tr} D is an 𝔸^{1}-weak equivalence.

Proof. Since ⊗_{L,ét}^{tr} commutes with cones and f is an 𝔸^{1}-weak equivalence if and only if its cone is in E_{𝔸}, it suffices to show that if C is in E_{𝔸}, then C ⊗_{L,ét}^{tr} D is in E_{𝔸} for any D.

If D = R_{tr}(X), consider the subcategory E of all C in D^{−} such that C⊗_{L,ét}^{tr}D is in E_{𝔸}. E is closed under direct sums and it is thick. Moreover, if Y is a smooth scheme, then E contains the cone of R_{tr}(Y × 𝔸^{1}) → R_{tr}(Y). Therefore E_{𝔸} ⊆ E.

Now fix C in E_{𝔸} and consider the full subcategory D of all D in D^{−} such that C ⊗_{L,ét}^{tr} D is in E_{𝔸}. D is closed under direct sums, it is thick and we have seen that it contains R_{tr}(X) for all X. By ^{\underline{l12et}} 9.3, we conclude that D = D^{−}.

DMetTTC Corollary 9.5. The product ⊗_{L,ét}^{tr} endows DM_{ét}^{eff,−}(k, R) with the structure of a tensor triangulated category.

Proof. Given ^{\underline{D-TTC}} 8.17, this follows from ^{\underline{86}} 9.4 and proposition ^{\underline{locTTC}} 8A.7.

DMet Remark 9.6. The category DM_{ét}^{−}(k, R) is obtained from DM_{ét}^{eff,−}(k, R) by inverting the Tate twist operation M ↦ M(1) = M ⊗_{L,ét}^{tr} R(1). It is also a tensor triangulated category by ^{\underline{TateTTC}} 8A.11 and ^{\underline{symtri}} 15.8 below.

If R = ℤ/m, then the Tate twist is already invertible by ^{\underline{Tateinv}} 8.19, so we have

DM_{ét}^{−}(k, ℤ/m) = DM_{ét}^{eff,−}(k, ℤ/m).

a1h Definition 9.7. Two morphisms F [figure: f; g] G of sheaves of R-modules with trans-

fer are called 𝔸^{1}-homotopic if there is a map h : F ⊗^{tr} R_{tr}(𝔸^{1}) → G so that

the restrictions of h along R [figure: 1; 0] R_{tr}(𝔸^{1}) coincide with f and g.

If G is an étale sheaf, h factors through (and is determined by) a map F_{ét} ⊗_{L,ét}^{tr} R_{tr}(𝔸^{1}) → G.

a1hebis Example 9.8. Suppose we are given two maps f, g : X → Y such that the induced maps ℤ_{tr}(X) → ℤ_{tr}(Y) are 𝔸^{1}-homotopic in the sense of ^{\underline{a1h}} 9.7. By the Yoneda lemma, this is equivalent to saying that f and g are restrictions of some h ∈ Cor(X × 𝔸^{1}, Y), i.e., that f and g are 𝔸^{1}-homotopic maps in the sense of \underline{a1he} 2.23.

a1eq Lemma 9.9. Let f, g : F → G be two maps between étale sheaves with transfers. If f and g are 𝔸^{1}-homotopic, then f = g in DM_{ét}^{eff,−}(k, R).

Proof. Any two sections of 𝔸^{1} → Spec k yield the same map R → R_{tr}(𝔸^{1}) in the localized category DM_{ét}^{eff,−}(k, R), namely the inverse of the 𝔸^{1}-weak equivalence R_{tr}(𝔸^{1}) → R. Therefore the maps:

F [figure: F ×0; F ×1] R_{tr}(𝔸^{1}) ⊗_{L,ét}^{tr} F ^{h} G

are the same in the localized category.

There is a mistake in the proof of the corresponding Lemma 3.2.5 in ^{\underline{tri}} [TrCat] as the proof there assumes that ℤ_{tr}(𝔸^{1}) is flat in Cor_{k}. If we replace ⊗ by ⊗_{L} in loc. cit., the proof goes through as written.

a1hea1we Corollary 9.10. Every 𝔸^{1}-homotopy equivalence is an 𝔸^{1}-weak equivalence.

Our next goal is to show that, F → C_{∗}F is always an 𝔸^{1}-weak equivalence (see ^{\underline{812}} 9.14 below). Hence F ≅ C_{∗}F in DM_{ét}^{eff,−}(k, R).

9.6 Lemma 9.11. Let B → B′ be a map of double complexes which are vertically bounded above in the sense that there is a Q so that B^{∗,q} = (B′)^{∗,q} = 0 for all q ≥ Q. Suppose that all rows are weak equivalences and that Tot(B) and Tot(B′) are bounded above.

Then Tot(B) → Tot(B′) is an 𝔸^{1}-weak equivalence.

Proof. Let S(n) be the double subcomplex of B consisting of the B^{pq} for q ≥ n. Then Tot S(n + 1) is a subcomplex of Tot S(n) whose cokernel is a shift of the n-th row of B. If S′(n) is defined similarly, then each Tot S(n) → Tot S′(n) is an 𝔸^{1}-weak equivalence by induction on n. Now Sh_{ét}(Cor_{k}, R) satisfies (AB4), meaning that ⊕, and hence Tot, is exact. Hence there is a short exact sequence of complexes

0  ⨁_{n=1}^{∞} Tot S(n) ^{id−shift} ⨁_{n=1}^{∞} Tot S(n)  Tot B  0

and similarly for B′. Since ⊕ Tot S(n) → ⊕ Tot S′(n) is an 𝔸^{1}-weak equivalence, so is Tot B → Tot B′.

8.10 Corollary 9.12. If f : C → C′ is a morphism of bounded above complexes, and f_{n} : C_{n} → C′_{n} is in W_{𝔸} for every n, then f is in W_{𝔸}.

Proof. This is a special case of ^{\underline{9.6}} 9.11.

811 Lemma 9.13. For every F and every n, the map F ^{s} \underline{Hom}(R_{tr}(Δ^{n}), F) = C_{n}(F) is an 𝔸^{1}-homotopy equivalence. A fortiori, it is an 𝔸^{1}-weak equivalence.

Proof. Since Δ^{n} is isomorphic to 𝔸^{n} as a scheme, we have C_{n}(F) ≅ C_{1}C_{n−1}(F). Thus we may suppose that n = 1. We define a map m : C_{1}F → C_{2}F as follows. For each X, the map

m_{X} : C_{1}(F)(X) = F (X × 𝔸^{1}) → F (X × 𝔸^{2}) = C_{2}(F)

is induced by the multiplication map 𝔸^{2} → 𝔸^{1} by crossing it with X and applying F. Since C_{2}F = \underline{Hom}(R_{tr}(𝔸^{1}), C_{1}F), the adjunction of ^{\underline{73}} 8.2 associates to m a map h : C_{1}F ⊗^{tr} R_{tr}(𝔸^{1}) → C_{1}F. Similarly the inclusions 𝔸^{1} ×{i} ⊂ 𝔸^{2} induce maps η_{i} : C_{2}F → C_{1}F, and the compositions η_{i}m : C_{1}F → C_{1}F are adjoint to the restriction of h along i : R → R_{tr}(𝔸^{1}). Hence h induces an 𝔸^{1}-homotopy between the identity (η_{1}m) and the composite

C_{1}F ^{∂_{0}} F ^{s} C_{1}F,

corresponding to η_{0}m. Since ∂_{0}s is the identity on F, s and ∂_{0} are inverse 𝔸^{1}-homotopy equivalences. They are 𝔸^{1}-weak equivalences by ^{\underline{a1hea1we}} 9.10.

812 Lemma 9.14. For every bounded above complex F of sheaves of R-modules with transfer, the morphism F → C_{∗}(F) is an 𝔸^{1}-weak equivalence. Hence F ≅ C_{∗}(F) in DM_{ét}^{eff,−}(k, R).

Proof. By ^{\underline{9.6}} 9.11, we may assume that F is a sheaf. Consider the diagram whose rows are chain complexes

[figure: ⋯; 0; 0; F]

[figure: ⋯; 0; F; 1; F; 0; F; =]

[figure: ⋯; C_{2}F; ≃_{𝔸^{1}}; C_{1}F; ≃_{𝔸^{1}}; F.; ≃_{𝔸^{1}}]

The first two rows are quasi-isomorphic. Now F ≃_{𝔸^{1}}C_{n}(F) by ^{\underline{811}} 9.13. Using ^{\underline{8.10}} 9.12, we see that the second and third rows are 𝔸^{1}-weak equivalent.

a-s Example 9.15. The identity map on O is 𝔸^{1}-homotopic to zero by ^{\underline{cstaro}} 2.21 and \underline{812} 9.14. Hence O is isomorphic to zero in DM_{ét}^{eff,−}(k). When char k = ℓ > 0 the Artin-Schrier sequence of étale sheaves ^{\underline{milne}} [Mil80, II 2.18(c)]

0  ℤ/ℓ  O ^{1−ϕ} O  0

shows that ℤ/ℓ ≅ 0 in DM_{ét}^{eff,−}(k). Here R may be either ℤ or ℤ/ℓ.

dfna1local Definition 9.16. An object L in D^{−} is called 𝔸^{1}-local if for all 𝔸^{1}-weak equivalences K → K′ the induced map Hom(K′, L) → Hom(K, L) is bijective.

localcrit Lemma 9.17. An object L in D^{−} is 𝔸^{1}-local if and only if Hom(R_{tr}(X)[n], L) → Hom(R_{tr}(X × 𝔸^{1})[n], L) is an isomorphism for all X and n.

Proof. Let K be the full subcategory of all K for which Hom(K[n], L) = 0 for all n. Clearly, K is a thick subcategory of D^{−} and it is closed under direct sums and shifts. Under the given hypothesis, K contains the cone of every map R_{tr}(X × 𝔸^{1}) → R_{tr}(X). By definition, E_{𝔸} is a subcategory of K, i.e., L is 𝔸^{1}-local.

9.2 Lemma 9.18. If f : K → K′ is an 𝔸^{1}-weak equivalence and K, K′ are 𝔸^{1}-local then f is an isomorphism in D^{−}, i.e., a quasi-isomorphism of complexes of étale sheaves with transfers.

Proof. By definition, f induces bijections Hom(K′, K) ≅ Hom(K, K) and Hom(K′, K′) ≅ Hom(K, K′). Hence there is a unique g : K′ → K so that fg = 1_{K}, and f(gf) = (fg)f = f implies that gf = 1_{K′}.

Corollary 9.19. If F is 𝔸^{1}-local then F ≅ C_{∗}F in D^{−}.

homdmd- Lemma 9.20. If Y is 𝔸^{1}-local then for every X in D^{−}

Hom_{DM_{ét}^{eff,−}(k,R)}(X, Y) = Hom_{D^{−}}(X, Y).

Proof. By the calculus of fractions ^{\underline{w}}[Wei94, 10.3.7], the left side consists of equivalence classes of diagrams X ^{s} K Y with s in W_{𝔸}. It suffices to show that if K → K′ is an 𝔸^{1}-weak equivalence then Hom(K′, Y) = Hom(K, Y). But this holds since Y is 𝔸^{1}-local.

def9.16 Definition 9.21. An étale sheaf with transfers F is strictly 𝔸^{1}-homotopy invariant if the map H_{ét}^{n}(X, F) → H_{ét}^{n}(X × 𝔸^{1}, F) is bijective for all smooth X and every n ∈ ℕ. In particular for n = 0 we must have that F is homotopy invariant (^{\underline{hinv}} 2.13).

sgalemma Lemma 9.22. (^{\underline{sga4}} [SGA4, XV 2.2]) If R is of torsion prime to char k then any locally constant sheaf of R-modules is strictly 𝔸^{1}-homotopy invariant.

star Lemma 9.23. Let F be an étale sheaf of R-modules with transfers. Then F is 𝔸^{1}-local if and only if F is strictly 𝔸^{1}-homotopy invariant.

Proof. By ^{\underline{etext}} 6.23 or ^{\underline{etextr}} 6.24, we have

Hom_{D^{−}}(R_{tr}(X), F [i]) = Ext _{Sh_{ét}(Cor_{k},R)}^{i}(R_{tr}(X), F) = H_{ét}^{i}(X, F)

for every smooth X. Since R_{tr}(X × 𝔸^{1})[n] → R_{tr}(X)[n] is an 𝔸^{1}-weak equivalence for all n, ^{\underline{localcrit}} 9.17 shows that F is 𝔸^{1}-local if and only if the induced map

H_{ét}^{−n}(X, F) = Hom(R_{tr}(X)[n], F) → Hom(R_{tr}(X×𝔸^{1})[n], F) = H_{ét}^{−n}(X×𝔸^{1}, F)

is an isomorphism, that is, if and only if F is strictly 𝔸^{1}-homotopy invariant.

Here is a special case of ^{\underline{star}} 9.23 which includes the sheaves μ_{n}^{⊗q}. It follows by combining ^{\underline{sgalemma}} 9.22 with ^{\underline{star}} 9.23.

starcor Corollary 9.24. Let M be a locally constant étale sheaf of torsion prime to char k. Then M is 𝔸^{1}-local.

We now make the running assumption that R is a commutative ring and that cd_{R}(k) < ∞, i.e., k is a field having finite étale cohomological dimension for coefficients in R. This assumption allows us to invoke a classic result from \underline{sga4} [SGA4].

classicsga Lemma 9.25. (^{\underline{sga4}} [SGA4], ^{\underline{milne}} [Mil80]) Let X be a scheme of finite type over k. If k has finite R-cohomological dimension d then cd_{R}(X) ≤ d + 2 dim_{k} X.

extet Corollary 9.26. Ext^{n}(R_{tr}(X), F) = 0 when n ≫ 0.

Proof. Ext^{n}(R_{tr}(X), F) ≅ H_{ét}^{n}(X, F) by ^{\underline{etextr}} 6.24.

If C is a chain complex of sheaves, each cohomology H^{n}(C) is a presheaf. We write a_{ét}H^{n}(C) for its associated sheaf.

hyperss Lemma 9.27. For every (bounded above) chain complex C there is a bounded, convergent spectral sequence:

E_{2}^{p,q} = Ext^{p}(R_{tr}(X), a_{ét}H^{q}(C)) ⟹ Hom_{D^{−}}(R_{tr}(X), C[p + q]).

Proof. This is well-known; see ^{\underline{w}}[Wei94, 5.7.9]. The spectral sequence is bounded, and hence converges, by ^{\underline{extet}} 9.26.

starc Proposition 9.28. Let C be a bounded above cochain complex of étale sheaves of R-modules with transfer, where cd_{R}(k) < ∞. If the sheaves a_{ét}H^{n}(C) are all strictly 𝔸^{1}-homotopy invariant, then C is 𝔸^{1}-local.

Proof. Let C be a complex of étale sheaves with transfers. By ^{\underline{localcrit}} 9.17, it suffices to prove that cone(f) is in this class when f is the projection R_{tr}(X × 𝔸^{1}) → R_{tr}(X). The map f induces a morphism between the spectral sequences of ^{\underline{hyperss}} 9.27 for X and X × 𝔸^{1}. Because the sheaves L = a_{ét}H^{q}C are strictly 𝔸^{1}-homotopy invariant, they are 𝔸^{1}-local by ^{\underline{star}} 9.23. Thus

Ext^{p}(R_{tr}(X), L) = Hom_{D^{−}}(R_{tr}(X)[−p], L)

≅ Hom_{D^{−}}(R_{tr}(X × 𝔸^{1})[−p], L) = Ext^{p}(R_{tr}(X × 𝔸^{1}), L).

Hence the morphism of spectral sequences is an isomorphism on all E_{2} terms. By the Comparison Theorem ^{\underline{w}}[Wei94, 5.2.12], f induces an isomorphism from Hom_{D^{−}}(R_{tr}(X)[n], C) to Hom_{D^{−}}(R_{tr}(X × 𝔸^{1})[n], C) for each n. Done.

9.22 Lemma 9.29. If K is a bounded above complex of étale sheaves of ℤ/n-modules with transfer and 1/n ∈ k, then Tot C_{∗}(K) is 𝔸^{1}-local.

Proof. Set C = Tot C_{∗}(K). By ^{\underline{aletter}} 2.17, each H^{i}C is an 𝔸^{1}-homotopy invariant presheaf of ℤ/n-modules with transfers. By the Rigidity Theorem ^{\underline{loccons}} 7.20, the sheaf a_{ét}H^{i}C is locally constant. By ^{\underline{sgalemma}} 9.22, a_{ét}H^{i}C is strictly 𝔸^{1}-homotopy invariant. Finally, ^{\underline{starc}} 9.28 lets us conclude that C is 𝔸^{1}-local.

9.23 Corollary 9.30. If 1/n ∈ k then ℤ/n(q) is 𝔸^{1}-local for all q.

Proof. Take K to be (ℤ/n)_{tr}𝔾_{m}^{∧q}[−q]; ℤ/n(q) = C_{∗}K by definition ^{\underline{motcompl}} 3.1.

callet Definition 9.31. If 1/n ∈ k, let L denote the full subcategory of D^{−} consisting of 𝔸^{1}-local complexes of ℤ/n-modules with transfer. If E and F are 𝔸^{1}-local, we set E ⊗_{L} F = Tot C_{∗}(E ⊗_{L,ét}^{tr} F). By ^{\underline{9.22}} 9.29, E ⊗_{L} F is 𝔸^{1}-local, so ⊗_{L} is a bifunctor from L × L → L.

Recall from ^{\underline{galois}} 6.9 that the category of locally constant étale sheaves of ℤ/nmodules is equivalent to the category Mod(G, ℤ/n) of discrete ℤ/n-modules over the Galois group G = Gal(k_{sep}/k). Let D^{−}(G, ℤ/n) denote the (bounded above) derived category of such modules. There is a triangulated functor π^{∗} from D^{−}(G, ℤ/n) to D^{−} = D^{−}(Sh_{ét}(Cor_{k}, ℤ/n)).

locconseff Theorem 9.32. If 1/n ∈ k, (L, ⊗_{L}) is a tensor triangulated category and the functors

D^{−}(G, ℤ/n) ^{π^{∗}} L D^{−}[W_{𝔸}^{−1}] = DM_{ét}^{eff,−}(k, ℤ/n)

are equivalences of tensor triangulated categories.

Proof. Clearly, L is a thick subcategory of D^{−}. By ^{\underline{homdmd-}} 9.20, the functor L → D^{−}[W_{𝔸}^{−1}] is fully faithful. By ^{\underline{9.22}} 9.29, every object of D^{−}[W_{𝔸}^{−1}] is isomorphic to an object of L. Hence L is equivalent to D^{−}[W_{𝔸}^{−1}] as a triangulated category.

By ^{\underline{DMetTTC}} 9.5, DM_{ét}^{eff,−}(k, ℤ/n) is a tensor triangulated category. Using the first part of this proof, we conclude that L is a tensor triangulated category as well. Moreover, if E and F are 𝔸^{1}-local, then E ⊗_{L} F is isomorphic to E ⊗_{L,ét}^{tr} F in D^{−}[W_{𝔸}^{−1}] by ^{\underline{812}} 9.14, so the induced tensor operation on L is isomorphic to ⊗_{L}.

Next we consider π^{∗}. It is easy to see from ^{\underline{galois}} 6.9 and ^{\underline{81}} 6.11 that π^{∗} induces an equivalence between D^{−}(G, ℤ/n) and the full subcategory of complexes of locally constant sheaves in D^{−}. By exercise ^{\underline{lctens}} 8.20, π^{∗} sends ⊗_{ℤ/n}^{𝕃} to ⊗_{L,ét}^{tr}. It suffices to show that every 𝔸^{1}-local complex F is isomorphic to such a complex. By ^{\underline{812}} 9.14, ^{\underline{9.22}} 9.29, and ^{\underline{9.2}} 9.18 F → C_{∗}F is a quasi-isomorphism. By ^{\underline{aletter}} 2.17, each a_{ét}H^{i}F is 𝔸^{1}-homotopy invariant. By ^{\underline{loccons}} 7.20 the sheaves a_{ét}H^{i}F are locally constant. Hence the canonical map F → π^{∗}π_{∗}F is a quasi-isomorphism of complexes of étale sheaves. But π_{∗}F is a complex of modules in Mod(G, ℤ/n).

Étale motivic cohomology and algebraic singular homology

algsinghom

There are two ways one might define an étale version of motivic cohomology. One way, which is natural from the viewpoint of these notes, is to use the morphisms in the triangulated category DM_{ét}^{−}, namely to define the integral cohomology group indexed by (p, q) as Hom_{DM}_{ét}^{−}(ℤ_{tr}(X), ℤ(q)[p]), and similarly for cohomology with coefficients in an A. The second approach, due to Lichtenbaum, is to take the étale hypercohomology of the complex ℤ(q).

hpql Definition 10.1. For any abelian group A, we define the étale (or Lichtenbaum) motivic cohomology of X as the hypercohomology of A(q):

H_{L}^{p,q}(X, A) = ℍ_{ét}^{p}(X, A(q)|_{X_{ét}}).

If q < 0 then H_{L}^{p,q}(X, A) = 0, because A(q) = 0. If q = 0 then H_{L}^{p,0}(X, A) ≅ H_{ét}^{p}(X, A), because A(0) = A.

The two definitions agree in some cases of interest. We will see in ^{\underline{etl}} 10.7 below that H_{L}^{p,q}(X, ℤ/n) ≅ Hom_{DM}_{ét}^{−}(ℤ_{tr}(X), ℤ/n(q)[p]) when 1/n ∈ k. Even further on, in ^{\underline{ratetmot}} 14.21, we will see that H_{L}^{p,q}(X, ℚ) ≅ Hom_{DM}_{ét}^{−}(ℤ_{tr}(X), ℚ(q)[p]). However, the two definitions do not agree for ℓ-torsion coefficients, for ℓ = char(k). Indeed, for q = 0 we have Hom_{DM}_{ét}^{−}(ℤ_{tr}(X), ℤ/ℓ[p]) = 0 in characteristic ℓ by

\underline{a-s} 9.15, yet the groups H_{L}^{p,0}(X, ℤ/ℓ) ≅ H_{ét}^{p}(X, ℤ/ℓ) can certainly be nonzero.

By proposition ^{\underline{muqiso}} 6.4 we have H_{L}^{p,1}(X, ℤ/n) ≅ H_{ét}^{p}(X, μ_{n}) when 1/n ∈ k. Here is the generalization to all q.

lichet Theorem 10.2. Let n be an integer prime to the characteristic of k then:

H_{L}^{p,q}(X, ℤ/n) = H_{ét}^{p}(X, μ_{n}^{⊗q}) q ≥ 0, p ∈ ℤ

By ^{\underline{muqiso}} 6.4 there is a quasi-isomorphism μ_{n} → ℤ/n(1) of complexes of étale sheaves. Because μ_{n} and the terms of ℤ/n(1) are flat as sheaves of ℤ/n-modules, there is a morphism μ_{n}^{⊗q} → (ℤ/n)(1)^{⊗q} in the category of complexes of étale sheaves of ℤ/n-modules. Combining with the multiplication of ^{\underline{42}} 3.10 gives a map

μ_{n}^{⊗q}  (ℤ/n)(1)^{⊗q}  (ℤ/n)(q).

We may now reformulate theorem ^{\underline{lichet}} 10.2 as follows.

tensqiso Theorem 10.3. The map μ_{n}^{⊗q} → ℤ/n(q) is a quasi-isomorphism of complexes of étale sheaves.

Proof. The theorem is true for q = 1 by ^{\underline{muqiso}} 6.4. By ^{\underline{starcor}} 9.24 and ^{\underline{9.22}} 9.29, both μ_{n}^{⊗q} and ℤ/n(q) are 𝔸^{1}-local. We will show that the map μ_{n}^{⊗q} → ℤ/n(q) is an 𝔸^{1}-weak equivalence in ^{\underline{munzn}} 10.6 below. By ^{\underline{9.2}} 9.18, it is also a quasi-isomorphism.

Let R be any commutative ring. Recall that R(n) = R ⊗_{ℤ} ℤ(n). Clearly, the multiplication map ℤ(m) ⊗_{ℤ} ℤ(n) → ℤ(m + n) of ^{\underline{42}} 3.10 induces a map R(m) ⊗_{R} R(n) → R(m + n).

diag7.6 Proposition 10.4. The multiplication map R(m) ⊗ R(n) → R(m + n) factors through a map μ : R(m) ⊗^{tr} R(n) → R(m + n).

R(m) ⊗_{R} R(n) \overline{\begin{matrix}mult. \\  \end{matrix}}R(m + n)

[figure: μ]

R(m) ⊗^{tr} R(n) {^{\underline{tpmap}} 8.9 \atop }

Proof. We first reinterpret the left vertical map in simplicial language. Recall that by definition ^{\underline{motcompl}} 3.1, R(n)[n] = C_{∗}(R_{tr}(𝔾_{m}^{∧n})). Let us write A_{•}^{n} for the underlying simplicial presheaf, viz., A_{•}^{n}(U) = ℤ_{tr}(𝔾_{m}^{∧n})(U × Δ^{•}), and write the associated unnormalized chain complex as A_{∗}^{n}. By ^{\underline{tpmap}} 8.9, we have a natural map of bisimplicial presheaves A_{•}^{m} ⊗_{R} A_{•}^{n} → A_{•}^{m} ⊗^{tr} A_{•}^{n}, and a map of their diagonal chain complexes, (A^{m}⊗_{R}A^{n})_{∗} → (A^{m}⊗^{tr}A^{n})_{∗}. As in ^{\underline{42}} 3.10, the Eilenberg-Zilber theorem yields quasi-isomorphisms ∇ fitting into a commutative diagram:

[figure: R(m) ⊗_{R} R(n)[m + n]; =; A_{∗}^{m} ⊗_{R} A_{∗}^{n}; ∇; (A^{m} ⊗_{R} A^{n})_{∗}]

[figure: R(m) ⊗^{tr} R(n)[m + n]; ^{\underline{tpmap}} 8.9; =; A_{∗}^{m} ⊗^{tr} A_{∗}^{n}; ^{\underline{tpmap}} 8.9; ∇; (A^{m} ⊗^{tr} A^{n})_{∗}.; ^{\underline{tpmap}} 8.9]

Comparing with ^{\underline{42}} 3.10, we see that if suffices to find a simplicial map for all X and Y,

diag(C_{•}R_{tr}(X) ⊗^{tr} C_{•}R_{tr}(Y)) ⟶ C_{•}R_{tr}(X × Y)  (10.4.1) eq:1041

compatible with the corresponding construction ^{\underline{tecnical}} 3.9 for ⊗_{R}. The map μ will be the composite of ∇ and the map induced by ^{\underline{eq:1041}} 10.4.1.

Let F be any presheaf with transfers. Definitions ^{\underline{73}} 8.2 and ^{\underline{cstar}} 2.12 imply that C_{n}(F) ≅ \underline{Hom}(R_{tr}(Δ^{n}), F) as presheaves and that C_{•}(F) ≅ \underline{Hom}(R_{tr}(Δ_{k}^{•}), F) as simplicial presheaves. Using these identifications, we define the map ^{\underline{eq:1041}} 10.4.1 in degree n as the composition:

C_{n}(R_{tr}(X)) ⊗^{tr} C_{n}(R_{tr}(Y)) =

\underline{Hom}(R_{tr}(Δ^{n}), R_{tr}(X)) ⊗^{tr} \underline{Hom}(R_{tr}(Δ^{n}), R_{tr}(Y)) ^{\underline{ex74} 8.5}

\underline{Hom}(R_{tr}(Δ^{n} × Δ^{n}), R_{tr}(X × Y)) ^{diagonal} \underline{Hom}(R_{tr}(Δ^{n}), R_{tr}(X × Y)) = C_{n}(R_{tr}(X × Y)).

Since \underline{Hom}(R_{tr}(Δ^{n} × Δ^{n}), R_{tr}(X × Y))(U) = R_{tr}(X × Y)(U × Δ^{n} × Δ^{n}), the above composition is the right vertical composition in the following commutative diagram (see ^{\underline{ex74}} 8.5):

R_{tr}(X)(U × Δ^{n}) ⊗_{R} R_{tr}(Y)(U × Δ^{n}) \overline{{^{\underline{tpmap}} 8.9 \atop }} (C_{n}R_{tr}(X) ⊗^{tr} C_{n}R_{tr}(Y))(U)

[figure: R_{tr}(X × Y)(U × U × Δ^{n} × Δ^{n}); ⊗; diag(U); R_{tr}(X × Y)(U × Δ^{n} × Δ^{n}); ^{\underline{ex74}} 8.5]

[figure: diag(U × Δ^{n})]

R_{tr}(X × Y)(U × Δ^{n}). \begin{matrix}diag(Δ^{n}) \\  \end{matrix}

Since the left composite is the degree n part of construction ^{\underline{tecnical}} 3.9, this shows that the triangle in ^{\underline{diag7.6}} 10.4 commutes.

tensa1we Proposition 10.5. The map ℤ/n(1)^{⊗_{L}^{tr}q} → ℤ/n(q) is an 𝔸^{1}-weak equivalence in D^{−}(Sh_{ét}(Cor_{k}, ℤ/n)).

Proof. The assertion follows from the diagram in figure ^{\underline{diagtensa1we}} 10.1, remembering that by definition ℤ/n(q) is C_{∗}(ℤ/n)_{tr}(𝔾_{m}^{∧n})[−q].

munzn Proposition 10.6. The map μ_{n}^{⊗q} → ℤ/n(q) is an 𝔸^{1}-weak equivalence in D^{−}(Sh_{ét}(Cor_{k}, ℤ/n)).

Proof. Consider the following diagram, in which ⊗^{tr} and ⊗_{L}^{tr} are to be under-

ℤ/n(1)^{⊗_{L}^{tr}q}  \overline{}ℤ/n(q)

[figure: (ℤ/n)_{tr}(𝔾_{m})[−1]^{⊗_{L}^{tr}q}; ≃_{𝔸^{1}} ^{\underline{86}} 9.4+^{\underline{812}} 9.14; (ℤ/n)_{tr}(𝔾_{m}^{∧q})[−q]; ^{\underline{812}} 9.14 ≃_{𝔸^{1}}]

                      [figure: ^{\underline{lem7.5}} 8.7; ≃; ^{\underline{wedgetens}} 8.10; =]

((ℤ/n)_{tr}(𝔾_{m}))^{⊗^{tr}q} [−q]

Figure 10.1: The factorization in proposition ^{\underline{tensa1we}} 10.5

                                                                            diagtensa1we

stood in ℤ/n-modules.

μ_{n}^{⊗_{L}^{tr}q}  \overline{≃} ℤ/n(1)^{⊗_{L}^{tr}q}

[figure: μ_{1} ∘ μ]

[figure: μ_{n}^{⊗q}; ≃; μ_{n}^{⊗^{tr}q}; ≃; ℤ/n(1)^{⊗^{tr}q}; μ_{1}; μ; ℤ/n(q)]

We already know that the top map is a quasi-isomorphism by ^{\underline{muqiso}} 6.4 and ^{\underline{etqiso}} 8.16. Lemma ^{\underline{83}} 8.13 proves that the bottom left map μ_{n}^{⊗q} → μ_{n}^{⊗^{tr}q} is a quasiisomorphism. Lemma ^{\underline{82}} 8.18 proves that the left vertical map is a quasiisomorphism. Hence the assertion follows from proposition ^{\underline{tensa1we}} 10.5.

Recall that when 1/n ∈ k we have DM_{ét}^{−} = DM_{ét}^{eff,−}(k, ℤ/n).

etl Proposition 10.7. If 1/n ∈ k then H_{L}^{p,q}(X, ℤ/n) ≅ Hom_{DM}_{ét}^{−}(ℤ_{tr}(X), ℤ/n(q)[p]).

Proof. Since A = ℤ/n(q) is 𝔸^{1}-local by ^{\underline{9.23}} 9.30, the right side is

Hom_{DM}_{ét}^{−}(ℤ_{tr}(X), ℤ/n(q)[p]) = Hom_{D^{−}}(ℤ_{tr}(X), ℤ/n(q)[p])
= Ext^{p}(ℤ_{tr}(X), ℤ/n(q)).

By ^{\underline{hexthcoho}} 6.25, this Ext group is H_{ét}^{p}(X, ℤ/n(q)), which is the left side.

As a bonus for all our hard work, we are able to give a nice interpretation of Suslin’s algebraic singular homology. Recall that R_{tr}(X) = ℤ_{tr}(X) ⊗ R.

Definition 10.8. We define the algebraic singular homology of X by:

H_{i}^{sing}(X, R) = H_{i}(C_{∗}(R_{tr}(X))(Spec k).

By remark ^{\underline{hsing}} 7.4, H_{0}^{sing}(X, ℤ) agrees with the group H_{0}^{sing}(X/ Spec k) of lecture ^{\underline{relpicandrig}} 7. As an exercise the reader should check that:

H^{p,q}(Spec k, R) = H_{q−p}^{sing}(𝔾_{m}^{∧q}, R)

Notice that R_{tr}(𝔾_{m}^{∧q}) is well-defined even though 𝔾_{m}^{∧q} is not a scheme.

The following theorem was first proven in ^{\underline{svin}} [SV96, 7.8] under the assumption of resolution of singularities on k. The proof we give here doesn’t need resolution of singularities, so it extends the result to fields of positive characteristic.

dual Theorem 10.9. Let k be a separably closed field and X a smooth scheme over k, and let l be a prime number different from char k. Then there exist natural isomorphisms for all i:

H_{i}^{sing}(X, ℤ/l)^{∗} ≅ H_{ét}^{i}(X, ℤ/l)

where the ^{∗} denotes the dual vector space over ℤ/l.

It is amusing to note that this implies that H_{ét}^{i}(X, ℤ/l) is finite, because it is a countable-dimensional dual module.

To prove ^{\underline{dual}} 10.9, we need one more lemma. To clarify the role of the coefficient ring R, we will write D_{R}^{−} for D^{−}(Sh_{ét}(Cor_{k}, R)), so that D_{ℤ}^{−} is just the usual derived category of Sh_{ét}(Cor_{k}).

9.7 Lemma 10.10. Let k be a separably closed field and C a bounded above chain complex of étale sheaves of R-modules with transfer. Assume that the cohomology sheaves of C are locally constant and projective (as R-modules).Then for any n ∈ ℤ we have:

Hom_{D_{R}^{−}}(C, R[n]) = Hom_{R−mod}(H^{n}(C)(Spec k), R)

Proof. For simplicity, let us write Ext^{∗} for Ext in the category Sh_{ét}(Cor_{k}, R). (There are enough injectives to define Ext by ^{\underline{etinj}} 6.19.)

If P is a summand of ⊕_{α}R, then Ext^{n}(P, R) injects into

Ext^{n}(⊕_{α}R, R) = ∏ Ext^{n}(R, R) = ∏ Ext^{n}(R_{tr}(Spec k), R).

But Ext^{n}(R_{tr}(Spec k), R) = H_{ét}^{n}(Spec k, R) by ^{\underline{etextr}} 6.24 and this vanishes if n ≠ 0 as k is separably closed. If n = 0, this calculation yields Ext^{0}(R, R) = R and Ext^{0}(P, R) = Hom_{R−mod}(P, R).

Now recall that Ext^{n}(F, R) = Hom_{D_{R}^{−}}(F, R[n]) for every sheaf F; see ^{\underline{w}}[Wei94, 10.7.5]. More generally, if R → I^{∗} is an injective resolution then the total Hom cochain complex ℝ Hom(C, R) of Hom^{∗}(C, I[n]) satisfies

H^{n}ℝ Hom(C, R) ≅ Hom_{D_{R}^{−}}(C, R[n]).

(See ^{\underline{w}}[Wei94, 10.7.4].) Since Hom^{∗}(C, I[n]) is a bounded double complex, it gives rise to a convergent spectral sequence which, as in ^{\underline{w}}[Wei94, 5.7.9], may be written

E_{2}^{pq} = Ext^{p}(H^{q}C, R) ⟹ H^{p+q}ℝ Hom(C, R) = Hom_{D_{R}^{−}}(C, R[p + q]).

The assumption on H^{q}C makes the spectral sequence collapse to yield Ext^{0}(H^{n}C, R) ≅ Hom_{D_{R}^{−}}(C, R[n]), whence the result.

Proof of ^{\underline{dual}} 10.9. Taking R = ℤ/l, this means that all R-modules are projective. Consider the diagram:

Hom_{D_{R}^{−}}(C_{∗}(R_{tr}(X)), R[n]) \overline{≅ ; ^{\underline{9.7}} 10.10 ; }Hom_{R−mod}(H_{n}^{sing}(X, R), R)

[figure: Hom_{D_{R}^{−}}(R_{tr}(X), R[n]); ^{\underline{starcor}} 9.24 ; ≅; ≅ ; ^{\underline{etextr}} 6.24; H_{ét}^{n}(X, R).]

By ^{\underline{aletter}} 2.17, each H^{n} = H^{n}C_{∗}R_{tr}(X) is a homotopy invariant presheaf of ℤ/lmodules with transfer. Hence the sheaves a_{ét}H^{n} are locally constant by the Rigidity Theorem ^{\underline{loccons}} 7.20. Hence the top map is an isomorphism by ^{\underline{9.7}} 10.10. Since R is 𝔸^{1}-local by ^{\underline{starcor}} 9.24, the left map is an isomorphism by ^{\underline{812}} 9.14. The bottom map is an isomorphism by ^{\underline{etextr}} 6.24.

Corollary 10.11. Let k be a separably closed field and X a smooth scheme over k, and let n be an integer relatively prime to char k. Then there exist natural isomorphisms for all i:

H_{i}^{sing}(X, ℤ/n)^{∗} ≅ H_{ét}^{i}(X, ℤ/n)

where the ^{∗} denotes the Pontrjagin dual ℤ/n-module.

Proof. Using the sequences 0 → ℤ/l → ℤ/lm → ℤ/m → 0, the 5-lemma shows that we may assume that n is prime.

Standard triples

For all of this lecture, F will be a homotopy invariant presheaf with transfers.

Our goal in this lecture is to prove the following result, which is one of the main properties of homotopy invariant presheaves with transfers. It (or rather its corollary ^{\underline{smsemi}} 11.2) will be used in subsequent lectures to promote results from the Nisnevich topology to the Zariski topology. It depends primarily upon the relative Picard group introduced in lecture ^{\underline{relpicandrig}} 7.

Recall that a subgroup A of an abelian group B is called pure if nA = nB∩A for every integer n. A homomorphism f : A → B of abelian groups is called pure injective if it is injective and f(A) is a pure subgroup of B.

prp1 Proposition 11.1. For any smooth semilocal S over k, any Zariski dense open subset V ⊂ S, and any homotopy invariant presheaf with transfers F, the map F (S) → F (V) is pure injective.

Proposition ^{\underline{prp1}} 11.1 is a consequence of a more precise result, proposition ^{\underline{prp1'}} 11.3, whose proof will take up most of this lecture.

Proof. The semilocal scheme S is the intersection of a family X_{α} of smooth varieties of finite type over k and V is the intersection of dense open subschemes V_{α} ⊂ X_{α}. Hence F (S) → F (V) is the filtered colimit of the maps F (X_{α}) → F (V_{α}). Since all these maps are injections by ^{\underline{prp1'}} 11.3, their colimit is an injection. If a ∈ F (X_{α}) equals nb ∈ F (V_{α}) for some b ∈ F (V_{α}), then the image of a in F (U_{α}), and hence in F (S), is n-divisible.

Passing to the direct limit over all such V, we see that F (S) injects (as a pure subgroup) into the direct sum of the F (Spec E_{i}), as E_{i} runs over the generic points of any semilocal S. In particular, we have:

smsemi Corollary 11.2. Let F be a homotopy invariant presheaf with transfers. If F (Spec E) = 0 for every field E over k, then F_{Zar} = 0.

Week 18 - March 22, 2000

prp1' Theorem 11.3. Let X be smooth of finite type over a field k and let V be a dense open subset. Then for every finite set of points x_{1}, …, x_{n} ∈ X there exists an open neighborhood U of these points such that F (X) → F (U) factors through F (X) → F (V). That is, there is a map F (V) → F (U) such that the following diagram commutes.

F (X)

[figure]

[figure: F (V); ∃; F (U)]

Example 11.4. If V ⫋ X is a dense open subset, then F = ℤ_{tr}(X)/ℤ_{tr}(V) is a presheaf with transfers, but F (X) → F (V) is not injective. (1_{X} is nonzero in F (X) but vanishes in F (V).) This shows that homotopy invariance is necessary in ^{\underline{prp1'}} 11.3.

To prepare for the proof of proposition ^{\underline{prp1'}} 11.3, we need a technical digression.

dfnstd3 Definition 11.5. A standard triple is a triple (X̄ ^{p̄} S, X_{∞}, Z) where p̄ is a proper morphism of relative dimension 1 and Z and X_{∞} are closed subschemes of X̄. The following conditions must be satisfied:

1. S is smooth and X̄ is normal,

2. X̄ − X_{∞} is quasi-affine and smooth over S,

3. Z ∩ X_{∞} = ∅,

4. X_{∞} ∪ Z lies in an affine open neighborhood in X̄.

Given a standard triple as above, we usually write X for X̄ − X_{∞}. Note that X̄ is a good compactification of both X and X − Z (see ^{\underline{defgc}} 7.8) by parts 2 and 4.

Conversely, if X̄ is a good compactification of a smooth quasi-affine curve X → S (see ^{\underline{defgc}} 7.8), then (X̄, X̄ − X, ∅) is a standard triple.

We will see in ^{\underline{teoremuzzo}} 11.17 below that any pair of smooth quasi-projective varieties Z ⊂ X is locally part of a standard triple, at least when k is infinite.

Remark 11.6. (Gabber) Parts 4 and 2 imply that S is affine, and that Z and X_{∞} are finite over S. Indeed, X_{∞} is finite and surjective over S by part 2, and affine by part 4, so Chevalley’s theorem (^{\underline{hart}} [Har77, III Ex.4.2]) implies that S is affine.

We will make use of the following observation. Recall from ^{\underline{relpicdef}} 7.10 that Pic(X̄, X_{∞}) is the group of isomorphism classes of pairs (L, s) where L is a line bundle on X̄ and s is a trivialization on X_{∞}.

Given a standard triple (X̄, X_{∞}, Z), any section x : S → X of p defines an element [x] of Pic(X̄, X_{∞}). Indeed, there is a homomorphism C_{0}(X/S) → Pic(X̄, X_{∞}). prp2 Remark 11.7. Let F be a homotopy invariant presheaf with transfers. Given

a standard triple (X̄, X_{∞}, Z), by ^{\underline{pairing}} 7.5 there is a pairing:

( , ) : Pic(X̄, X_{∞}) ⊗ F (X) → F (S).

Let x : S → X be a section of p. If [x] is the class of x in Pic(X̄, X_{∞}), then ([x], f) = F (x)(f) for all f ∈ F (X).

Lemma 11.8. Let (X̄, X_{∞}, Z) be a standard triple over S and X = X̄ − X_{∞}. Then there is a commutative diagram for every homotopy invariant presheaf with transfers F.

Pic(X̄, X_{∞} ∐ Z) ⊗ F (X) \overline{}Pic(X̄, X_{∞} ∐ Z) ⊗ F (X − Z)

[figure: Pic(X̄, X_{∞}) ⊗ F (X); F (S)]

Proof. Be definition, X̄ is a good compactification of both X and X − Z. Thus the pairings exist by ^{\underline{pairing}} 7.5 (or ^{\underline{relpic}} 7.16) and are induced by the transfers pairing Cor_{k}(S, X) ⊗ F (X) → F (S). Commutativity of the diagram is a restatement of the fact that any presheaf with transfers is a functor on Cor_{k}.

16.8 Corollary 11.9. If x : S → X is a section and [x] ∈ Pic(X̄, X_{∞}) lifts to λ ∈ Pic(X̄, X_{∞} ∐ Z), there is a commutative diagram:

F (X) \overline{}F (X − Z)

[figure: λ]

F (S). \begin{aligned}[x] \\ &  \end{aligned}

Moreover, if λ′ ∈ C_{0}(X − Z/S) ⊂ Cor(S, X − Z) is any representative of λ (see ^{\underline{relpic}} 7.16 and ^{\underline{corscork}} 1A.10), the composition of λ′ with the inclusion X − Z ⊂ X is 𝔸^{1}-homotopic to x in Cor(S, X).

manylifts Exercise 11.10. Use example ^{\underline{picn}} 7.14 with F = O^{∗} to show that there can be

more than one lift λ : F (X − Z) → F (S).

More generally, observe that any unit s of O(Z) gives a trivialization of O(X̄) on Z; combining this with the trivialization 1 on X_{∞} gives an element σ(s) = (O, 1 ∐ s) of Pic(X̄, X_{∞} ∐ Z). Show that λ + σ(s) is also a lift of [x] to Pic(X̄, X_{∞} ∐ Z), and that every other lift has this form for some s ∈ O^{∗}(Z).

split3 Definition 11.11. A standard triple is split over an open subset U ⊂ X if L_{Δ}|_{U×_{S}Z} is trivial, where L_{Δ} is the line bundle on U ×_{S} X̄ corresponding to the graph of the diagonal map.

P1split Example 11.12. For any affine S, the standard triple (S × ℙ^{1}, S × ∞, S × 0) is split over any U in X = S × 𝔸^{1}. Indeed, the line bundle L_{Δ} is trivial on all of X × X.

xsplit Exercise 11.13. Let X̄ be a smooth projective curve over k, with affine open X = Spec(A) and set X_{∞} = X̄ − X. Then (X̄, X_{∞}, Z) is a standard triple for every finite Z in X. Let P_{1}, ... be the prime ideals of A defining the points of Z, and suppose for simplicity that A/P_{i} ≅ k for all i. Show that the standard triple splits over D(f) if and only if each P_{i} becomes a principal ideal in the ring A[1/f].

In particular, if X̄ = ℙ^{1}, the triple splits over all X because in this case A is a principal ideal domain.

lemmuzzo Lemma 11.14. Any finite set of points in X has an open neighborhood U such that the triple is split over U.

Proof. The map f : X ×_{S} Z → X is finite, as Z is finite over S. Given points x_{i} ∈ X, each f^{−1}(x_{i}) is finite. Now the line bundle L_{Δ} is trivial in some neighborhood V of ∪_{i}f^{−1}(x_{i}), because every line bundle on a semilocal scheme is trivial. But every such V contains an open of the form U ×_{S} Z, and the triple is split over such a U.

prp1'' Proposition 11.15. Consider a standard triple split over an affine U. Then there is an 𝔸^{1}-equivalence class of finite correspondences λ : U → (X − Z) such that the composite of λ with (X − Z) ⊂ X is 𝔸^{1}-homotopic to the inclusion U ⊂ X.

In particular, F (X) → F (U) factors through λ : F (X − Z) → F (U):

F (X) \overline{}F (X − Z)

[figure: ∃λ]

F (U).

Proof. Pulling back yields a standard triple (U ×_{S} X̄, U ×_{S} X_{∞}, U ×_{S} Z) over the affine U. The diagonal Δ : U → U ×_{S} X is a section and its class in Pic(U ×_{S} X̄, U ×_{S} X_{∞}) is represented by the line bundle L_{Δ}. If the triple is split over an affine U, then L_{Δ} has a trivialization on U ×_{S} Z as well, so [Δ] lifts to a class λ in Pic(U ×_{S} X̄, U ×_{S} (X_{∞} ∐ Z)). By ^{\underline{h0singa1he}} 7.2 and ^{\underline{relpic}} 7.16, λ is an 𝔸^{1}-equivalence class of maps in Cor(U, X −Z). By ^{\underline{16.8}} 11.9 we have a commutative diagram

[figure: qU; λ; U ×_{S} (X − Z); [pr]; X − Z]
[figure: [Δ]]
[figure: U ×_{S} X; [pr]; X]

and it suffices to observe that pr ∘ Δ : U → U ×_{S} X → X is the inclusion.

A different splitting (trivialization on U ×_{S} Z) may yield a different lifting λ′. By exercise ^{\underline{manylifts}} 11.10, λ′ = λ + σ(s) for some unit s of O(U ×_{S} Z).

C0rep Exercise 11.16. Suppose that λ is represented by an element D of Cor(U, X − Z) = C_{0}(U ×(X −Z)/U), as in exercise ^{\underline{C0lift}} 7.15. Show that the element D−[Δ(U)] of Cor(U, X) is represented by a principal divisor (f) on U × X̄, with f equal to 1 on U × X_{∞}.

teoremuzzo Theorem 11.17. Let W be a connected quasi-projective smooth scheme over an infinite field k, Y a proper closed subset of W and y_{1}, …, y_{n} ∈ Y. Then there is an affine open neighborhood X of these points in W and a standard triple (X̄ → S, X_{∞}, Z) such that (X, X ∩ Y) ≅ (X̄ − X_{∞}, Z).

Proof. (Mark Walker) We may assume that W is affine, a closed (d + 1)- dimensional subscheme of 𝔸^{n}. Embed 𝔸^{n} in 𝔸^{N} by

(x_{1}, …, x_{n}) ↦ (x_{1}, …, x_{n}, x_{1}^{2}, x_{1}x_{2}, …, x_{i}x_{j}, …, x_{n}^{2}).

Given a closed point x ∈ W, Bertini’s Theorem (see ^{\underline{sga4}} [SGA4, XI.2.1]) implies that the general linear projection p : 𝔸^{N} → 𝔸^{d} is smooth near each point of W lying on p^{−1}(p(x)). It is also finite when restricted to Y, because Y has dimension ≤ d.

Let W̄ denote the closure of W in ℙ^{N}, H = ℙ^{N} − 𝔸^{N}, and W_{∞} = W̄ ∩ H. The general projection defines a rational map p : W̄ ......... ℙ^{d} whose center C is finite, because C lies in the intersection of W_{∞} with a codimension d linear subspace of H. Let X̄_{1} be the closure of the graph of p : (W̄ − C) → ℙ^{d} in W̄ × ℙ^{d}. Then W is naturally an open subscheme of X̄_{1} and X̄_{1} − W has finite fibers over 𝔸^{d}.

The singular points Σ of the projection X̄_{1} → ℙ^{d} are closed, and finite over each p(y_{i}) because p is smooth near W ∩ p^{−1}(p(y_{i})). Therefore there is an affine open neighborhood S in 𝔸^{d} of {p(y_{i})} over which Σ is finite and disjoint from Y. Define X to be p^{−1}(S) ∩ W − Σ; by construction p : X → S is smooth. Define X̄ ⊂ X̄_{1} to be the inverse image of S, and X_{∞} = X̄ − X. Then X ∩ Y → S and X_{∞} → S are both finite.

It remains to show that X_{∞} ∐(X ∩ Y) lies in an affine open neighborhood of X̄. As X̄ is projective over S, there is a global section of some very ample line bundle L whose divisor D misses all of the finitely many points of X_{∞} and X ∩ Y over any p(y_{i}). Because L is very ample and S is affine, X̄ − D is affine. Replacing S by a smaller affine neighborhood of the p(y_{i}), we can assume that D misses X_{∞} and X ∩ Y, i.e., that X_{∞} and X ∩ Y lie in X̄ − D, as desired.

tmuzzofinite Porism 11.18. If k is finite, the proof shows that there is a finite extension k′ and an affine open X′ of the points in W ×_{k} Spec k′ so that (X′, X′ ∩ Y′) comes from a standard triple over k′, where Y′ = Y ×_{k} Spec k′. In fact, for each prime l we can assume that [k′ : k] is a power of l.

Finally, we will use ^{\underline{prp1''}} 11.15, ^{\underline{lemmuzzo}} 11.14 and ^{\underline{teoremuzzo}} 11.17 to prove ^{\underline{prp1'}} 11.3.

Proof of ^{\underline{prp1'}} 11.3. We first assume that k is infinite. Since we may replace V by V − {x_{1}, …, x_{n}}, we may assume that the closed points x_{1}, …, x_{n} of X lie in Z = X − V. We can use ^{\underline{teoremuzzo}} 11.17 to shrink X about these points to assume that there exists a standard triple with X = X̄ − X_{∞}. By ^{\underline{lemmuzzo}} 11.14 the triple splits over an open neighborhood U of the points. As X is quasi-projective, we may shrink U to make it affine. By ^{\underline{prp1''}} 11.15 we get the map F (X − Z) → F (U) factoring F (X) → F (U).

If k is finite, we proceed as follows. We see by porism ^{\underline{tmuzzofinite}} 11.18 that there is an open X′ of X ×_{k} Spec(k′) fitting into a standard triple over k′. The argument above shows that there is an open neighborhood U of x_{1}, …, x_{n} (depending on k′) such that if U′ = U×_{k}Spec(k′) and V′ = V ×_{k}Spec(k′), then F (X′) → F (U′) factors through a map Φ′ : F (V′) → F (U′). Let Φ(k′) : F (V) → F (U) be the composite of Φ′ and the transfer F (U′) → F (U). By ^{\underline{galcor}} 1.11, [k′ : k] times F (X) → F (U) factors through Φ(k′). By ^{\underline{tmuzzofinite}} 11.18, we can choose two such extensions k′,k′′ with [k′ : k] and [k′′ : k] relatively prime. Shrinking U, we may assume that F (U) is the target of both Φ(k′) and Φ(k′′). But then F (X) → F (U) factors through a linear combination of Φ(k′) and Φ(k′′).

Nisnevich sheaves

We have already mentioned the Nisnevich topology several times in previous lectures, as an alternative to the étale and Zariski topologies. In this lecture we develop some of its more elementary properties.

We begin by recalling the definition of the Nisnevich topology (see ^{\underline{nistop}} [Nis89]). A family of étale morphisms {p_{i} : U_{i} → X} is said to be a Nisnevich covering of X if it has the Nisnevich lifting property:

• for all x ∈ X, there is an i and a u ∈ U_{i} so that p_{i}(u) = x and the induced map k(x) → k(u) is an isomorphism.

It is easy to check that this notion of cover satisfies the axioms for a Grothendieck topology (in the sense of ^{\underline{milne}} [Mil80, I.1.1], or pre-topology in the sense of ^{\underline{sga4}} [SGA4]). The Nisnevich topology is the class of all Nisnevich coverings.

Example 12.1. Here is an example to illustrate the arithmetic nature of a Nisnevich cover. When char k ≠ 2, the two morphisms U_{0} = 𝔸^{1} − {a} ^{⊂} ^{j} 𝔸^{1}

and U_{1} = 𝔸^{1} − {0} ^{z↦z^{2}} 𝔸^{1} form a Nisnevich covering of 𝔸^{1} if and only if a ∈ (k^{∗})^{2}. They form an étale covering of 𝔸^{1} for any nonzero a ∈ k.

nisfields Example 12.2. Let k be a field. The small Nisnevich site on Spec k consists of the étale U over Spec k, together with their Nisnevich coverings. Every étale U over Spec k is a finite disjoint union ∐ Spec l_{i} with the l_{i} finite and separable over k; to be a Nisnevich cover, one of the l_{i} must equal k. Thus a Nisnevich sheaf F on Spec k merely consists of a family of sets F (l), natural in the finite separable extension fields l of k. In fact, each such l determines a “point” of (Spec k)_{Nis} in the sense of ^{\underline{sga4}} [SGA4, IV 6.1].

From this description it follows that Spec k has Nisnevich cohomological dimension zero. This implies that the Nisnevich cohomological dimension of any Noetherian scheme X is at most dim X; see ^{\underline{ks}} [KS86].

sect Lemma 12.3. If {U_{i} → X} is a Nisnevich covering then there is a nonempty open V ⊂ X and an index i such that U_{i}|_{V} → V has a section.

Proof. For each generic point x of X, there is a generic point u ∈ U_{i} so that k(x) ≅ k(u). Hence U_{i} → X induces a rational isomorphism between the corresponding components of U_{i} and X, i.e., U_{i} → X has a section over an open subscheme V of X containing x.

Example 12.4. A Hensel local ring (R, 𝔪) is a local ring such that any finite R-algebra S is a product of local rings. It is well-known (see ^{\underline{milne}} [Mil80, I.4.2]) that if S is finite and étale over R, and if R/𝔪 ≅ S/𝔪_{i} for some maximal ideal 𝔪_{i} of S, then R → S splits; one of the factors of S is isomorphic to R. If {U_{i} → Spec R} is a Nisnevich covering then some U_{i} is finite étale, so U_{i} → Spec R splits. Thus every Nisnevich covering of Spec R has the trivial covering as a refinement. Consequently, the Hensel local schemes Spec R determine “points” for the Nisnevich topology.

As with any Grothendieck topology, the category Sh_{Nis}(Sm/k) of Nisnevich sheaves of abelian groups is abelian, and sheafification F ↦ F_{Nis} is an exact functor. We know that exactness in Sh_{Nis}(Sm/k) may be tested at the Hensel local rings O_{X,x}^{h} of all smooth X at all points x (see ^{\underline{nistop}} [Nis89, 1.17]). That is, for every presheaf F:

• F_{Nis} = 0 if and only if F (Spec O_{X,x}^{h}) = 0 for all (X, x);

• F_{Nis}(Spec O_{X,x}^{h}) = F (Spec O_{X,x}^{h}).

By abuse of notation, we shall write F (O_{X,x}^{h}) for F (Spec O_{X,x}^{h}), and refer to it as the stalk of F_{Nis} at x.

we refer to ^{\underline{uds}} 12.5

                                                                    in ^{\underline{step2.1}} 12.7, ^{\underline{morstd3}} 20.1
                                                                    \underline{step2.1.5} 20.12 and ^{\underline{step2.2}} 21.2

uds Definition 12.5. A commutative square Q = Q(X, Y, A) of the form

[figure: B; i; Y; A; f; i; X; f]

is called upper distinguished if B = A ×_{X} Y, f is étale, i : A → X is an open embedding and (Y −B) → (X −A) is an isomorphism. Clearly, any upper distinguished square determines a Nisnevich covering of X: {Y → X, A → X}.

niscurve Exercise 12.6. If dim X ≤ 1 show that any Nisnevich cover of X admits a refinement {U, V } such that Q(X, U, V) is upper distinguished. Show that this fails if dim X ≥ 2.

By definition, F (Q) is a pullback square if and only if F (X) is the pullback F (Y) ×_{F (B)} F (A), i.e., the kernel of f − i : F (Y) × F (A) → F (B).

we use ^{\underline{step2.1}} 12.7 in the proof of

                                                                    \underline{step2.2} 21.2

step2.1 Lemma 12.7. A presheaf F is a Nisnevich sheaf if and only if F (Q) is a pullback square for every upper distinguished square Q.

Proof. For the “if” part, suppose that each F (Q) is a pullback square. To prove that F is a Nisnevich sheaf, fix a Nisnevich covering {U_{i} → X}. Let us say that an open subset V ⊂ X is good (for the covering) if

F (V) ∏ F (U_{i} ×_{X} V) [figure] ∏ F (U_{i} ×_{X} U_{j} ×_{X} V)

is an equalizer diagram. We need to show that X itself is good.

By Noetherian induction, we may assume that there is a largest good V ⊂ X. Suppose that V ≠ X and let Z = X − V. By lemma ^{\underline{sect}} 12.3, there is a nonempty open W ⊂ Z and an index i such that U_{i}|_{W} → W splits. Let X′ ⊂ X be the complement of the closed set Z − W. Then V and U′_{i} = U_{i}|_{X′} form an upper distinguished square Q over X′. Pulling back along each U′_{j} = U_{j}|_{X′} also yields an upper distinguished square. Thus we have pullback squares

[figure: F (X′); F (U′_{i}); F (U′_{j}); F (U′_{i} ×_{X} U′_{j}); F (V); F (U′_{i}|_{V}); F (U′_{j}|_{V}); F (U′_{i} ×_{X} U′_{j}|_{V}).]

A diagram chase shows that X′ is also good, contradicting the assumption that V ≠ X. Hence X is good for each cover, i.e., F is a Nisnevich sheaf.

For “only if”, we assume that F is a Nisnevich sheaf and Q is upper distinguished and need to prove that the map F (X) → F (Y) ×_{F (B)} F (A) is an isomorphism. We already know the map is monic because {A, Y } is a Nisnevich cover of X. For the surjectivity, note that the sheaf axiom for this covering yields the equalizer sequence

F (X) → F (Y) × F (A)\overrightarrow{→}F (B) × F (A) × F (Y ×_{X} Y).

Since {Δ(Y), B ×_{A} B} is a cover of Y ×_{X} Y, we have an injection F (Y ×_{X} Y) → F (Y) × F (B ×_{A} B). Now (a, y) ∈ F (A) × F (Y) lies in F (A) ×_{F (B)} F (Y) if the two restrictions to F (B) are the same. The two maps to F (A) and F (Y) are the same, so it suffices to consider the maps from F (Y) to F (B ×_{A} B). These both factor through F (B), so the images of y are the same as the images of a. But by construction the two maps F (A)\overrightarrow{→}F (B ×_{A} B) are the same.

Porism 12.8. Suppose more generally that F is a sheaf for some Grothendieck topology, and that Q = Q(X, Y, A) is a pullback square whose horizontal maps are monomorphisms. If {A, Y } is a cover of X and {B ×_{A} B, Y } is a cover of Y ×_{X} Y, the proof of ^{\underline{step2.1}} 12.7 shows that F (Q) is a pullback square.

starmodn Exercise 12.9. Write O^{∗}/O^{∗l} for the presheaf U ↦ O^{∗}(U)/O^{∗l}(U), and O^{∗}/l for the Zariski sheaf associated to O^{∗}/O^{∗l}. Show that there is an exact sequence

0 → O^{∗}(U)/O^{∗l}(U) → O^{∗}/l(U) → Pic(U) ^{l} Pic(U)

for all smooth U. Then show that O^{∗}/l is a Nisnevich sheaf on Sm/k. If 1/l ∈ k, this is an example of a Nisnevich sheaf which is not an étale sheaf. In fact, (O^{∗}/l)_{ét} = 0.

plainnisres Exercise 12.10. If F is a Nisnevich sheaf, consider the presheaf E^{0}(F) defined

by:

E^{0}(F)(X) = ∏_{\begin{matrix}closed \\ x∈X \end{matrix}} F (O_{X,x}^{h}).

Show that E^{0}(F) is a Nisnevich sheaf, and that the canonical map F → E^{0}(F) is an injection. Using ^{\underline{nisfields}} 12.2, show that E^{0}(F) is a flasque sheaf, i.e., that it has no higher cohomology (see ^{\underline{sga4}} [SGA4, V.4.1]). Iteration of this construction yields the canonical flasque resolution 0 → F → E^{0}(F) → ⋯ of a Nisnevich sheaf, which may be used to compute the cohomology groups H_{Nis}^{∗}(X, F).

znisdef Definition 12.11. Consider the presheaf sending U to ℤ[Hom_{Sm/k}(U, X)]. We write ℤ(X) for ℤ[Hom(−, X)]_{Nis}, its sheafification with respect to the Nisnevich topology. It is easily checked that ℤ[Hom(−, X)]_{Zar}(U) = ℤ[Hom(−, X)](U) for every connected open U. We do not know if this is true for ℤ(X).

By the Yoneda lemma, Hom(ℤ(X), G) = G(X) for every sheaf G. Since ℤ_{tr}(X) is a Nisnevich sheaf by ^{\underline{etsh}} 6.2, we see that ℤ(X) is a subsheaf of ℤ_{tr}(X).

Let D_{Nis}^{−} denote the derived category of cohomologically bounded above complexes in Sh_{Nis}(Sm/k). If F and G are Nisnevich sheaves, it is well known that Ext_{Nis}^{n}(F, G) = Hom_{D_{Nis}^{−}}(F, G[n]) (see ^{\underline{w}}[Wei94, 10.7.5]).

l20 Lemma 12.12. Let G be any Nisnevich sheaf. Then for all X:

Ext_{Nis}^{n}(ℤ(X), G) = H_{Nis}^{n}(X, G).

Proof. If G → I^{∗} is a resolution by injective Nisnevich sheaves, then the nth cohomology of G is H^{n} of I^{∗}(X). But by ^{\underline{w}}[Wei94, 10.7.4] we know that the left side is H^{n} of Hom_{Sh_{Nis}(Sm/k)}(ℤ(X), I^{∗}) = I^{∗}(X).

l12 Lemma 12.13. The smallest class in D_{Nis}^{−} which contains all the ℤ(X) and is closed under quasi-isomorphisms, direct sums, shifts, and cones is all of D_{Nis}^{−}.

Proof. The proof of ^{\underline{l12et}} 9.3 goes through using ℤ(X) in place of R_{tr}(X).

For the rest of this lecture, we shall write ⊗ for the presheaf tensor product, (F ⊗ G)(U) = F (U) ⊗_{ℤ} G(U), and ⊗_{Nis} for the tensor product of Nisnevich sheaves, i.e., the sheafification of ⊗. Note that if a sheaf F is flat as a presheaf then F is also flat as a sheaf. This is true for example of the sheaves ℤ(X).

l10 Lemma 12.14. ℤ(X × Y) = ℤ(X) ⊗_{Nis} ℤ(Y).

Proof. Since Hom(U, X × Y) = Hom(U, X) × Hom(U, Y), we see that ℤ[Hom(U, X ×Y)] = ℤ[Hom(U, X)]⊗ℤ[Hom(U, Y)]. Thus ℤ[Hom(−, X ×Y)] ≅ ℤ[Hom(−, X)] ⊗ ℤ[Hom(−, Y)] as presheaves. Now sheafify. l1 Lemma 12.15. Let G be a Nisnevich sheaf on Sm/k such that H_{Nis}^{n}(−, G) is

homotopy invariant for all n. Then for all n and all bounded above C:

Hom_{D_{Nis}^{−}}(C, G[n]) ≅ Hom_{D_{Nis}^{−}}(C ⊗_{Nis} ℤ(𝔸^{1}), G[n]).

Proof. By ^{\underline{l20}} 12.12, our assumption yields Ext^{n}(ℤ(X), G) ≅ Ext^{n}(ℤ(X × 𝔸^{1}), G) for all X. Since ℤ(X × 𝔸^{1}) = ℤ(X) ⊗_{Nis} ℤ(𝔸^{1}) by ^{\underline{l10}} 12.14, the conclusion holds for C = ℤ(X). If C and C′ are quasi-isomorphic, then so are C ⊗_{Nis} ℤ(𝔸^{1}) and C ⊗_{Nis} ℤ(𝔸^{1}), because ℤ(𝔸^{1}) is a flat sheaf. But the class of all complexes C for which the conclusion holds is closed under direct sums and cones, so by \underline{l12} 12.13 the conclusion holds for all C.

We borrow yet another topological definition: deformation retract. For each F, note that the presheaf F ⊗ ℤ[Hom(−, Spec k)] is just F.

Definition 12.16. An injection of presheaves i : F → G is called a (strong) deformation retract if there is a map r : G → F such that r ∘ i = id_{F} and a homotopy h : G ⊗ ℤ[Hom(−, 𝔸^{1})] → G so that the restriction h|_{F} is the projection F ⊗ ℤ[Hom(−, 𝔸^{1})] F, h(G ⊗ 0) = i ∘ r and h(G ⊗ 1) = id.

If F and G are sheaves, the condition in the definition is equivalent to the condition that there is a sheaf map h : G⊗_{Nis} ℤ(𝔸^{1}) → G so that the restriction h|_{F} is the projection F ⊗_{Nis} ℤ(𝔸^{1}) → F, h(G ⊗ 0) = i ∘ r and h(G ⊗ 1) = id.

For example, the zero-section Spec k ^{0} 𝔸^{1} induces a deformation retract ℤ → ℤ(𝔸^{1}); the homotopy map h is induced by the multiplication 𝔸^{1}×𝔸^{1} → 𝔸^{1} using ^{\underline{l10}} 12.14. If I^{1} is the quotient presheaf ℤ(𝔸^{1})/ℤ, so that ℤ(𝔸^{1}) ≅ ℤ ⊕ I^{1}, then 0 ⊂ I^{1} is also a deformation retract.

l21 Lemma 12.17. If F → G is a deformation retract, then the quotient presheaf G/F is a direct summand of G/F ⊗ I^{1}.

Proof. The inclusion 0 ⊂ G/F is a deformation retract, whose homotopy is induced from h. Therefore we may assume that F = 0.

Let K denote the kernel of h. Since the evaluation “t = 1” : G = G ⊗ ℤ → G ⊗ ℤ(𝔸^{1}) is a section of both h and the projection G ⊗ ℤ(𝔸^{1}) → G, we see that K is isomorphic to G ⊗ I^{1}. But “t = 0”: G → G ⊗ ℤ(𝔸^{1}) embeds G as a summand of K.

For every presheaf F we define C̃_{m}(F) to be the quotient presheaf C_{m}(F)/F. That is, C̃_{m}(F)(U) is F (U × 𝔸^{m})/F (U). Thus we have split exact sequences 0 → F → C_{m}(F) → C̃_{m}(F) → 0.

l2 Corollary 12.18. C̃_{m}(F) is a direct summand of C̃_{m}(F) ⊗ I^{1} for all m ≥ 0.

Proof. It is easy to see that F → C_{m}F is a deformation retract, so ^{\underline{l2}} 12.18 is a special case of ^{\underline{l21}} 12.17. prop2 Proposition 12.19. Let G be a Nisnevich sheaf on Sm/k such that H_{Nis}^{n}(−, G)

is homotopy invariant for all n. Then for all n and for all presheaves F, there is an isomorphism

Hom_{D_{Nis}^{−}}((C_{∗}F)_{Nis}, G[n]) ^{≅} Hom_{D_{Nis}^{−}}(F_{Nis}, G[n]).

Proof. Write Ext^{n}(C, G) for Hom_{D_{Nis}^{−}}(C, G[n]). For each complex C, lemma

\underline{l1} 12.15 implies that Ext^{q}(C ⊗_{Nis} I^{1}, G) = 0 for all q. For C = (C̃_{p}F)_{Nis}, ^{\underline{l2}} 12.18 yields Ext^{q}((C̃_{p}F)_{Nis}, G) = 0.

Note that Ext^{q}(C, G) = H^{q}ℝ Hom(C, G) for any C; see ^{\underline{w}}[Wei94, 10.7.4]. As in the proof of ^{\underline{9.7}} 10.10, a resolution G → I^{∗} yields a first quadrant Hom double complex Hom((C̃_{∗}F)_{Nis}, I^{∗}) and hence a first quadrant spectral sequence

E_{1}^{p,q} = Ext^{q}((C̃_{p}F)_{Nis}, G) ⇒ Ext^{p+q}((C̃_{∗}F)_{Nis}, G)

(see ^{\underline{w}}[Wei94, 5.6.1]). Since every E_{1}^{p,q} vanishes, this implies that Ext^{n}((C̃_{∗}F)_{Nis}, G) = 0 for all n. In turn, this implies the conclusion of ^{\underline{prop2}} 12.19, viz., Ext^{n}((C_{∗}F)_{Nis}, G) ≅ Ext^{n}(F, G) for all n.

Nisnevich sheaves with transfers

lecnishxfer

We now consider the category Sh_{Nis}(Cor_{k}) of Nisnevich sheaves with transfers. As with étale sheaves, we say that a presheaf with transfers F is a Nisnevich sheaf with transfers if its underlying presheaf is a Nisnevich sheaf on Sm/k. Clearly, every étale sheaf with transfers is a Nisnevich sheaf with transfers.

nisshxfer Theorem 13.1. Let F be a presheaf with transfers, and write F_{Nis} for the sheafification of the underlying presheaf. Then F_{Nis} has a unique structure of presheaf with transfers such that F → F_{Nis} is a morphism of presheaves with transfers.

Consequently, Sh_{Nis}(Cor_{k}) is an abelian category, and the forgetful functor Sh_{Nis}(Cor_{k}) ^{⊂} PST(k) has a left adjoint (F ↦ F_{Nis}) which is exact and commutes with the forgetful functor to (pre)sheaves on Sm/k.

Finally, Sh_{Nis}(Cor_{k}) has enough injectives.

Proof. The Nisnevich analogue of ^{\underline{etcov}} 6.16, is valid; just replace ‘étale cover’ by cf. ^{\underline{zarcov}} 13.14 ‘Nisnevich cover’ in the proof. As explained after ^{\underline{ztruu}} 6.12, the Čech complex ℤ_{tr}(Ǔ) is a Nisnevich resolution of ℤ_{tr}(X). With these two observations, the proofs of \underline{ettr} 6.17, ^{\underline{exact}} 6.18, and ^{\underline{etinj}} 6.19 go through for the Nisnevich topology.

Exercise 13.2. By theorem ^{\underline{zreg}} 4.1, ℤ(1) ≃ O^{∗}[−1] as complexes of Nisnevich sheaves with transfers. By ^{\underline{starmodn}} 12.9, O^{∗}/l = O^{∗} ⊗_{Nis} ℤ/l. Since ℤ/l(1) = ℤ(1) ⊗_{Nis}^{𝕃} ℤ/l, it follows that there is a distinguished triangle of Nisnevich sheaves with transfers for each l:

μ_{l} → ℤ/l(1) → O^{∗}/l[−1] → μ_{l}[1].

Since (O^{∗}/l)_{ét} = 0, this recovers ^{\underline{ztret}} 4.8: μ_{l} ≃ ℤ/l(1)_{ét}.

nisres Exercise 13.3. If F is a Nisnevich sheaf with transfers, modify example ^{\underline{6.18}} 6.20 to show that the sheaf E^{0}(F) defined in ^{\underline{plainnisres}} 12.10 is a Nisnevich sheaf with transfers, and that the canonical flasque resolution F → E^{∗}(F) is a complex of Nisnevich sheaves with transfers.

nistr Lemma 13.4. Let F be a Nisnevich sheaf with transfers. Then:

1. Its cohomology sheaves H_{Nis}^{n}(−, F) are presheaves with transfers;

2. For any smooth X, we have F (X) ≅ Hom_{Sh_{Nis}(Cor_{k})}(ℤ_{tr}(X), F);

3. For any smooth X and any n ∈ ℤ,

H_{Nis}^{n}(X, F) ≅ Ext_{Sh_{Nis}(Cor_{k})}^{n}(ℤ_{tr}(X), F).

Proof. (Cf. ^{\underline{homshpst}} 6.3, ^{\underline{6.20}} 6.21, and ^{\underline{etext}} 6.23.) Assertion 2 is immediate from ^{\underline{nisshxfer}} 13.1 and the Yoneda isomorphism F (X) ≅ Hom_{PST}(ℤ_{tr}(X), F). Now consider the canonical flasque resolution F → E^{∗}(F) in Sh_{Nis}(Sm/k). By ^{\underline{nisres}} 13.3, this is a resolution of sheaves with transfers. Since H_{Nis}^{n}(−, F) is the cohomology of E^{∗}(F) as a presheaf, and hence as a presheaf with transfers, we get part 1.

For part 3, it suffices by part 2 to show that if F is an injective sheaf with transfers and n > 0, then H_{Nis}^{n}(−, F) = 0. Since F → E^{0}(F) must split in Sh_{Nis}(Cor_{k}), H_{Nis}^{n}(X, F) is a summand of H_{Nis}^{n}(X, E^{0}(F)) = 0, and must vanish.

nisextcoho Exercise 13.5. (Cf. ^{\underline{hexthcoho}} 6.25.) Let K be any complex of Nisnevich sheaves of R-modules with transfer. Generalize ^{\underline{nistr}} 13.4 to show that its hyperext and hypercohomology agree in the sense that for any smooth X and n ∈ ℤ:

Ext^{n}(R_{tr}(X), K) ≅ ℍ_{Nis}^{n}(X, K).

The following result allows us to bootstrap quasi-isomorphism results from the field level to the sheaf level.

boot Proposition 13.6. Let A → B be a morphism of complexes of presheaves with transfers. Assume that their cohomology presheaves H^{∗}A and H^{∗}B are homotopy invariant, and that A(Spec E) → B(Spec E) is a quasi-isomorphism for every field E over k. Then A_{Zar} → B_{Zar} is a quasi-isomorphism in the Zariski topology.

Proof. Let C be the mapping cone. By the 5-lemma, each H^{i}C is a homotopy invariant presheaf with transfers, which vanishes on Spec E for every field E over k. Corollary ^{\underline{smsemi}} 11.2 states that (H^{i}C)_{Zar} = 0. This implies that C_{Zar} is acyclic as a complex of Zariski sheaves, i.e., that A_{Zar} and B_{Zar} are quasi-isomorphic in the Zariski topology.

The main result of this lecture, ^{\underline{theorem'}} 13.11, as well as the next few lectures, depend upon the following result, whose proof will not be completed until ^{\underline{restatement}} 23.1. Theorem ^{\underline{theorem''}} 13.7 allows us to bypass the notion of strictly 𝔸^{1}-homotopy invariance (see ^{\underline{def9.16}} 9.21) used in lecture ^{\underline{leca1we}} 9. The case n = 0 of ^{\underline{theorem''}} 13.7, that F_{Nis} is homotopy invariant, will be completed in ^{\underline{n0}} 21.3. theorem'' Theorem 13.7. Let k be a perfect field and F a homotopy invariant presheaf

with transfers. Then each presheaf H_{Nis}^{n}(−, F_{Nis}) is homotopy invariant.

The proofs of the following results are all based upon a combination of theorem ^{\underline{theorem''}} 13.7, lemma ^{\underline{nistr}} 13.4, and proposition ^{\underline{boot}} 13.6.

znis Proposition 13.8. Let k be a perfect field. If F is a homotopy invariant Nisnevich sheaf with transfers, then for all n and all smooth X:

H_{Zar}^{n}(X, F) ≅ H_{Nis}^{n}(X, F).

We will prove in ^{\underline{azartr}} 21.15 that F_{Zar} is a presheaf with transfer. This would simplify the proof of ^{\underline{znis}} 13.8.

Proof. For n = 0 we have H_{Nis}^{0}(X, F) = H_{Zar}^{0}(X, F) = F (X) for every sheaf. By the Leray spectral sequence, it now suffices to prove that H_{Nis}^{n}(S, F) = 0 for all n > 0 when S is a local scheme. By ^{\underline{nistr}} 13.4 and ^{\underline{theorem''}} 13.7, each H_{Nis}^{n}(−, F) is a homotopy invariant presheaf with transfers. By ^{\underline{smsemi}} 11.2, it suffices to show that H_{Nis}^{n}(Spec E, F) = 0 for every field E over k. But fields are Hensel local rings, and as such have no higher cohomology, i.e., H_{Nis}^{n}(Spec E, −) = 0 for n > 0.

a1henis Proposition 13.9. Let C be a be bounded above complex of Nisnevich sheaves with transfer, whose cohomology sheaves are homotopy invariant. Then its Zariski and Nisnevich hypercohomology agree:

ℍ_{Zar}^{n}(X, C) ≅ ℍ_{Nis}^{n}(X, C) for all smooth X and for all n.

Proof. We will proceed by descending induction on n − p, where C^{i} = 0 for i > p. If dim X = d, then ℍ_{Zar}^{n}(X, C) = ℍ_{Nis}^{n}(X, C) = 0 for all n > p + d, because cd_{Zar}(X) and cd_{Nis}(X) are at most d. By ^{\underline{nisshxfer}} 13.1, both the good Nisnevich truncation τC and the p^{th}-cohomology sheaf H^{p} = (C/τC)_{Nis} are Nisnevich sheaves with transfers. Setting m = n − p, we have a diagram

[figure: H_{Zar}^{m−1}(X, H^{p}); H_{Zar}^{n}(X, τC); H_{Zar}^{n}(X, C); H_{Zar}^{m}(X, H^{p}); H_{Zar}^{n+1}(X, τC); H_{Nis}^{m−1}(X, H^{p}); ≅; H_{Nis}^{n}(X, τC); ≅; H_{Nis}^{n}(X, C); H_{Nis}^{m}(X, H^{p}); ≅; H_{Nis}^{n+1}(X, τC).; ≅]

The four outer verticals are isomorphisms, by induction and ^{\underline{znis}} 13.8. The statement now follows from the 5-lemma.

mcpst Example 13.10. The motivic complex ℤ(i) is bounded above, and has homotopy invariant cohomology by ^{\underline{aletter}} 2.17. If A is any abelian group, the same is true for A(i) = A ⊗_{ℤ} ℤ(i). By ^{\underline{a1henis}} 13.9, the motivic cohomology of a smooth X could be computed using Nisnevich hypercohomology:

H^{n,i}(X, A) = ℍ_{Zar}^{n}(X, A(i)) = ℍ_{Nis}^{n}(X, A(i)).

This is the definition of motivic cohomology used in ^{\underline{orange}} [VSF00]. Note that the motivic cohomology groups H^{n,i}(X, A) are presheaves with transfers by ^{\underline{nisextcoho}} 13.5. theorem' Theorem 13.11. Let k be a perfect field and F a presheaf with transfers such

that F_{Nis} = 0. Then (C_{∗}F)_{Nis} ≃ 0 in the Nisnevich topology, and (C_{∗}F)_{Zar} ≃ 0 in the Zariski topology.

Proof. Let F be a presheaf with transfers such that F_{Nis} = 0. We will first prove that (C_{∗}F)_{Nis} ≃ 0 or, equivalently, that the homology presheaves H_{i} = H_{i}C_{∗}F satisfy (H_{i})_{Nis} = 0 for all i. For i < 0 this is trivial; C_{i}F = 0 implies that H_{i}C_{∗}F = 0. Since (H_{0})_{Nis} is a quotient of F_{Nis} = 0, it is also true for i = 0.

We shall proceed by induction on i, so we assume that (H_{j})_{Nis} = 0 for all j < i. That is, we assume that τ(C_{∗}F)_{Nis} ≃ (C_{∗}F)_{Nis}, where τ(C_{∗}F)_{Nis} denotes the subcomplex of (C_{∗}F)_{Nis} obtained by good truncation at level i:

τ(C_{∗}F)_{Nis} is ⋯ → (C_{i+1}F)_{Nis} → (C_{i}F)_{Nis} → d(C_{i}F)_{Nis} → 0.

There is a canonical morphism τ(C_{∗}F)_{Nis} → (H_{i})_{Nis}[i] and hence a morphism f : (C_{∗}F)_{Nis} → (H_{i})_{Nis}[i] in the derived category D_{Nis}^{−}. Since f induces an isomorphism on the ith homology sheaves, it suffices to prove that f = 0.

The presheaf with transfers H_{i} is homotopy invariant by ^{\underline{aletter}} 2.17, so by ^{\underline{nisshxfer}} 13.1 and \underline{theorem''} 13.7 the sheaf G = (H_{i})_{Nis} satisfies the hypothesis of ^{\underline{prop2}} 12.19. Since F_{Nis} = 0, \underline{prop2} 12.19 yields

Hom_{D_{Nis}^{−}}((C_{∗}F)_{Nis}, (H_{i})_{Nis}[i]) ≅ Hom_{D_{Nis}^{−}}(F_{Nis}, (H_{i})_{Nis}[i]) = 0.

Hence f = 0 in D_{Nis}^{−}, and this implies that (H_{i})_{Nis} = 0.

We can now prove that C_{∗}F_{Zar} ≃ 0. Each cohomology presheaf H^{i} = H^{i}C_{∗}F is a homotopy invariant presheaf with transfers by ^{\underline{aletter}} 2.17. Since (C_{∗}F)_{Nis} ≃ 0, we have C_{∗}(F)(Spec E) ≃ 0 for every finitely generated field extension E of k (and hence for every field over k). Indeed, E is O_{X,x}^{h} for the generic point of some smooth X. Now apply ^{\underline{boot}} 13.6 to C_{∗}F → 0.

Here is a stalkwise restatement of theorem ^{\underline{theorem'}} 13.11.

13.8 Corollary 13.12. Let k be a perfect field and F a presheaf with transfers so that F (Spec O_{X,x}^{h}) = 0 for all smooth X and all x ∈ X. Then (C_{∗}F)(Spec O_{X,x}) ≃ 0 for all X and all x ∈ X.

cor2 Corollary 13.13. Let f : C_{1} → C_{2} be a map of bounded above cochain complexes of presheaves with transfers. If f induces a quasi-isomorphism over all Hensel local rings Spec O_{X,x}^{h}, then Tot(C_{∗}C_{1}) → Tot(C_{∗}C_{2}) induces a quasiisomorphism over all local rings.

Proof. Let K = cone(f) denote the mapping cone of f. By assumption, each H^{p}K is a presheaf with transfers which vanishes on all Hensel local schemes, i.e., K_{Nis} ≃ 0. By ^{\underline{theorem'}} 13.11, C_{∗}H^{p}K ≃ 0 in the Zariski topology.

Since K is a bounded above cochain complex, the double complex C_{∗}(K) is bounded. Hence the usual spectral sequence of a double complex (see ^{\underline{w}}[Wei94, 5.6.2]) converges to H_{∗} Tot C_{∗}(K). Since C_{q}K(X) = K(X × Δ^{q}) we have H^{p}C_{q}K = C_{q}H^{p}K for all p and q, and we have seen that each H^{q}C_{∗}H^{p}K vanishes on every local scheme X. The resulting collapse in the spectral sequence shows that H_{∗} Tot C_{∗}(K) vanishes on every local scheme, which yields the result.

If U = \{U_{1}, …, U_{n}\} is a Zariski covering of X, we saw in ^{\underline{ztruu}} 6.12 that the Čech complex

ℤ_{tr}(Ǔ) : 0 → ℤ_{tr}(U_{1} ∩ … ∩ U_{n}) → … → ⊕_{i}ℤ_{tr}(U_{i}) → 0

is a resolution of ℤ_{tr}(X) in the étale topology (and the Nisnevich topology). Surprisingly, this gets even better when we apply C_{∗}.

zarcov Proposition 13.14. If U is a Zariski covering of X then the Čech resolution Tot C_{∗}ℤ_{tr}(Ǔ) → C_{∗}ℤ_{tr}(X) is a quasi-isomorphism in the Zariski topology.

Proof. Apply ^{\underline{cor2}} 13.13 to ^{\underline{13.5}} 6.14.

p1cover Example 13.15. Applying ^{\underline{zarcov}} 13.14 to the usual cover of ℙ^{1} (by ℙ^{1} − \{0\} and ℙ^{1} − {∞}) allows us to deduce that C_{∗} (ℤ_{tr}(ℙ^{1})/ℤ) ≃ C_{∗}ℤ_{tr}(𝔾_{m})[1] = ℤ(1)[2] for the Zariski topology, because C_{∗}ℤ_{tr}(𝔸^{1})/ℤ ≃ 0 by ^{\underline{217half}} 2.22. This was already observed in example ^{\underline{6.14}} 6.15 for the étale topology. This example will be generalized in theorem ^{\underline{thm4}} 15.2 below.

The category of motives

lecmot

In this lecture, we define the triangulated category of (effective) motives over k, and the motive of a scheme in this category. The construction of DM_{Nis}^{eff,−}(k, R) is parallel to the construction of DM_{ét}^{eff,−}(k, R) in ^{\underline{defa1we}} 9.2, and we will see that the categories are equivalent if ℚ ⊆ R.

Write D^{−} for D^{−}Sh_{Nis}(Cor_{k}, R), and let E_{𝔸} denote the smallest thick subcategory of D^{−} containing every R_{tr}(X ×𝔸^{1}) → R_{tr}(X) and closed under direct sums. (See ^{\underline{9.1}} 9.1 and ^{\underline{defa1we}} 9.2.) The quotient D^{−}/E_{𝔸} is the localization D^{−}[W_{𝔸}^{−1}], where W_{𝔸} = W_{E_{𝔸}} is the class of maps in D^{−} whose cone is in E_{𝔸}. A map in W_{𝔸} is called an 𝔸^{1}-weak equivalence.

DMnis Definition 14.1. The triangulated category of motives over k is defined to be the localization DM_{Nis}^{eff,−}(k, R) = D^{−}[W_{𝔸}^{−1}] of D^{−} = D^{−}Sh_{Nis}(Cor_{k}, R). (Cf. \underline{defa1we} 9.2.) If X is a smooth scheme over k, we write M(X) for the class of ℤ_{tr}(X) in DM_{Nis}^{eff,−}(k, ℤ) and call it the motive of X.

In ^{\underline{D-TTC}} 8.17, we showed that the derived category D^{−}(Sh_{ét}(Cor_{k}, R)) is a tensor triangulated category. The same argument works in the Nisnevich topology for D^{−}Sh_{Nis}(Cor_{k}, R). Here are the details.

D-tens Definition 14.2. If C and D are bounded above complexes of presheaves with transfers, we write C ⊗_{L,Nis}^{tr} D for (C ⊗_{L}^{tr} D)_{Nis}. Because ^{\underline{ztruu}} 6.12 holds for the Nisnevich topology, the Nisnevich analogues of ^{\underline{8.x}} 8.14, ^{\underline{8.4}} 8.15, ^{\underline{etqiso}} 8.16, ^{\underline{D-TTC}} 8.17, and ^{\underline{82}} 8.18 hold. In particular, the derived category D^{−} of bounded above complexes of Nisnevich sheaves with transfer is a tensor triangulated category under ⊗_{L,Nis}^{tr}.

Given ^{\underline{D-tens}} 14.2, the proofs of ^{\underline{86}} 9.4 and ^{\underline{DMetTTC}} 9.5 go through to show that the tensor ⊗_{L,Nis}^{tr} on D^{−} also endows the localization DM_{Nis}^{eff,−}(k, R) of D^{−}Sh_{Nis}(Cor_{k}, R) with the structure of a tensor triangulated category.

Our definitions of DM_{Nis}^{eff,−}(k, R) and M(X) are equivalent to the definitions in ^{\underline{tri}} [TrCat, p. 205]. This follows by comparing the definition in loc. cit., to theorem ^{\underline{motiviccat}} 14.10 below, using the following lemma. reflect Lemma 14.3. For every bounded above complex K of sheaves of R-modules

with transfer, the morphism K → Tot C_{∗}(K) is an 𝔸^{1}-weak equivalence. Hence K ≅ Tot C_{∗}(K) in DM_{Nis}^{eff,−}(k, R).

Proof. The proof of lemmas ^{\underline{9.6}} 9.11 and ^{\underline{812}} 9.14 go through in this setting.

dmd-nis Remark 14.4. As is ^{\underline{dfna1local}} 9.16, we say that an object L of D^{−} is called 𝔸^{1}-local (for the Nisnevich topology) if Hom_{D^{−}}(−, L) sends 𝔸^{1}-weak equivalences to isomorphisms. The proof of ^{\underline{homdmd-}} 9.20 goes through in the Nisnevich setting to show that if Y is 𝔸^{1}-local then

Hom_{DM}{eff,− \atop Nis}(X, Y) ≅ Hom_{D^{−}}(X, Y).

Let F be a Nisnevich sheaf with transfers. Then F is 𝔸^{1}-local if and only if F is homotopy invariant, because the proof of ^{\underline{star}} 9.23 goes through using ^{\underline{nistr}} 13.4 and \underline{theorem''} 13.7. This is the easy case of the following proposition.

nisa1local Proposition 14.5. Let k be a perfect field and K a bounded above cochain complex of Nisnevich sheaves of R-modules with transfer. Then K is 𝔸^{1}-local if and only if the sheaves a_{Nis}(H^{n}K) are all homotopy invariant.

Proof. Suppose first that the cohomology sheaves of K are homotopy invariant. By ^{\underline{theorem''}} 13.7 applied to F = a_{Nis}(H^{q}K), the presheaves H_{Nis}^{n}(−, F) are homotopy invariant. As in the proof of ^{\underline{star}} 9.23, this implies that each a_{Nis}(H^{q}K) is 𝔸^{1}-local. Because cd_{Nis}(X) < ∞, the hyperext spectral sequence (see ^{\underline{w}}[Wei94, 5.7.9])

E_{2}^{pq}(X) = Ext^{p}(R_{tr}(X), a_{Nis}H^{q}K) ⟹ Hom_{D^{−}}(R_{tr}(X), K[p + q])

is bounded and converges. The map f induces a morphism from it to the corresponding spectral sequence for X × 𝔸^{1}. By the Comparison Theorem (^{\underline{w}}[Wei94, 5.2.12]), f induces an isomorphism from Hom_{D^{−}}(R_{tr}(X)[n], K) to Hom_{D^{−}}(R_{tr}(X × 𝔸^{1})[n], K) for each n. By ^{\underline{localcrit}} 9.17, K is 𝔸^{1}-local.

Now suppose that K is 𝔸^{1}-local. The cohomology presheaves of K′ = Tot C_{∗}(K) are homotopy invariant by ^{\underline{aletter}} 2.17. Theorem ^{\underline{theorem''}} 13.7 applied to the cohomology presheaves H^{q}K′ shows that the sheaves a_{Nis}(H^{q}K′) are homotopy invariant. The first part of this proof shows that K′ is 𝔸^{1}-local. By lemma \underline{reflect} 14.3, the canonical map K → K′ is an 𝔸^{1}-weak equivalence. By ^{\underline{9.2}} 9.18, which goes through for the Nisnevich topology, K → K′ is an isomorphism in D^{−}. Hence the sheaves a_{Nis}(H^{n}K) ≅ a_{Nis}(H^{n}K′) are homotopy invariant.

14.6 Corollary 14.6. Let k be a perfect field and K a bounded above cochain complex of Nisnevich sheaves of R-modules with transfer. If the presheaves H^{n}(K) are all homotopy invariant, then K is 𝔸^{1}-local.

Proof. Combine ^{\underline{theorem''}} 13.7 and ^{\underline{nisa1local}} 14.5.

Example 14.7. Here is an example to show that the converse does not hold in \underline{14.6} 14.6. Consider the complex K of example ^{\underline{6.14}} 6.15:

0 → ℤ_{tr}(𝔾_{m}) → 2ℤ_{tr}(𝔸^{1}, 1) → ℤ_{tr}(ℙ^{1}, 1) → 0.

Evaluating at Spec(k) and at 𝔸^{1}, it is easy to see that the cohomology presheaf H^{2}K is not homotopy invariant (consider an embedding of 𝔸^{1} in ℙ^{1} whose image contains both 0 and 1). On the other hand K is 𝔸^{1}-local, because its cohomology sheaves a_{Nis}H^{∗}(K) vanish by ^{\underline{13.5}} 6.14.

mvtri Exercise 14.8 (Mayer-Vietoris). For each open cover {U, V } of a smooth scheme X, show that the “Mayer-Vietoris” triangle

ℤ_{tr}(U ∩ V) → ℤ_{tr}(U) ⊕ ℤ_{tr}(V) → ℤ_{tr}(X) → ℤ_{tr}(U ∩ V)[1]

is distinguished in DM. Hint: Use a Nisnevich analogue of ^{\underline{13.5}} 6.14 to show that the associated Čech complex is isomorphic to zero in DM. Cf. ^{\underline{tri}} [TrCat, 3.2.6].

Definition 14.9. Let L_{Nis} denote the full subcategory of D^{−} consisting of complexes with homotopy invariant cohomology sheaves. By ^{\underline{nisa1local}} 14.5, it is also the category of 𝔸^{1}-local complexes. If E and F are in L, then we define E ⊗_{L} F = Tot C_{∗}(E ⊗_{L,Nis}^{tr} F). By ^{\underline{aletter}} 2.17 and ^{\underline{14.6}} 14.6, E ⊗_{L} F is 𝔸^{1}-local.

motiviccat Theorem 14.10. The category (L_{Nis}, ⊗_{L}) is a tensor triangulated category, and the canonical functor

L_{Nis} → D^{−}[W_{𝔸}^{−1}] = DM_{Nis}^{eff,−}(k, R)

is an equivalence of tensor triangulated categories.

Proof. The category L = L_{Nis} is a thick subcategory of D^{−}. By ^{\underline{dmd-nis}} 14.4 and ^{\underline{nisa1local}} 14.5, the functor L → D^{−}[W_{𝔸}^{−1}] is fully faithful. By ^{\underline{reflect}} 14.3, every object K of D^{−}[W_{𝔸}^{−1}] is isomorphic to Tot C_{∗}(K). By ^{\underline{theorem''}} 13.7, Tot C_{∗}(K) is in L. Hence L is equivalent to D^{−}[W_{𝔸}^{−1}].

It follows that L is a tensor triangulated category, because D^{−}[W_{𝔸}^{−1}] is. If E and F are 𝔸^{1}-local, we have seen that the tensor product E⊗_{L,Nis}^{tr}F is naturally isomorphic to E ⊗_{L} F in D^{−}[W_{𝔸}^{−1}]. That is, ⊗_{L} is isomorphic to the induced tensor operation on L.

In ^{\underline{tri}} [TrCat, p. 210], the tensor structure on L_{Nis} was defined using ⊗_{L}.

Next, recall from ^{\underline{a1h}} 9.7 that two parallel morphisms f and g of sheaves are said to be 𝔸^{1}-homotopic if there is a map F ⊗_{L}^{tr} ℤ_{tr}(𝔸^{1}) → G whose restrictions along 0 and 1 coincide with f and g, respectively. The proof of ^{\underline{a1eq}} 9.9 shows that 𝔸^{1}-homotopic morphisms between Nisnevich sheaves became equal in DM_{Nis}^{eff,−}(k, R).

a1henisbis Proposition 14.11. Let C and D be bounded above complexes of Nisnevich sheaves with transfer, whose cohomology sheaves are homotopy invariant. If C and D are 𝔸^{1}-local, then 𝔸^{1}-homotopic maps f, g : C → D induce the same maps on hypercohomology: used in ^{\underline{symtri}} 15.8

f = g : ℍ_{Zar}^{∗}(X, C) → ℍ_{Zar}^{∗}(X, D).

Proof. To prove the proposition, write DM for DM_{Nis}^{eff,−}(k). Combining ^{\underline{a1henis}} 13.9 with ^{\underline{nisextcoho}} 13.5, we see that

H_{Zar}^{n}(X, C) ≅ Hom_{D^{−}}(ℤ_{tr}(X), C[n]).

If C is 𝔸^{1}-local, this equals Hom_{DM}(ℤ_{tr}(X), C[n]) by ^{\underline{dmd-nis}} 14.4. Since f and g agree in DM, they induce the same map from H^{n}(X, C) ≅ Hom_{DM}(ℤ_{tr}(X), C[n]) to H^{n}(X, D) ≅ Hom_{DM}(ℤ_{tr}(X), D[n]), as asserted.

We will need the following elementary result for R = ℚ in ^{\underline{dmetdmnis}} 14.22 below. It is proven by replacing ℤ(X) by R_{tr}(X) in the proof of ^{\underline{l12}} 12.13.

l12bis Lemma 14.12. The smallest class in D^{−} which contains all the R_{tr}(X) and is closed under quasi-isomorphisms, direct sums, shifts, and cones is all of D^{−}.

We now consider the case when the coefficient ring R contains ℚ. Our first goal is to identify étale and Nisnevich motivic cohomology (^{\underline{etnismotcoho}} 14.16). We will then describe DM_{ét}^{eff,−}(k, R) (in ^{\underline{13a.15}} 14.19), and finally show that DM_{Nis}^{eff,−}(k, R) ≅ DM_{ét}^{eff,−}(k, R) (^{\underline{dmetdmnis}} 14.22).

21.5 Lemma 14.13. Let F be a homotopy invariant Zariski sheaf of ℚ-modules with transfers. Then F is also an étale sheaf with transfers.

Proof. It suffices to show that the presheaf kernel and cokernel of F → F_{ét} vanish. By ^{\underline{ettr}} 6.17, these are presheaves with transfers. Thus we may suppose that F_{ét} = 0. If F ≠ 0 then there is a point x ∈ X and a nonzero element c ∈ F (S), S = Spec O_{X,x}. Since F_{ét} = 0, there is a finite étale map S′ → S with c|_{S′} = 0. As in ^{\underline{galcor}} 1.11, the composition of the transfers and inclusion

F (S) → F (S′) → F (S)

is multiplication by d, the degree of S′ → S. Hence this composition is an isomorphism. Since it sends c to zero, we have c = 0. This contradiction shows that F = 0, as desired.

fhiniset Corollary 14.14. If F is an homotopy invariant presheaf of ℚ-modules with transfer, then F_{Nis} = F_{ét}.

Proof. By ^{\underline{theorem''}} 13.7, F_{Nis} is homotopy invariant, so ^{\underline{21.5}} 14.13 applies.

22.2 Proposition 14.15. If F is an étale sheaf of ℚ-modules, then

H_{ét}^{n}(−, F) = H_{Nis}^{n}(−, F).

Proof. We need to prove that H_{ét}^{n}(S, F) = 0 for n > 0 when S is Hensel local. Given this, the result will follow from the Leray spectral sequence. Pick c ∈ H_{ét}^{n}(S, F). Then there is a finite étale map S′ → S with c|_{S′} = 0. We may assume that S′ is connected, of degree d. The composition of the étale trace map and the inclusion

H_{ét}^{n}(S, F)  H_{ét}^{n}(S′, F)  H_{ét}^{n}(S, F)

is multiplication by d (see ^{\underline{milne}} [Mil80, V.1.12]) and so an isomorphism. Since it sends c to zero, we have c = 0. Hence H^{n}(S, F) = 0.

Recall from ^{\underline{hpql}} 10.1 that the étale (or Lichtenbaum) motivic cohomology H_{L}^{p,q}(X, ℚ) is defined to be the étale hypercohomology of the complex ℚ(q).

etnismotcoho Theorem 14.16. Let k be a perfect field. If C is a complex of presheaves of ℚ-modules with transfer with homotopy invariant cohomology, then

ℍ_{ét}^{∗}(X, C_{ét}) = ℍ_{Nis}^{∗}(X, C_{Nis})

for every X in Sm/k. In particular

H_{L}^{p,q}(X, ℚ) = H^{p,q}(X, ℚ).

Proof. Consider F = H^{q}C. By ^{\underline{fhiniset}} 14.14, F_{Nis} = F_{ét}. By ^{\underline{22.2}} 14.15, we have isomorphisms H_{ét}^{i}(X, F_{ét}) → H_{Nis}^{i}(X, F_{Nis}). Comparing the hypercohomology spectral sequences for the Nisnevich and the étale topology yields the result.

In particular, the result applies to C = ℚ(q) by ^{\underline{mcpst}} 13.10.

21.7 Lemma 14.17. Let F be any homotopy invariant étale sheaf of ℚ-modules with transfers. Then F is strictly 𝔸^{1}-homotopy invariant in the sense of ^{\underline{def9.16}} 9.21. need ^{\underline{theorem''}} 13.7

Proof. By ^{\underline{22.2}} 14.15 and ^{\underline{theorem''}} 13.7, we have H_{ét}^{p}(X, F) ≅ H_{ét}^{p}(X × 𝔸^{1}, F) for all smooth X, i.e., F is strictly homotopy invariant.

For clarity, let us say that a complex K is étale 𝔸^{1}-local if it is 𝔸^{1}-local for the étale topology (as in ^{\underline{dfna1local}} 9.16), and Nisnevich 𝔸^{1}-local if it is 𝔸^{1}-local for the Nisnevich topology (as in ^{\underline{dmd-nis}} 14.4).

We will write D_{ét}^{−} for D^{−}(Sh_{ét}(Cor_{k}, R)).

etqa1local Proposition 14.18. Let k be a perfect field and suppose that ℚ ⊆ R. If K is a bounded above cochain complex of étale sheaves of R-modules with transfer, then K is étale 𝔸^{1}-local if and only if the sheaves a_{ét}(H^{n}K) are homotopy invariant.

In particular, each ℚ(j) is an étale 𝔸^{1}-local complex.

Proof. Suppose first that the sheaves a_{ét}H^{n}(K) are homotopy invariant. By \underline{21.7} 14.17 and ^{\underline{star}} 9.23, they are étale 𝔸^{1}-local. Since ℚ ⊆ R, we have cd_{R}(k) = 0, so K is étale 𝔸^{1}-local by ^{\underline{starc}} 9.28.

Conversely, suppose that K is étale 𝔸^{1}-local and set K′ = Tot C_{∗}(K). By \underline{aletter} 2.17, each H^{n}(K′) is 𝔸^{1}-homotopy invariant. By theorem ^{\underline{theorem''}} 13.7 and ^{\underline{21.5}} 14.13, each étale sheaves a_{ét}(H^{n}K′) is homotopy invariant. The first part of this proof shows that K′ is étale 𝔸^{1}-local. By ^{\underline{812}} 9.14, K → K′ is an (étale) 𝔸^{1}-weak equivalence, and an isomorphism in DM_{ét}^{eff,−}(k, R). By ^{\underline{9.2}} 9.18, K → K′ is an isomorphism in D_{ét}^{−}. Hence each sheaf a_{ét}(H^{n}K) is isomorphic to a_{ét}(H^{n}K′), and is therefore also homotopy invariant.

Let L_{ét} denote the full subcategory of D_{ét}^{−} consisting of complexes with homotopy invariant cohomology sheaves. By ^{\underline{etqa1local}} 14.18, it is also the subcategory of étale 𝔸^{1}-local complexes.

13a.15 Theorem 14.19. The natural functor L_{ét} → DM_{ét}^{eff,−}(k, R) is an equivalence of triangulated categories if ℚ ⊆ R.

Proof. The functor is full and faithful by ^{\underline{homdmd-}} 9.20 and ^{\underline{etqa1local}} 14.18. Since every K in D_{ét}^{−} becomes isomorphic to Tot C_{∗}(K) in DM_{ét}^{eff,−} by ^{\underline{812}} 9.14, and Tot C_{∗}(K) is in L_{ét} by ^{\underline{aletter}} 2.17, the functor is an equivalence.

Remark 14.20. Theorem ^{\underline{13a.15}} 14.19 implies that L_{ét} is a tensor triangulated category. As in the proof of ^{\underline{locconseff}} 9.32 and ^{\underline{motiviccat}} 14.10, ^{\underline{reflect}} 14.3 and ^{\underline{14.6}} 14.6 show that the tensor operation of L_{ét} is isomorphic to the operation ⊗_{L} defined in ^{\underline{callet}} 9.31.

ratetmot Corollary 14.21. For every smooth X, the étale motivic cohomology groups H_{L}^{p,q}(X, ℚ) may be computed in DM_{ét}^{−}:

H_{L}^{p,q}(X, ℚ) ≅ Hom_{DM}_{ét}^{−}(ℚ_{tr}(X), ℚ(q)[p]).

Proof. Write D_{ét}^{−} for D^{−}(Sh_{et}(Cor_{k}, ℚ)) and write DM for DM_{et}^{−}(k, ℚ). By \underline{homdmd-} 9.20, we have

Hom_{DM}(ℚ_{tr}(X), ℚ(q)[p]) = Hom_{D_{ét}^{−}}(ℚ_{tr}(X), ℚ(q)[p]) = Ext^{p}(ℚ_{tr}(X), ℚ(q)).

By ^{\underline{hexthcoho}} 6.25, this Ext group is H_{L}^{p,q}(X, ℚ) = H_{ét}^{p}(X, ℚ(q)).

dmetdmnis Theorem 14.22. If ℚ ⊆ R, then DM_{Nis}^{eff,−}(k, R) → DM_{ét}^{eff,−}(k, R) is an equivalence of tensor triangulated categories.

Proof. Because sheafification is exact, it induces a triangulated functor from D_{Nis}^{−} = D^{−}(Sh_{Nis}(Cor_{k}, R)) to D_{ét}^{−} = D^{−}(Sh_{ét}(Cor_{k}, R)). By definitions \underline{812} 9.14 and ^{\underline{D-tens}} 14.2, we have (K ⊗_{L,Nis}^{tr} L)_{ét} = K ⊗_{L,ét}^{tr} L. Comparing definitions, we see that D_{Nis}^{−} → D_{ét}^{−} sends Nisnevich 𝔸^{1}-weak equivalences to étale 𝔸^{1}-weak equivalences, so it induces a tensor triangulated functor σ from DM_{Nis}^{eff,−}(k, R) to DM_{ét}^{eff,−}(k, R). Clearly, D_{Nis}^{−} → D_{ét}^{−} and DM_{Nis}^{eff,−} → DM_{ét}^{eff,−} are onto on objects.

It remains to show that the functor σ is full and faithful, i.e., that we have Hom_{DM}{eff,− \atop Nis}(K, L) ≅ Hom_{DM}{eff,− \atop ét}(K_{ét}, L_{ét}). By theorem ^{\underline{motiviccat}} 14.10, we may assume that L is in L_{Nis}. The class of objects K so that Hom_{DM}{eff,− \atop Nis}(K, L[n]) ≅ Hom_{DM}{eff,− \atop ét}(K_{ét}, L_{ét}[n]) for all n is closed under quasi-isomorphisms, direct sums, shifts, and cones. By ^{\underline{l12bis}} 14.12, it suffices to show that each R_{tr}(X) is in this class. But then by ^{\underline{dmd-nis}} 14.4, ^{\underline{nisextcoho}} 13.5, ^{\underline{homdmd-}} 9.20, and ^{\underline{hexthcoho}} 6.25, we have

Hom_{DM}{eff,− \atop Nis}(R_{tr}(X), L[n]) ≅ Hom_{D_{Nis}^{−}}(R_{tr}(X), L[n]) ≅ H_{Nis}^{n}(X, L)

and

Hom_{DM}{eff,− \atop ét}(R_{tr}(X), L_{ét}[n]) ≅ Hom_{D_{ét}^{−}}(R_{tr}(X), L_{ét}[n]) ≅ H_{ét}^{n}(X, L_{ét}).

These groups are isomorphic by ^{\underline{etnismotcoho}} 14.16, as required.

The complex ℤ(n) and ℙ^{n}

The goal of this lecture is to interpret the motivic complex ℤ(n) in terms of ℤ_{tr}(ℙ^{n}) and use this to show that the product on motivic cohomology is gradedcommutative. We begin by observing that M(ℙ^{n} − {0}) ≅ M(ℙ^{n−1}).

lem31 Lemma 15.1. There is a chain homotopy equivalence:

C_{∗}ℤ_{tr}(ℙ^{n} − {0}) ≃ C_{∗}ℤ_{tr}(ℙ^{n−1}).

Proof. Consider the projection (ℙ^{n} − 0) → ℙ^{n−1} sending (x_{0} : ⋯: x_{n}) to (x_{1} : ⋯: x_{n}), where 0 is (1 : 0 : ⋯: 0). This map has affine fibers. The self homotopy λ(x_{0} : ⋯: x_{n}) → (λx_{0} : x_{1} : ⋯: x_{n}) is well defined on ℙ^{n} − {0} × 𝔸^{1}, even for λ = 0, because one of x_{1}, …, x_{n} is always non zero. Hence the projection and the section (x_{1} : ⋯: x_{n}) ↦ (0 : x_{1} : ⋯: x_{n}) are inverse 𝔸^{1}-homotopy equivalences. The lemma now follows from ^{\underline{cor1}} 2.24.

thm4 Theorem 15.2. If k is a perfect field, there is a quasi-isomorphism of Zariski sheaves for each n:

C_{∗} (ℤ_{tr}(ℙ^{n})/ℤ_{tr}(ℙ^{n−1})) ≃ C_{∗}ℤ_{tr}(𝔾_{m}^{∧n})[n] = ℤ(n)[2n].

In particular, C_{∗} (ℤ_{tr}(ℙ^{n})/ℤ_{tr}(ℙ^{n−1})) (X) ≃ ℤ(n)[2n](X) for any smooth local X.

Our proof will use theorem ^{\underline{theorem'}} 13.11, whose proof depended upon theorem ^{\underline{theorem''}} 13.7, a result whose proof we have postponed until lecture ^{\underline{lechic}} 23. The requirement that k be perfect is only needed for ^{\underline{theorem''}} 13.7 (and hence ^{\underline{theorem'}} 13.11).

Week 15 part a - February 23, 2000

Proof. Let U be the usual cover of ℙ^{n} by (n + 1) copies of 𝔸^{n} and note that n of these form a cover V of ℙ^{n} − {0}. The intersection of i + 1 of these 𝔸^{n} is 𝔸^{n−i} ×(𝔸^{1} −\{0\})^{i}. By ^{\underline{13.5}} 6.14, we have quasi-isomorphisms ℤ_{tr}(Ǔ) → ℤ_{tr}(ℙ^{n}) and ℤ_{tr}(V̌) → ℤ_{tr}(ℙ^{n} − 0) of complexes of Nisnevich sheaves with transfers. Hence the quotient complex Q_{∗} = ℤ_{tr}(Ǔ)/ℤ_{tr}(V̌) is a resolution of ℤ_{tr}(ℙ^{n})/ℤ_{tr}(ℙ^{n} − 0) as a Nisnevich sheaf. By ^{\underline{cor2}} 13.13 and ^{\underline{lem31}} 15.1, or by ^{\underline{zarcov}} 13.14, Tot C_{∗}Q_{∗} is quasiisomorphic to C_{∗} (ℤ_{tr}(ℙ^{n})/ℤ_{tr}(ℙ^{n} − 0)) and hence to C_{∗}(ℤ_{tr}(ℙ^{n})/ℤ_{tr}(ℙ^{n−1})) for the Zariski topology.

On the other hand, we know from ^{\underline{les}} 2.11 that for T = 𝔸^{1} − 0 the sequence

0 → ℤ_{tr}(𝔾_{m}^{∧n}) → ℤ_{tr}(T^{n}) → ⊕_{i}ℤ_{tr}(T^{n−1}) → ⊕_{i,j}ℤ_{tr}(T^{n−2}) →

⋯ → ⊕_{i,j}ℤ_{tr}(T^{2}) → ⊕_{i}ℤ_{tr}(T) → ℤ → 0

is split exact. Rewriting this as 0 → ℤ_{tr}(𝔾_{m}^{∧n}) → R_{n} → R_{n−1} → ⋯ → R_{0} → 0, with R_{n} = ℤ_{tr}(T^{n}), R_{n−1} = ⊕_{i}ℤ_{tr}(T^{n−1}), and R_{0} = ℤ, we may regard it as a chain homotopy equivalence ℤ_{tr}(𝔾_{m}^{∧n})[n] → R_{∗}. With this indexing there is a natural map Q_{∗} → R_{∗} whose typical term is a direct sum of the projections

ℤ_{tr}(𝔸^{n−i} × T^{i}) → ℤ_{tr}(T^{i}).

These are 𝔸^{1}-homotopy equivalences (see ^{\underline{a1he}} 2.23). Applying C_{∗} turns them into quasi-isomorphisms by ^{\underline{cor1}} 2.24. Hence we have quasi-isomorphisms of total complexes of presheaves with transfers

Tot C_{∗}Q_{∗} ^{≃} Tot C_{∗}R_{∗} ^{≃} C_{∗}ℤ_{tr}(𝔾_{m}^{∧n})[n].

Combining with Tot C_{∗}Q_{∗} ≃ C_{∗} (ℤ_{tr}(ℙ^{n})/ℤ_{tr}(ℙ^{n−1})) yields the result in the Zariski topology.

If n = 1, it is easy to see that the isomorphisms of ^{\underline{p1cover}} 13.15 and ^{\underline{thm4}} 15.2 agree. Figure ^{\underline{bigdiagram}} 15.1 illustrates the proof of theorem ^{\underline{thm4}} 15.2 when n = 2. We have written ‘X’ for C_{∗}ℤ_{tr}(X) in order to save space, and ‘𝔸^{1}−h.e.’ for 𝔸^{1}-homotopy equivalence.

anless0 Corollary 15.3. For each n there is a quasi-isomorphism for the Zariski topol-

ogy

C_{∗} (ℤ_{tr}(𝔸^{n} − 0)/ℤ) ≃ ℤ(n)[2n − 1].

Proof. Applying ^{\underline{zarcov}} 13.14 and ^{\underline{lem31}} 15.1 to the cover of ℙ^{n} by 𝔸^{n} and ℙ^{n} − 0, we see that the sequence

0 → C_{∗}ℤ_{tr}(𝔸^{n} − 0) → C_{∗}ℤ_{tr}(𝔸^{n}) ⊕ C_{∗}ℤ_{tr}(ℙ^{n−1}) → C_{∗}ℤ_{tr}(ℙ^{n}) → 0

becomes exact for the Zariski topology. The result now follows from theorem \underline{thm4} 15.2, since C_{∗}ℤ_{tr}(𝔸^{n}) ≃ C_{∗}ℤ_{tr}(Spec k) ≃ ℤ by ^{\underline{217half}} 2.22 and ^{\underline{cstar}} 2.12.

cstarzi Exercise 15.4. Show that the map C_{∗}ℤ_{tr}(ℙ^{i}) → ℤ(i)[2i] of theorem ^{\underline{thm4}} 15.2 factors through the natural inclusion C_{∗}ℤ_{tr}(ℙ^{i}) → C_{∗}ℤ_{tr}(ℙ^{n}) for all n > i. Hint: First construct ℤ_{tr}(Ǔ) → ℤ(1)[2] vanishing on ℤ_{tr}(U_{0}), and form

ℤ_{tr}(Ǔ) ^{Δ} ℤ_{tr}(Ǔ) ⊗ ⋯ ⊗ ℤ_{tr}(Ǔ) → ℤ(1)[2]^{⊗i} → ℤ(i)[2i].

motivepn Corollary 15.5. There is a quasi-isomorphism

M(ℙ^{n}) = C_{∗}ℤ_{tr}(ℙ^{n}) ^{≃} ℤ ⊕ ℤ(1)[2] ⊕ ⋯ ⊕ ℤ(n)[2n].

[figure: 0; 0; 𝔸^{1} × (𝔸^{1} − 0); 2𝔸^{2}; ℙ^{2} − 0]

[figure: 0; (𝔸^{1} − 0)^{2}; 3 (𝔸^{1} × (𝔸^{1} − 0)); 3𝔸^{2}; ℙ^{2}]

[figure: 0; (𝔸^{1} − 0)^{2}; =; 2 (𝔸^{1} × (𝔸^{1} − 0)); 𝔸^{2}; ℙ^{2}/(ℙ^{2} − 0)]

[figure: 𝔾_{m}^{∧2}; (𝔸^{1} − 0)^{2}; =; 2(𝔸^{1} − 0); 𝔸^{1}−h.e.; pt; 𝔸^{1}−h.e.; 0]

Figure 15.1: The case n = 2 of theorem ^{\underline{thm4}} 15.2  bigdiagram

Proof. We proceed by induction, the case n = 1 being ^{\underline{p1cover}} 13.15. By exercise ^{\underline{cstarzi}} 15.4, the maps ℤ_{tr}(ℙ^{n−1}) → ℤ_{tr}(ℙ^{n}) is split injective in DM, because the quasiisomorphism ℤ_{tr}(ℙ^{n−1}) → ⊕ℤ(i)[i] factors through it. Hence the distinguished triangle

C_{∗}ℤ_{tr}(ℙ^{n−1})  C_{∗}ℤ_{tr}(ℙ^{n})  ℤ(n)[n]  C_{∗}ℤ_{tr}(ℙ^{n−1})[1]

splits.

Our re-interpretation of the motivic complexes allows us to show that the product in motivic cohomology is skew-commutative. This will be a consequence of the following construction, and some linear algebra.

tau Example 15.6. Consider the reflection automorphism τ of ℙ^{n}, n ≥ 1, sending (x_{0} : x_{1} : ⋯: x_{n}) to (−x_{0} : x_{1} : ⋯: x_{n}). We claim that the induced automorphism of C_{∗}ℤ_{tr}(ℙ^{n}) is 𝔸^{1}-homotopic to the identity map, so that it is the identity map in DM_{Nis}^{eff,−} (see ^{\underline{DMnis}} 14.1 and ^{\underline{a1eq}} 9.9).

To see this, consider the elementary correspondence from ℙ^{n} × 𝔸^{1} (parametrized by x_{0}, ..., x_{n} and t) to ℙ^{n} (parametrized by y_{0}, ..., y_{n}) given by Z, the subvariety of ℙ^{n} × 𝔸^{1} × ℙ^{n} defined by the homogeneous equation

y_{1}(x_{0}y_{1} + tx_{1}y_{0}) = (t^{2} − 1)x_{1}y_{0}^{2}

together with x_{2} = y_{2}, ..., x_{n} = y_{n} if n ≥ 2. (Exercise: check that this is an elementary correspondence!) The restrictions along t = ±1 yield two finite correspondences from ℙ^{n} to itself, whose difference is id_{ℙ^{n}} − τ.

The symmetric group Σ_{n} acts canonically on 𝔸^{n} by permuting coordinates. By inspection, this induces a Σ_{n}-action on the sheaf with transfers ℤ_{tr}(𝔾_{m}^{∧n}) and hence on the motivic complexes ℤ(n).

permute Proposition 15.7. The action of the symmetric group Σ_{n} on C_{∗}ℤ_{tr}(𝔸^{n} − 0) is 𝔸^{1}-homotopic to the trivial action.

Proof. Because the action is induced from an embedding Σ_{n} ↪ GL_{n}(k), and every transposition acts as τ times an element of SL_{n}(k), we see from example \underline{tau} 15.6 that it suffices to show that the action of SL_{n}(k) on C_{∗}ℤ_{tr}(𝔸^{n} −0) is chain homotopic to the trivial action.

Since every matrix in SL_{n}(k) is a product of elementary matrices, it suffices to consider one elementary matrix e_{ij}(a), a ∈ k. But multiplication by this matrix is 𝔸^{1}-homotopic to the identity of 𝔸^{n} − 0, by the homotopy (x, t) ↦ e_{ij}(at)x (see ^{\underline{a1hebis}} 9.8). In particular it is an 𝔸^{1}-homotopy equivalence (see ^{\underline{a1he}} 2.23). By \underline{cor1} 2.24, the resulting endomorphism of C_{∗}ℤ_{tr}(𝔸^{n} − 0) is chain homotopic to the identity.

We say that the action of a group G on a complex C is cohomologically trivial (for the Zariski topology) if the actions on the cohomology presheaves ℍ_{Zar}(−; C) are trivial.

symtri Corollary 15.8. The action of the symmetric group Σ_{n} on ℤ(n) is cohomologi-

cally trivial. Hence it is trivial on the motivic cohomology groups H^{p,q}(X, ℤ(n)). we use this in

                                                                    \underline{DMet} 9.6

Proof. The action of Σ_{n} on 𝔸^{n} extends to an action on ℙ^{n} fixing ℙ^{n−1}. In fact, all the constructions in the proof of theorem ^{\underline{thm4}} 15.2 and corollary ^{\underline{anless0}} 15.3 are equivariant. By ^{\underline{anless0}} 15.3, it suffices to show that the action of Σ_{n} on C_{∗}ℤ_{tr}(𝔸^{n} − 0) is cohomologically trivial. This follows from ^{\underline{permute}} 15.7 and ^{\underline{a1henisbis}} 14.11.

Recall from ^{\underline{42}} 3.10 that there is a pairing of presheaves ℤ(i) ⊗ ℤ(j) → ℤ(i + j). By inspection of ^{\underline{tecnical}} 3.9, this pairing is compatible with the action of the subgroup Σ_{i} × Σ_{j} of Σ_{i+j}, as well as with the permutation τ interchanging the first i and last j coordinates of ℤ_{tr}(𝔾_{m}^{∧i+j}).

skew Theorem 15.9. The pairing defined in ^{\underline{mult}} 3.11 is skew-commutative:

H_{Zar}^{p}(X, ℤ(i)) ⊗ H_{Zar}^{q}(X, ℤ(j)) → H_{Zar}^{p+q}(X, ℤ(i + j)).

Proof. As in ^{\underline{twist}} 8A.2, the permutation τ fits into the commutative diagram

[figure: H^{p}(X, ℤ(i)) ⊗ H^{q}(X, ℤ(j)); H^{p+q}(X, ℤ(i) ⊗ ℤ(j)); H^{p+q}(X, ℤ(i + j)); H^{q}(X, ℤ(j)) ⊗ H^{p}(X, ℤ(i)); (; − 1)^{pq}; H^{p+q}(X, ℤ(j) ⊗ ℤ(i)); τ; H^{p+q}(X, ℤ(j + i)); τ]

and the right vertical map is the identity by proposition ^{\underline{symtri}} 15.8.

We conclude this lecture with a generalization of the decomposition ^{\underline{motivepn}} 15.5 of M(ℙ^{n}) to a projective bundle theorem.

22.8 Construction 15.10. Let ℙ = ℙ(E) → X be a projective bundle associated to a vector bundle E of rank n + 1. From ^{\underline{smallpq}} 4.2, ^{\underline{mcpst}} 13.10, and ^{\underline{nisextcoho}} 13.5 we have an isomorphism

Pic(ℙ) ≅ H_{Nis}^{2}(ℙ, ℤ(1)) ≅ Hom_{D^{−}}(ℤ_{tr}(ℙ), ℤ(1)[2]).

Therefore the canonical line bundle yields a canonical map τ : ℤ_{tr}(ℙ) → ℤ(1)[2] in D^{−}. Recall from ^{\underline{diag7.6}} 10.4 that there are multiplication maps for all i ≥ 1, from ℤ(1)^{⊗^{tr}i} = ℤ(1) ⊗^{tr} ⋯ ⊗^{tr} ℤ(1) to ℤ(i). For i > 1, we let τ^{i} denote the composite

ℤ_{tr}(ℙ) ^{Δ} ℤ_{tr}(ℙ × ⋯ × ℙ) ^{≅} ℤ_{tr}(ℙ)^{⊗^{tr}i} ^{τ^{⊗i}} ℤ(1)[2]^{⊗^{tr}i} ℤ(i)[2i].

Finally, we extend the structure map σ_{0} : ℤ_{tr}(ℙ) → ℤ_{tr}(X) to a canonical family of maps in D^{−}

σ_{i} : ℤ_{tr}(ℙ) ^{Δ} ℤ_{tr}(ℙ) ⊗^{tr} ℤ_{tr}(ℙ) ^{σ_{0}⊗τ^{i}} ℤ_{tr}(X) ⊗ ℤ(i)[2i].

22.9 Exercise 15.11. Show that the canonical map in ^{\underline{22.8}} 15.10 induces the isomorphism ℤ_{tr}(ℙ_{k}^{n}) ≅ ⊕_{i=0}^{n}ℤ(i)[2i] of ^{\underline{motivepn}} 15.5. Hint: Use exercise ^{\underline{cstarzi}} 15.4.

motivepe Theorem 15.12 (Projective Bundle Theorem). Let ℙ(E) → X be a projective bundle associated to a vector bundle E of rank n + 1. Then the canonical map

⊕_{i=0}^{n}ℤ_{tr}(X)(i)[2i] → ℤ_{tr}(ℙ(E))

is an isomorphism in DM.

Proof. Using induction on the number of open subsets in a trivialization of E, together with the Mayer-Vietoris triangles ^{\underline{mvtri}} 14.8, we are reduced to the case when ℙ(E) = X × ℙ^{n}. Since ℤ_{tr}(X × ℙ^{n}) ≅ ℤ_{tr}(X) ⊗^{tr} ℤ_{tr}(ℙ^{n}), we may even assume X = Spec(k). This case is given by exercise ^{\underline{22.9}} 15.11.

Equidimensional cycles

In this lecture we introduce the notion of an equidimensional cycle, and use it to construct the Suslin-Friedlander chain complex ℤ^{SF} (i). We then show (in ^{\underline{thm2}} 16.7) that ℤ^{SF} (i) is quasi-isomorphic to ℤ(i). In lecture ^{\underline{mchcg}} 19 (^{\underline{equisubset}} 19.4) we shall compare ℤ^{SF} (i) to the complex defining higher Chow groups.

Let Z be a scheme of finite type over S such that every irreducible component of Z dominates a component of S. We say that Z is equidimensional over S of relative dimension m if for every point s of S, either Z_{s} is empty or each of its components have dimension m. If S′ → S is any map, the pullback S′ ×_{S} Z is equidimensional over S′ of relative dimension m.

dfnzequi Definition 16.1. Let T be any scheme of finite type over k and m ≥ 0 an integer. The presheaf z_{equi}(T, m) on Sm/k is defined as follows. For each smooth S, z_{equi}(T, m)(S) is the free abelian group generated by the closed and irreducible subvarieties Z of S × T which are dominant and equidimensional of relative dimension m over a component of S. If S′ → S is any map, the pullback of equidimensional cycles (see ^{\underline{a4}} 1A.5) induces the required natural map z_{equi}(T, m)(S) → z_{equi}(T, m)(S′).

It is not hard to see that z_{equi}(T, m) is a Zariski sheaf, and even an étale sheaf, for each T and m. One can also check that each z_{equi}(T, m) is contravariant for flat maps in T, and covariant for proper maps in T, both with the appropriate change in the dimension index m, (see ^{\underline{relchow}} [RelCy, 3.6.2 and 3.6.4]); see \underline{acahkt} [Blo86, 1.3]. In particular, if T′ ↪ T is a closed immersion, there are canonical inclusions z_{equi}(T′, m) ↪ z_{equi}(T, m) for all m.

ztrzequi Example 16.2. The case m = 0 is of particular interest, since z_{equi}(T, 0)(U) is free abelian on the irreducible Z ⊂ U × T which are quasi-finite and dominant over U. Hence ℤ_{tr}(T)(U) ⊆ z_{equi}(T, 0)(U), because ℤ_{tr}(T)(U) is the free abelian group of cycles in U ×T which are finite and surjective over U. In fact, ℤ_{tr}(T) is a sub-sheaf of z_{equi}(T, 0) because the structure morphisms associated to V → U are compatible: ℤ_{tr}(T)(U) → ℤ_{tr}(T)(V) is also the pullback of cycles (see ^{\underline{a8}} 1A.9).

Week 11 - January 19, 2000

If T is projective, then ℤ_{tr}(T) = z_{equi}(T, 0). Indeed, each closed subvariety Z ⊂ U × T is proper over U, so Z is quasi-finite over U if and only if Z is finite over U (see ^{\underline{hart}} [Har77, Ex. III.11.2]).

We now define transfer maps for z_{equi}(T, m) which are compatible with the transfers in ℤ_{tr}(T) when m = 0. Given an elementary correspondence V from X to Y and a cycle Z in z_{equi}(T, m)(Y), the pullback Z_{V} is a well-defined cycle of V × T by ^{\underline{a4}} 1A.5 and ^{\underline{a7}} 1A.8. We define ϕ_{V} (Z) ∈ z_{equi}(T, m)(X) to be the pushforward of Z_{V} along the finite map V ×T → X ×T. This gives a homomorphism ϕ_{V} : z_{equi}(T, m)(Y) → z_{equi}(T, m)(X).

If m = 0, the restriction of ϕ_{V} to ℤ_{tr}(T) is the transfer map constructed in \underline{elcor} 1.1 and ^{\underline{a8}} 1A.9, as we see from ^{\underline{corc}} 1A.11.

We leave the verification of the following to the reader; cf. ^{\underline{biv}} [BivCy, 5.7].

equipst Exercise 16.3. If T is a smooth scheme, show that ϕ makes each z_{equi}(T, m) into a presheaf with transfers. If U → T is flat, show that the pull-back z_{equi}(T, m) → z_{equi}(U, m) is a morphism of presheaves with transfer.

Example 16.4. For each X, there is a natural map z_{equi}(𝔸^{i}, 0)(X) → CH^{i}(𝔸^{i} × X) ≅ CH^{i}(X), sending a subvariety Z of 𝔸^{i} × X, quasi-finite over X, to its cycle. Comparing the transfer map for z_{equi}(𝔸^{i}, 0) to the transfer map for CH^{i}(X) defined in ^{\underline{chipst}} 2.5, we see that z_{equi}(𝔸^{i}, 0) → CH^{i}(−) is a morphism of presheaves with transfers.

We define the Suslin-Friedlander motivic complexes ℤ^{SF} (i) by:

ℤ^{SF} (i) = C_{∗}z_{equi}(𝔸^{i}, 0)[−2i].

We regard ℤ^{SF} (i) as a bounded above cochain complex, whose top term is z_{equi}(𝔸^{i}, 0) in cohomological degree 2i. As in ^{\underline{motcompl}} 3.1, C_{∗}(F) stands for the chain complex of presheaves associated to the simplicial presheaf U ↦ F (U × Δ^{•}).

Example 16.5. It follows from ^{\underline{ztrzequi}} 16.2 and ^{\underline{equipst}} 16.3 that there is a morphism of presheaves with transfers from ℤ_{tr}(ℙ^{i}) = z_{equi}(ℙ^{i}, 0) to z_{equi}(𝔸^{i}, 0), with kernel ℤ_{tr}(ℙ^{i−1}). Applying C_{∗} gives an exact sequence of complexes of presheaves with transfers 0 → C_{∗}ℤ_{tr}(ℙ^{i−1}) → C_{∗}ℤ_{tr}(ℙ^{i}) → ℤ^{SF} (i)[2i].

zequike Exercise 16.6. Let E be the function field of a smooth variety over k. Show that the stalk at Spec E of the sheaf z_{equi}(𝔸_{k}^{i}, 0) on Sm/k is equal to the global sections z_{equi}(𝔸_{E}^{i}, 0)(Spec E) of the sheaf z_{equi}(𝔸_{E}^{i}, 0) on Sm/E. Similarly, show that the stalk of C_{m}z_{equi}(𝔸_{k}^{i}, 0) at Spec E equals C_{m}z_{equi}(𝔸_{E}^{i}, 0)(Spec E).

Conclude that the stalk of ℤ^{SF} (i) at Spec E equals ℤ^{SF} (i)(Spec E), and is independent of the choice of k.

Here are the two main results in this lecture. Figure ^{\underline{week15fig}} 16.1 gives the scheme of the proof of ^{\underline{thm2}} 16.7. It shows how this result ultimately depends on theorem \underline{theorem'} 13.11, whose proof will be completed in lectures ^{\underline{leccov}} 20-^{\underline{lechic}} 23 below.

thm2 Theorem 16.7. There is a quasi-isomorphism in the Zariski topology:

ℤ(i) ≃ ℤ^{SF} (i).

In particular, H^{n,i}(X, ℤ) ≅ ℍ^{n}(X, ℤ^{SF} (i)) for all n and i.

Week 11 - January 19, 2000

             \underline{theorem'} 13.11^{\underline{cor2}} 13.13  ^{\underline{lem33}} 16.12^{\underline{lem32}} 16.10^{\underline{lem31}} 15.1  ^{\underline{thm3}} 16.8^{\underline{thm4}} 15.2  ^{\underline{thm2}} 16.7

Figure 16.1: Scheme of the proof of ^{\underline{thm2}} 16.7  week15fig

Proof. As ℙ^{i} is proper, z_{equi}(ℙ^{i}, 0) = ℤ_{tr}(ℙ^{i}) by ^{\underline{ztrzequi}} 16.2. Hence ^{\underline{thm2}} 16.7 follows from combining ^{\underline{thm4}} 15.2 and ^{\underline{thm3}} 16.8.

thm3 Theorem 16.8. There is a quasi-isomorphism in the Zariski topology:

C_{∗} [z_{equi}(ℙ^{i}, 0)/z_{equi}(ℙ^{i−1}, 0)] ^{≃} C_{∗}z_{equi}(𝔸^{i}, 0).

We now prepare for the proof of ^{\underline{thm3}} 16.8.

Let F_{i}(U) denote the (free abelian) subgroup of z_{equi}(𝔸^{i}, 0)(U) generated by the cycles in U × 𝔸^{i} which do not touch U × 0. By inspection, the transfers z_{equi}(𝔸^{i}, 0)(U) → z_{equi}(𝔸^{i}, 0)(V) send F_{i}(U) to F_{i}(V). Hence F_{i} is a subpresheaf with transfers of z_{equi}(𝔸^{i}, 0).

52 Lemma 16.9. There is a commutative diagram with exact rows in PST(k)

[figure: 0; ℤ_{tr}(ℙ^{i} − 0)/ℤ_{tr}(ℙ^{i−1}); ℤ_{tr}(ℙ^{i})/ℤ_{tr}(ℙ^{i−1}); coker_{1}; 0]

[figure: 0; F_{i}; ∩; z_{equi}(𝔸^{i}, 0); φ; ∩; coker_{2}; ∩; 0.]

All three vertical maps are injections.

Proof. By example ^{\underline{ztrzequi}} 16.2, there is a natural map from ℤ_{tr}(ℙ^{i}) = z_{equi}(ℙ^{i}, 0) to z_{equi}(𝔸^{i}, 0) with kernel ℤ_{tr}(ℙ^{i−1}). Thus φ is an injection; by exercise ^{\underline{equipst}} 16.3, φ is a morphism of presheaves with transfers. Now the inclusion ℤ_{tr}(ℙ^{i} − 0) ⊂ ℤ_{tr}(ℙ^{i}) is a morphism in PST by the Yoneda lemma; see ^{\underline{defztr}} 2.7. Since ℤ_{tr}(ℙ^{i} − 0)(U) consists of cycles Z ⊂ U × (ℙ^{i} − 0) finite over U, their restriction belongs to the subgroup F_{i}(U), i.e., φ sends ℤ_{tr}(ℙ^{i} − 0) to F_{i}. Hence the diagram commutes.

By inspection, coker_{1}(X) is free abelian on the elementary correspondences Z ⊂ X × ℙ^{i} which touch X × 0 and coker_{2}(X) is free abelian on the equidimensional W ⊂ X × 𝔸^{i} which touch X × 0. Since Z ↦ φ(Z) is a monomorphism on these generators, it follows that coker_{1}(X) → coker_{2}(X) is an injection for all X.

lem32 Lemma 16.10. C_{∗}(F_{i}) is chain contractible as a complex of presheaves.

Proof. Recall that F_{i}(X) is a subgroup of the group of cycles on X × 𝔸^{i}. Let h_{X} : F_{i}(X) → F_{i}(X × 𝔸^{1}) be the pullback of cycles along μ : X × 𝔸^{i} × 𝔸^{1} → X × 𝔸^{i} defined by (x, r, t) → (x, r · t). This is a good pullback because the map μ is flat over X ×(𝔸^{i} −{0}). By construction, the following diagram commutes.

[figure: X × 𝔸^{i}; t = 1; X × 𝔸^{i} × 𝔸^{1}; t = 0; X × 𝔸^{i}]

[figure: id; id × 0]

X × 𝔸^{i} \begin{aligned}μ \\ &  \end{aligned}

It follows that F_{i}(t = 1)h_{X} is the identity and F_{i}(t = 0)h_{X} = 0. Thus the requirements of lemma ^{\underline{thm6}} 2.20 are satisfied for F_{i}.

cokhens Lemma 16.11. For every Hensel local scheme S, the map coker_{1}(S) → coker_{2}(S) in diagram ^{\underline{52}} 16.9 is an isomorphism.

Proof. Since coker_{1}(S) → coker_{2}(S) is injective by ^{\underline{52}} 16.9, it suffices to prove that it is surjective. Let Z be a equidimensional correspondence from S to 𝔸^{i}. As Z is quasi-finite over a Hensel scheme, the projection decomposes Z into the disjoint union of Z_{0} (which doesn’t contain any point over the closed point of the Hensel scheme) and Z_{1} (which is finite over the base). We claim that the Z_{0} part comes from F_{i}. Take Z_{0} and consider its irreducible components. The intersection Z_{0} ∩ {0} must be empty, otherwise it would project to the closed point of the base. Hence Z_{0} is zero in the cokernel. But now Z_{1} comes from ℤ_{tr}(ℙ^{i})/ℤ_{tr}(ℙ^{i−1}).

lem33 Lemma 16.12. The map C_{∗}(coker_{1}) → C_{∗}(coker_{2}) is a quasi-isomorphism of complexes of Zariski sheaves.

Proof. Let φ′ be the map between the cokernels in ^{\underline{52}} 16.9. By ^{\underline{cokhens}} 16.11, φ′ is an isomorphism on all Hensel local schemes. By ^{\underline{cor2}} 13.13, φ′ induces quasi-isomorphisms C_{∗} coker_{1}(X) ≃ C_{∗} coker_{2}(X) for all local X.

Proof of ^{\underline{thm3}} 16.8. Applying C_{∗} to the diagram in ^{\underline{52}} 16.9 yields a commutative diagram of chain complexes with exact rows. The left two complexes are acyclic by ^{\underline{lem31}} 15.1 and ^{\underline{lem32}} 16.10. The right two complexes are quasi-isomorphic by ^{\underline{lem33}} 16.12. Theorem ^{\underline{thm3}} 16.8 now follows from the 5-lemma.

Higher Chow groups

During the first part of this series of lectures we defined motivic cohomology and we studied its basic properties. We also established relations with some classic objects of algebraic geometry, such as Milnor K-Theory, ^{\underline{motmil}} 5.1, and étale cohomology, ^{\underline{lichet}} 10.2.

The goal of the next few lectures is to find a relation between motivic cohomology and the classical Chow groups CH^{i}, generalizing the isomorphism H^{2,1}(X, ℤ) ≅ Pic(X) = CH^{1}(X) of ^{\underline{smallpq}} 4.2. That is, we will prove that:

H^{2i,i}(X, ℤ) ≅ CH^{i}(X)

for any smooth variety X. There are at least three ways to prove this. The original approach, which needs resolution of singularities, was developed in the book “Cycles, Transfers and Motivic Homology Theories” ^{\underline{orange}} [VSF00]. A second recent approach is to use the Cancellation Theorem of ^{\underline{Cancellation}} [Voe02] and the Gersten resolution ^{\underline{gersten}} 23.11 for motivic cohomology sheaves.

A third approach, which is the one we shall develop here, uses Bloch’s higher Chow groups CH^{i}(X, m) to establish the more general isomorphism H^{n,i}(X, ℤ) ≅ CH^{i}(X, 2i − n). This approach uses the equidimensional cycle groups of the previous lecture, but does not use resolution of singularities.

The main goal of this lecture is to prove that the higher Chow groups are presheaves with transfers. (See theorem ^{\underline{hcgpst}} 17.20.) In particular, they are functorial for maps between smooth schemes. (We will give a second proof of this in ^{\underline{hyperfunct}} 19.15.)

We begin with Bloch’s definition of higher Chow groups (see ^{\underline{acahkt}} [Blo86]).

hcgdef Definition 17.1. Let X be an equidimensional scheme. We write z^{i}(X, m) for the free abelian group generated by all codimension i subvarieties on X × Δ^{m} which intersect all faces X × Δ^{j} properly for all j < m (in the sense of ^{\underline{intmult}} 17A.1).

Each face X × Δ^{j} is defined by a regular sequence, and intersection of cycles defines a map z^{i}(X, m) → z^{i}(X, j) (see ^{\underline{intmult}} 17A.1, or ^{\underline{fulton}} [Ful84, Example 7.1.2]). We write z^{i}(X, •) for the resulting simplicial abelian group m ↦ z^{i}(X, m). We write z^{i}(X, ∗) for the chain complex associated to z^{i}(X, •).

The higher Chow groups of X are defined to be the groups:

CH^{i}(X, m) = π_{m}(z^{i}(X, •)) = H_{m}(z^{i}(X, ∗)).

If X is any scheme, it is easy to check that CH^{i}(X, 0) is the classical Chow group CH^{i}(X) (see ^{\underline{CHagree}} 17.3). Indeed, z^{i}(X, 0) is the group of all codimension i cycles on X while z^{i}(X, 1) is generated by those codimension i subvarieties Z on X × 𝔸^{1} which intersect both X × {0} and X × {1} properly. Moreover the maps ∂_{j} : z^{i}(X, 1) [figure] z^{i}(X, 0) send Z to Z ∩ (X × {j}).

zequizi Example 17.2. If i ≤ d = dim X, then z_{equi}(X, d − i)(Δ^{m}) ⊆ z^{i}(X, m) for every m, because cycles in X × Δ^{m} which are equidimensional over Δ^{m} must meet every face properly. By ^{\underline{1a12}} 1A.12, the inclusion is compatible with the face maps, which are defined in ^{\underline{dfnzequi}} 16.1 and ^{\underline{hcgdef}} 17.1, so this yields an inclusion of simplicial groups, z_{equi}(X, d − i)(Δ^{•}) ⊆ z^{i}(X, •).

CHagree Exercise 17.3. (a) If d = dim X, show that every irreducible cycle in z^{d}(X, 1) is either disjoint from X × {0, 1} or else is quasi-finite over 𝔸^{1}. Use this to describe z^{d}(X, 1) → z^{d}(X, 0) explicitly and show that CH^{d}(X, 0) ≅ CH^{d}(X). (The group CH^{d}(X) is defined in ^{\underline{fulton}} [Ful84, 1.6].)

(b) Show that C_{∗}ℤ_{tr}(X)(Spec k) is a subcomplex of z^{d}(X, ∗). On homology, this yields maps H_{m}^{sing}(X/k) → CH^{d}(X, m). For m = 0, show that this is the surjection H_{0}^{sing}(X/k) → CH_{0}(X) = CH^{d}(X) of ^{\underline{h0ztr}} 2.19, which is an isomorphism when X is projective. By ^{\underline{hsingline}} 7.3, it is not an isomorphism when X is 𝔸^{1} or 𝔸^{1} − 0.

The push-forward of cycles makes the higher Chow groups covariant for finite morphisms (see ^{\underline{switchsm}} 17A.9). It also makes them covariant for proper morphisms (with the appropriate change in codimension index i; see ^{\underline{acahkt}} [Blo86, 1.3]).

At the chain level, it is easy to prove that the complexes z^{i}(−, ∗), and hence Bloch’s higher Chow groups, are functorial for flat morphisms. However, the complexes z^{i}(−, ∗) are not functorial for all maps. We will see in ^{\underline{hcgpst}} 17.20 below that the higher Chow groups are functorial for maps between smooth schemes.

We will need the following non-trivial properties of higher Chow groups:

hcghi 1. Homotopy Invariance: The projection p : X × 𝔸^{1} → X induces an isomor-

phism

CH^{i}(X, m) ^{≅} CH^{i}(X × 𝔸^{1}, m)

for any scheme X over k. The proof is given in ^{\underline{acahkt}} [Blo86, 2.1].

2. Localization Theorem: For any U ⊂ X open, the cokernel of z^{i}(X, •) → z^{i}(U, •) is acyclic. This is proven by Bloch in ^{\underline{hmov}} [Blo94]. (Cf. ^{\underline{acahkt}} [Blo86, 3.3].) If the complement Z = X − U has pure codimension c, it is easy to see that we have an exact sequence of simplicial abelian groups (and also of complexes of abelian groups):

0 → z^{i−c}(Z, •) → z^{i}(X, •) → z^{i}(U, •) → coker → 0.

Thus the localization theorem yields long exact sequences of higher Chow groups. The fact that we need to use Bloch’s Localization Theorem is unfortunate, because its proof is very hard and complex.

Transfers maps associated to correspondences are not defined on all of z^{i}(Y, ∗). We need to restrict to a subcomplex on which W^{∗} may be defined.

hcgtrans Definition 17.4. Let W be a finite correspondence from X to Y. Write z^{i}(Y, m)_{W} for the subgroup of z^{i}(Y, m) generated by the irreducible subvarieties T ⊂ Y × Δ^{m} such that the pullback X × T intersects W × Δ^{j} properly in X × Y × Δ^{m} for every face Δ^{j} ↪ Δ^{m}. By construction, z^{i}(Y, ∗)_{W} is a subcomplex of z^{i}(Y, ∗).

The proof of the following proposition, which is a refinement of the results in ^{\underline{mm}} [Lev98], is due to Marc Levine.

levineprop Proposition 17.5. Let W be a finite correspondence from X to Y, with Y affine. Then the inclusion z^{i}(Y, ∗)_{W} ⊂ z^{i}(Y, ∗) is a quasi-isomorphism.

Proof. (Levine) Let w : W → Y be a morphism of schemes with Y smooth, and W locally equidimensional but not necessarily smooth. Write z^{i}(Y, m)_{w} for the subgroup of z^{i}(Y, m) generated by the irreducible subvarieties T ⊂ Y × Δ^{m} for which every component of w^{−1}(T) has codimension at least i in W × Δ^{m} and intersects every face properly. By construction, z^{i}(Y, ∗)_{w} is a subcomplex of z^{i}(Y, ∗).

For example, if W is the support of a finite correspondence W, let w : W → Y be the natural map. Then W is locally equidimensional, and the group z^{i}(Y, m)_{w} is the same as the group z^{i}(Y, m)_{W} of ^{\underline{hcgtrans}} 17.4.

Levine proves that z^{i}(Y, m)_{w} ↪ z^{i}(Y, m) is a quasi-isomorphism on p.102 of ^{\underline{mm}} [Lev98] (in I.II.3.5.14), except that W is required to also be smooth in order to cite lemma I.II.3.5.2 of op. cit.. In loc. cit., a finite set {C_{j}} of locally closed irreducible subsets of Y and a sequence of integers m_{j} ≤ i is constructed, with the property that T is in z^{i}(Y, m)_{w} if and only if T is in z^{i}(Y, m) and the intersections of T with C_{j} × Δ^{p} have codimension at least m_{j} for all j and for every face Δ^{p} of Δ^{m}. A reading of the proof of lemma I.II.3.5.2 shows that in fact W need only be locally equidimensional.

ch Definition 17.6. Let W be a finite correspondence between two smooth schemes X and Y. For each cycle Y in z^{i}(Y, m)_{W}, we define the cycle W^{∗}(Y) on X × Δ^{m} to be:

W^{∗}(Y) = π_{∗}((W × Δ^{m}) · (X × Y)).

Here π : X × Y × Δ^{m} → X × Δ^{m} is the canonical projection.

For each W, it is clear that W^{∗} defines a homomorphism from the group z^{i}(Y, m)_{W} to the group of all cycles on X × Δ^{n}.

usualpb Example 17.7. Let f : X → Y be a morphism of smooth varieties, and let Γ_{f} be its graph. For Y in z^{i}(Y, 0)_{Γ_{f}}, Γ_{f}^{∗}(Y) is just the classical pullback of cycles f^{∗}(Y) defined in ^{\underline{serre}} [Ser65, V-28] (see ^{\underline{fpullback}} 17A.3). wstaragree Remark 17.8. The map W^{∗} of ^{\underline{ch}} 17.6 is compatible with the map W^{∗} defined in

\underline{wupperstar} 17A.7 in the following sense. Given W in Cor(X, Y), W × diag(Δ^{m}) is a finite correspondence from X × Δ^{m} to Y × Δ^{m}. If Y is a cycle in z^{i}(Y, m)_{W}, we may regard it as a cycle in Y × Δ^{m}. The projection formula ^{\underline{pf}} 17A.10 says that the following diagram commutes:

[figure: z^{i}(Y, m)_{W}; ⊂; z^{i}(Y × Δ^{m})_{W}; z^{i}(X, m); W^{∗}; ⊂; z^{i}(X × Δ^{m}).; (W × diag(Δ^{m}))^{∗}]

chain Lemma 17.9. The maps W^{∗} define a chain map z^{i}(Y, ∗)_{W} → z^{i}(X, ∗).

Proof. Let ∂_{j} : Δ^{m} ^{⊂} Δ^{m+1} be a face, and consider the following diagram, whose vertical compositions are W^{∗}:

z(Y × Δ^{m+1}) \overline{∂_{j}^{∗}} z(Y × Δ^{m})

[figure: z(X × Y × Δ^{m+1}); f^{∗}; ∂_{j}^{∗}; z(X × Y × Δ^{m}); f^{∗}]

[figure: z(X × Y × Δ^{m+1}); W × Δ^{m+1} · −; ∂_{j}^{∗}; z(X × Y × Δ^{m}); W × Δ^{m} · −]

[figure: z(X × Δ^{m+1}); π_{∗}; ∂_{j}^{∗}; z(X × Δ^{m}).; π_{∗}]

The horizontal maps ∂_{j}^{∗} are only defined for cycles meeting the face properly (see \underline{pullint} 17A.4) and the intersection products in the middle are only defined on cycles in good position for W. The top square commutes because of the functoriality of Bloch’s complex for flat maps, and the bottom square commutes by ^{\underline{switchsm}} 17A.9.

Suppose that Z is a cycle in X × Y × Δ^{m+1} which intersects the face X × Y × Δ^{m} as well as W × Δ^{m+1} and W × Δ^{m} properly. By ^{\underline{aps}} 17A.2:

W × Δ^{m} · ((X × Y × Δ^{m}) · Z) = X × Y × Δ^{m} · ((W × Δ^{m+1}) · Z).

That is, the middle square commutes for Z. Finally, if Y ∈ z^{i}(Y, m + 1)_{w}, the cycle (W × Δ^{m+1}) · f^{∗}Y is finite over X × Δ^{m+1}, so π_{∗} may be applied to it. A diagram chase now shows that W^{∗} is a morphism of chain complexes.

chWstar Corollary 17.10. If Y is affine, any finite correspondence W from X to Y induces maps W^{∗} : CH^{i}(Y, m) → CH^{i}(X, m) for all m.

Proof. On homology, ^{\underline{levineprop}} 17.5 and W^{∗} give: CH^{i}(Y, m) ≅ H_{m}(z^{i}(Y, ∗)_{W}) → H_{m}(z^{i}(X, ∗)) = CH^{i}(X, m). chaffine Example 17.11. If f : X → Y is a morphism of smooth varieties, and Y is

affine, we will write f^{∗} for the map Γ_{f}^{∗} from z^{i}(Y, m)_{Γ_{f}} to z^{i}(X, m), and also for the induced map from CH^{i}(Y, m) to CH^{i}(X, m). It agrees with Levine’s map f^{∗} (see pp. 67 and 102 of ^{\underline{mm}} [Lev98]). This is not surprising, since we are using lemma ^{\underline{levineprop}} 17.5, which is taken from p. 102 of ^{\underline{mm}} [Lev98]. The map f^{∗} may also be obtained from ^{\underline{acahkt}} [Blo86, 4.1] using ^{\underline{hmov}} [Blo94].

If f is flat, then f^{∗} is just the flat pullback of cycles. That is, if Y = [V] then f^{∗}(Y) is the cycle associated to the scheme f^{−1}(V). This fact is a special case of ^{\underline{pullint}} 17A.4.

We can now show that the higher Chow groups are functors on the subcategory of affine schemes in Cor_{k}.

affcomp Lemma 17.12. Let X, Y and Z be smooth affine schemes. Given finite correspondences W_{1} in Cor(X, Y) and W_{2} in Cor(Y, Z), then

(W_{2} ∘ W_{1})^{∗} = W_{1}^{∗}W_{2}^{∗} : CH^{i}(Z, m) → CH^{i}(X, m).

In particular, if f_{1} : X → Y and f_{2} : Y → Z are morphisms, then (f_{2}∘f_{1})^{∗} = f_{1}^{∗}f_{2}^{∗}.

Proof. By ^{\underline{levineprop}} 17.5 and ^{\underline{chaffine}} 17.11, it suffices to show that (W_{2} ∘W_{1})^{∗} = W_{1}^{∗}W_{2}^{∗} as maps from z^{i}(Z, m)_{W} → z^{i}(X, m), where W ∈ Cor(Y ∐ X, Z) is the coproduct of W_{2} and W_{2} ∘ W_{1}. An element of z^{i}(Z, m)_{W} is a cycle in z^{i}(Z, m) which is in good position with respect to both W_{2} and W_{2} ∘ W_{1}. Hence the result follows from theorem ^{\underline{composition}} 17A.13, given the reinterpretation in ^{\underline{wstaragree}} 17.8.

We now extend the definition of the transfer map W^{∗} from affine varieties to all smooth varieties using Jouanolou’s device ^{\underline{j}}[Jou73, 1.5] and ^{\underline{wh}} [Wei89, 4.4]: over every smooth variety X there is a vector bundle torsor X′ → X with X′ affine.

torsorhcg Lemma 17.13. Let X be a variety and p : X′ → X a vector bundle torsor. Then p^{∗} : CH^{∗}(X, ∗) → CH^{∗}(X′, ∗) is an isomorphism.

Proof. By definition, there is a dense open U in X so that p^{−1}(U) ≅ U × 𝔸^{n}. There is a commutative diagram

[figure: 0; z^{∗}(X′ − p^{−1}(U)); z^{∗}(X′, ∗); z^{∗}(p^{−1}(U)); 0; z^{∗}(X − U); z^{∗}(X, ∗); z^{∗}(U).]

By homotopy invariance of the higher Chow groups (see p. ^{\underline{hcghi}} 111), the right vertical arrow is a quasi-isomorphism. By Noetherian induction, the result is true for X − U, i.e., the left vertical arrow is a quasi-isomorphism. By the Localization Theorem and and the five lemma, p^{∗} : CH^{∗}(X, ∗) → CH^{∗}(X′, ∗) is an isomorphism. jouansurj Lemma 17.14. Let p : Y′ → Y be a vector bundle torsor and let X be affine.

• Every morphism f : X → Y has a lift f′ : X → Y′ such that pf′ = f.

• Every finite correspondence has a lift, i.e., p_{∗} : Cor(X, Y′) → Cor(X, Y) is surjective.

Proof. Clearly, X ×_{Y} Y′ → X is a vector bundle torsor. But X is affine and therefore every vector bundle torsor over X is a vector bundle (see ^{\underline{wh}} [Wei89, 4.2]). Define f′ : X → Y′ to be the composition of the zero-section of X×_{Y} Y′ followed by the projection. Clearly, pf′ = f.

Now suppose that W ⊂ X × Y is an elementary correspondence. Since W is finite over X, it is affine. By the first part of this proof, the projection p : W → Y lifts to a map p′ : W → Y′. Together with the projection W → X, p′ induces a lift i : W → X × Y′ of W ⊂ X × Y. Then i(W) is an elementary correspondence from X to Y′ whose image under p_{∗} is W.

jouan Lemma 17.15. Let X and Y be two smooth varieties over k and let p : X′ → X and q : Y′ → Y be vector bundle torsors with X′ and Y′ affine. Then for every finite correspondence W from X to Y, there exists a correspondence W′ from X′ to Y′ so that q ∘ W′ = W ∘ p in Cor_{k}(X′, Y).

[figure: X′; W′; Y′; X; p; W; Y; q]

Proof. Since Cor(X′, Y′) → Cor(X′, Y) is onto by ^{\underline{jouansurj}} 17.14, W∘p has a lift W′.

pstch Definition 17.16. Let X and Y be two smooth varieties over k and let W be a finite correspondence from X to Y. We define W^{∗} : CH^{i}(Y, m) → CH^{i}(X, m) as follows.

By Jouanolou’s device ^{\underline{j}}[Jou73, 1.5], there exist vector bundle torsors p : X′ → X and q : Y′ → Y where X′ and Y′ are affine. Both X′ and Y′ are smooth, because X and Y are. By lemma ^{\underline{torsorhcg}} 17.13, p^{∗} and q^{∗} are isomorphisms. By ^{\underline{jouan}} 17.15 there is a finite correspondence W′ from X′ to Y′ so that q ∘ W′ = W ∘ p in Cor_{k}(X′, Y). Since Y′ is affine, the map W′^{∗} : CH^{∗}(Y′, m) → CH^{∗}(X′, m) was defined in ^{\underline{chWstar}} 17.10. We set W^{∗} = (p^{∗})^{−1}W′^{∗}q^{∗} : CH^{∗}(Y, m) → CH^{∗}(X, m).

[figure: CH^{∗}(X′, ∗); W′^{∗}; CH^{∗}(Y′, ∗); CH^{∗}(X, ∗); p^{∗} ≅; W^{∗}; CH^{∗}(Y, ∗); q^{∗} ≅]

If f : X → Y is a morphism, we define f^{∗} : CH^{i}(Y, m) → CH^{i}(X, m) to be Γ_{f}^{∗}, that is, f^{∗} = (p^{∗})^{−1}(f′)^{∗}q^{∗}, where f′ : X′ → Y′ lies over f. affWagree Lemma 17.17. If X and Y are affine, the map defined in ^{\underline{pstch}} 17.16 agrees with

the map W^{∗} defined in ^{\underline{chWstar}} 17.10.

Proof. By ^{\underline{affcomp}} 17.12, the map defined in ^{\underline{pstch}} 17.16 equals:

(p^{∗})^{−1}W′^{∗}q^{∗} = (p^{∗})^{−1}(q ∘ W′)^{∗} = (p^{∗})^{−1}(W ∘ p)^{∗} = (p^{∗})^{−1}p^{∗}W^{∗} = W^{∗}.

indep Lemma 17.18. The definition of W^{∗} in ^{\underline{pstch}} 17.16 is independent of the choices.

Proof. Suppose given affine torsors X′′ → X and Y′′ → Y and a lift W′′ ∈ Cor(X′′, Y′′) of W. We have to show that W′ and W′′ induce the same map.

We begin by making two reductions. First, we may assume that X′ = X′′ and Y′ = Y′′, by passing to X′ ×_{X} X′′ and Y′ ×_{Y} Y′′ and choosing lifts of W′ and W′′. (This reduction uses ^{\underline{affWagree}} 17.17.)

We may also assume that X is affine and that X′ = X, by replacing X by X′. Thus we need to show that for any two lifts W_{0} and W_{1} of W, W_{0}^{∗}q^{∗} = W_{1}^{∗}q^{∗}.

By lemma ^{\underline{w1w2a1h}} 17.19, there is a finite correspondence W̃ so that the following diagram commutes:

X × 𝔸^{1}  \overline{W̃} Y′

[figure: W_{0}; W_{1}]

[figure: X; s_{0}; s_{1}; W; Y.; q]

Since s_{0} and s_{1} are both inverses to the projection p : X × 𝔸^{1} → X, we have s_{0}^{∗}p^{∗} = s_{1}^{∗}p^{∗} by ^{\underline{affcomp}} 17.12. Since higher Chow groups are homotopy invariant, p^{∗} is an isomorphism and we get s_{0}^{∗} = s_{1}^{∗}. Since X and Y′ are affine, and W_{i} = W̃∘s_{i}, we may apply ^{\underline{affcomp}} 17.12 again to get

W_{0}^{∗} = s_{0}^{∗}W̃^{∗} = s_{1}^{∗}W̃^{∗} = W_{1}^{∗}.

Recall from ^{\underline{a1he}} 2.23 that two correspondences W_{0} and W_{1} from X to Y are said to be 𝔸^{1}-homotopic, written W_{0} ≃ W_{1}, if they are the restrictions of an element of Cor(X × 𝔸^{1}, Y) along X × 0 and X × 1.

w1w2a1h Lemma 17.19. Let W be a finite correspondence between a smooth affine scheme X and a smooth Y. If q : Y′ → Y is a vector bundle torsor, then any two lifts W_{0} and W_{1} are 𝔸^{1}-homotopic.

Proof. Let V be the image of the union of the supports of W_{0} and W_{1} in X ×Y, and let V′ denote the fiber product of V and Y′ over Y; p : V′ → V is a vector bundle torsor. Since X is affine and the induced map V → X is finite, V is affine too. Hence p : V′ → V is a vector bundle. Fix a section s : V → V′.

[figure: V′; X × Y′; Y′; V; s; p; ⊂; X × Y; Y; q] Clearly, p is an 𝔸^{1}-homotopy equivalence (in the sense of ^{\underline{a1he}} 2.23) with inverse s, that is, sp is 𝔸^{1}-homotopic to the identity.

Both W_{0} and W_{1} induce correspondences W̃_{0} and W̃_{1} from X to V′. Now the composition g ∘(W̃_{i} ×𝔸^{1}) ∈ Cor(X ×𝔸^{1}, V′) is an 𝔸^{1}-homotopy from spW̃_{i} to W̃_{i}, for i = 0, 1. Since pW̃_{0} = pW̃_{1}, we have

W̃_{0} ≃ spW̃_{0} = spW̃_{1} ≃ W̃_{1}.

Since W_{i} is the composition of W̃_{i} with the map V′ → Y, W_{0} is 𝔸^{1}-homotopic to W_{1}.

At last, we have the tools to show that the higher Chow groups are presheaves with transfers, i.e., functors on the category Cor_{k} of smooth separated schemes over k.

hcgpst Theorem 17.20. The maps W^{∗} defined in ^{\underline{pstch}} 17.16 give the higher Chow groups

CH^{i}(−, m) the structure of presheaves with transfers.

That is, for any two finite correspondences W_{1} and W_{2} from X to Y and from Y to Z, respectively, and for all α ∈ CH^{i}(Z, m):

W_{1}^{∗}(W_{2}^{∗}(α)) = (W_{2} ∘ W_{1})^{∗}(α).

In particular, if f_{1} : X → Y and f_{2} : Y → Z are morphisms, then (f_{2} ∘ f_{1})^{∗} = f_{1}^{∗}f_{2}^{∗}.

Proof. By ^{\underline{jouan}} 17.15, there is a commutative diagram in Cor_{k} of the form

[figure: X′; W′_{1}; Y′; W′_{2}; Z′; X; p; W_{1}; Y; q; W_{2}; Z; r]

where the vertical maps are affine vector bundle torsors. By ^{\underline{affcomp}} 17.12, we have W′_{1}^{∗}W′_{2}^{∗} = (W′_{2} ∘ W′_{1})^{∗}. Since the definitions of W_{i}^{∗} and (W_{2} ∘ W_{1})^{∗} are independent of the choices by ^{\underline{indep}} 17.18, the statement now follows from an unwinding of ^{\underline{pstch}} 17.16:

W_{1}^{∗}W_{2}^{∗} = (p^{∗})^{−1}W′_{1}^{∗}q^{∗}(q^{∗})^{−1}W′_{2}^{∗}r^{∗} = (p^{∗})^{−1}(W′_{2} ∘ W′_{1})^{∗}r^{∗} = (W_{2} ∘ W_{1})^{∗}.

Appendix 17A- Cycle maps

If W is a finite correspondence from X to Y, we can define a map W^{∗} from “good” cycles on Y to cycles on X. The formula is to pull the cycle back to X × Y, intersect it with W, and push forward to X. In this appendix, we will make this precise, in ^{\underline{wupperstar}} 17A.7. First we must explain what makes a cycle “good”.

intmult Definition 17A.1. Two subvarieties Z_{1} and Z_{2} of X are said to intersect properly if every component of Z_{1} ∩ Z_{2} has codimension codim Z_{1} + codim Z_{2} in X. This is vacuously true if Z_{1} ∩ Z_{2} = ∅.

If the ambient variety X is regular, the intersection cycle Z_{1} · Z_{2} is defined to be the sum ∑ n_{j}[W_{j}], where the indexing is over the irreducible components W_{j} of Z_{1} ∩ Z_{2}, and the n_{j} are their (local) intersection multiplicities. Following Serre ^{\underline{serre}} [Ser65], the multiplicity n_{j} is defined as follows. If A is the local ring of X at the generic point of W_{j}, and I_{l} are the ideals of A defining Z_{l}, then

n_{j} = ∑_{i}(−1)^{i}length Tor _{i}^{A}(A/I_{1}, A/I_{2}).

If X is not regular, the multiplicity will only make sense when only finitely many Tor-terms are non-zero.

We say that two equidimensional cycles V = ∑ m_{i}V_{i} and W = ∑ n_{j}W_{j} intersect properly if each V_{i} and W_{j} intersect properly. In this case, the intersection cycle V · W is defined to be ∑ m_{j}n_{j}(V_{i} · W_{j}).

aps Exercise 17A.2. Let V_{1}, V_{2} and V_{3} be three cycles on a smooth scheme X. Show that (V_{1} · V_{2}) · V_{3} = V_{1} · (V_{2} · V_{3}) whenever both sides are defined. (This is proven in ^{\underline{serre}} [Ser65, V-24].)

fpullback Definition 17A.3. Suppose that f : X → Y is a morphism with X and Y regular, and that Y is a codimension i cycle on Y. We say that f^{∗}(Y) is defined if each component of f^{−1}(Supp(Y)) has codimension ≥ i in X. As in ^{\underline{serre}} [Ser65, V-28], we define the cycle f^{∗}(Y) to be Γ_{f} · (X × Y) (see ^{\underline{intmult}} 17A.1), identifying the graph Γ_{f} with X.

\underline{week17a.tex}

As noted in ^{\underline{serre}} [Ser65, V-29], the intersection cycle makes sense even if X is not regular, since the multiplicities may be computed over Y by flat base change for Tor (see ^{\underline{w}}[Wei94, 3.2.9]). pullint Example 17A.4. If f is flat and Y = [V], then f^{∗}(Y) is the cycle associated

to the scheme f^{−1}(V). If X is a subvariety of Y, then the cycle f^{∗}(Y) on X is the same as the cycle X ·Y considered as a cycle on X. If X ^{⊂} Y is a regular embedding, the coefficients of f^{∗}(Y) agree with the intersection multiplicities defined in ^{\underline{fulton}} [Ful84, 7.1.2].

goodpos Definition 17A.5. Let f : Y′ → Y be a morphism of smooth varieties and W a cycle on Y′. We say that a cycle Y on Y is in good position for W (relative to f) if the cycle f^{∗}(Y) is defined, and intersects W properly on Y′. If Y is in good position for W, the intersection product W · f^{∗}Y is defined (see ^{\underline{intmult}} 17A.1). If the map f is flat, the cycle f^{∗}(Y) is always defined.

Let W be an irreducible subvariety of Y′ and let w be the composition W → Y′ → Y. By ^{\underline{intmult}} 17A.1 and ^{\underline{fpullback}} 17A.3, a codimension i cycle Y is in good position for W if and only if codim_{W} w^{−1}(Supp(Y)) ≥ i, that is, if w^{∗}(Y) is defined.

As a special case, we will say that a cycle Y is in good position for a finite correspondence W from X to Y if Y is in good position for the cycle underlying W, relative to the projection X × Y → Y.

push Remark 17A.6. Let f : X → Y be a morphism of smooth varieties and let Z be a cycle on X, supported on a closed subscheme Z so that the composition Z → X → Y is a proper map. It is clear that f_{∗}(Z) is well-defined even though f is not proper.

wupperstar Definition 17A.7. Let W be a finite correspondence between two smooth schemes X and Y. For every cycle Y on Y in good position for W, we define

W^{∗}(Y) = π_{∗}(W · f^{∗}Y),

where f : X × Y → Y and π : X × Y → X are the canonical projections. The intersection and the push-forward are well-defined by ^{\underline{goodpos}} 17A.5 and ^{\underline{push}} 17A.6. The map W^{∗} induces the transfer map for Chow groups, see ^{\underline{chWstar}} 17.10 and ^{\underline{pstch}} 17.16.

For any smooth T, W × T is a finite correspondence from X × T to Y × T over T. By abuse of notation, we shall also write W^{∗} for (W × T)^{∗}.

Example 17A.8. We can now reinterpret the composition of correspondences. If W_{1} and W_{2} are finite correspondences from X to Y and from Y to Z, respectively, we have:

W_{2} ∘ W_{1} = (W_{1} × Z)^{∗}(W_{2}) = (X × W_{2})^{∗}(W_{1}).

Here are two formulas which are useful in the study of W^{∗}.

switchsm Lemma 17A.9. Consider the following diagram of varieties

[figure: X′; g′; X; Y′; f′; g; Y; f]

where the square is fiber and both X and Y are smooth. Let X be a cycle on X whose support is finite over Y and for which (g′)^{∗}X is defined. Then g^{∗}f_{∗}X is defined and g^{∗}f_{∗}X = f′_{∗}(g′)^{∗}X.

Proof. If V is a component of X, then the map f : V → f(V) is finite. Hence f′ : (g′)^{−1}(V) → g^{−1}(f(V)) is finite too, so that codim(g′)^{−1}(V) = codim g^{−1}(f(V)). By hypothesis, codim(g′)^{−1}(V) ≥ i, which proves that g^{∗}f_{∗}X is defined. The equality now follows from ^{\underline{fultsing}} [Ful75, 2.2(4)].

pf Lemma 17A.10 (Projection Formula). Let f : X → Y be a morphism of smooth schemes. Suppose given a cycle X on X, whose support is finite over Y, and a cycle Y on Y which is in good position for X (see ^{\underline{goodpos}} 17A.5). Then f_{∗}X and Y intersect properly, and the projection formula holds:

f_{∗}(X · f^{∗}Y) = f_{∗}X · Y.

Proof. Since the restriction of f to the support of X is finite, it is clear that f_{∗}(X) and Y intersect properly too. The result is now a consequence of the basic identity 2.2(2) of ^{\underline{fultsing}} [Ful75], or ^{\underline{serre}} [Ser65, V-30].

17a.8 Exercise 17A.11. Let i be the inclusion of a closed subvariety W in a smooth scheme X and let f : X → Y be a map of smooth schemes. Prove that if Y is a cycle on Y so that both f^{∗}Y and (fi)^{∗}(Y) are defined, then i_{∗}(fi)^{∗}(Y) = W · f^{∗}Y. Hint: Use ^{\underline{serre}} [Ser65, V-30] or ^{\underline{fultsing}} [Ful75, 2.2(2)].

Recall from ^{\underline{a7}} 1A.8 that if V → Y is a morphism with Y regular, then the pull-back Z_{V} of a relative cycle Z in T × Y is a well defined cycle on T × V with integer coefficients.

17a.9 Lemma 17A.12. Let T and Y be regular and let Z be a cycle in T × Y which is dominant equidimensional over Y. If f : V → Y is a morphism, then the pull-back Z_{V} agrees with the pull-back cycle (f × T)^{∗}(Z). used in ^{\underline{nut}} 19.6

Proof. Note that Z is a relative cycle by ^{\underline{a4}} 1A.5, so that Z_{V} is defined. Its coefficients are characterized by the equalities (Z_{V})_{v} = Z_{f(v)} for every v ∈ V. By ^{\underline{relchow}} [RelCy, 3.5.8 and 3.5.9], the coefficients of Z_{V} are the same as the multiplicities in ^{\underline{intmult}} 17A.1, i.e., the coefficients of (f × T)^{∗}(Z) given by ^{\underline{fpullback}} 17A.3.

composition Theorem 17A.13. Let W_{1} and W_{2} be two finite correspondences from X to Y and from Y to Z, respectively. Suppose that Z is a cycle on Z which is in good position with respect to both W_{2} and W_{2} ∘ W_{1}. Then

(W_{2} ∘ W_{1})^{∗}(Z) = W_{1}^{∗}(W_{2}^{∗}(Z)).

The term W_{1}^{∗}(W_{2}^{∗}(Z)) makes sense by the following lemma.

goodposcomp Lemma 17A.14. Let Z be in good position for W_{2} and W_{2} ∘W_{1}. Then W_{2}^{∗}(Z) is in good position with respect to W_{1}.

Proof. We may assume that the correspondences are elementary, i.e., W_{1} and W_{2} are subvarieties W_{1} and W_{2} of X ×Y, and Y ×Z, respectively. In this spirit, we will write W_{2} ∘ W_{1} for the subvariety of X × Z which is the support of the composition of correspondences, W_{2} ∘ W_{1}. Consider the following diagram:

W_{2} ∘ W_{1}

[figure: c]

[figure: W_{1} ×_{Y} W_{2}; u; e; W_{2}; d; Z]

[figure: W_{1}; q; b; Y; p]

By hypothesis, codim d^{−1}(Z) ≥ codim Z and codim c^{−1}(Z) ≥ codim Z.

We claim that codim b^{−1}pd^{−1}Z ≥ codim Z. Since the central square is cartesian, b^{−1}p = qe^{−1}. Since q is finite, this yields

codim b^{−1}pd^{−1}Z = codim qe^{−1}d^{−1}Z = codim e^{−1}d^{−1}Z.

But e^{−1}d^{−1} = u^{−1}c^{−1}, and u is finite, so:

codim e^{−1}d^{−1}Z = codim u^{−1}c^{−1}Z = codim c^{−1}Z.

But codim c^{−1}Z ≥ codim Z by hypothesis, as claimed.

Proof of ^{\underline{composition}} 17A.13. The right side is defined by ^{\underline{goodposcomp}} 17A.14. We will follow the notation established in figure ^{\underline{compcorr}} 17A.1, where we have omitted the factor Δ^{n} in every entry to simplify notation. Note that the central square is cartesian.

By definition ^{\underline{ch}} 17.6, we have

W_{1}^{∗}(W_{2}^{∗}(Z)) = r_{∗}(W_{1} · b^{∗}(p_{∗}(W_{2} · d^{∗}Z))).

Since the central square is cartesian, we have b^{∗}p_{∗} = q_{∗}e^{∗} by ^{\underline{switchsm}} 17A.9. Since the pull-back e^{∗} is a ring homomorphism, we have

b^{∗}(p_{∗}(W_{2} · d^{∗}Z)) = q_{∗}(e^{∗}(W_{2} · d^{∗}Z)) = q_{∗} (e^{∗}(W_{2}) · e^{∗}d^{∗}Z) .

Consider the two cycles X = e^{∗}(W_{2}) · (de)^{∗}(Z) and Y = W_{1} and the function q. The intersection X · q^{∗}Y = e^{∗}(W_{2}) · (de)^{∗}(Z) · q^{∗}(W_{1}) is proper because Z is

X × Z

[figure: u; c; v]

[figure: X × Y × Z; e; Y × Z; d; Z]

[figure: X × Y; q; b; Y; p]

X\begin{matrix}r \\  \end{matrix}
Figure 17A.1: Composition of correspondences  compcorr

in good position with respect to W_{2} ∘ W_{1}. Therefore the conditions for ^{\underline{pf}} 17A.10 are satisfied, and the projection formula yields Y · q_{∗}X = q_{∗}(q^{∗}Y · X), i.e.,

W_{1}^{∗}(W_{2}^{∗}(Z)) = r_{∗}q_{∗} (q^{∗}(W_{1}) · (e^{∗}(W_{2}) · e^{∗}d^{∗}Z)) .

Since the push-forward and pullback are functorial, we have r_{∗}q_{∗} = v_{∗}u_{∗} and e^{∗}d^{∗} = u^{∗}c^{∗}. Our cycle then becomes

v_{∗}u_{∗} (q^{∗}(W_{1}) · e^{∗}(W_{2}) · u^{∗}c^{∗}Z) .

We may use the projection formula (^{\underline{pf}} 17A.10) once again, this time for u^{∗}, with X = q^{∗}(W_{1}) · e^{∗}(W_{2}) and Y = c^{∗}Z (the conditions are satisfied by the same argument we used above). This yields u_{∗}(X · u^{∗}Y) = (u_{∗}X) · Y, i.e.,

W_{1}^{∗}(W_{2}^{∗}(Z)) = v_{∗} (u_{∗}(q^{∗}(W_{1}) · e^{∗}(W_{2})) · c^{∗}Z) .

Since the composition of W_{1} and W_{2} as correspondences is exactly u_{∗}(q^{∗}(W_{1}) · e^{∗}(W_{2})), the last equation becomes

W_{1}^{∗}(W_{2}^{∗}(Z)) = v_{∗}((W_{2} ∘ W_{1}) · c^{∗}Z) = (W_{2} ∘ W_{1})^{∗}(Z).

This concludes the proof of ^{\underline{composition}} 17A.13.

Higher Chow groups and equidimensional cycles

The next step in the proof of theorem ^{\underline{10.2}} 19.1 (that motivic cohomology and higher Chow groups agree) is the reduction to equidimensional cycles. The main references for this lecture are ^{\underline{higher}} [HiCG] and ^{\underline{FS}} [FS00].

dfnzequim Definition 18.1. For an equidimensional X, and i ≤ dim X, we write z_{equi}^{i}(X, m) for z_{equi}(X, dim X − i)(Δ^{m}), the free abelian group generated by all codimension i subvarieties on X × Δ^{m} which are dominant and equidimensional over Δ^{m} (of relative dimension dim X − i). We write z_{equi}^{i}(X, •) and z_{equi}^{i}(X, ∗) for the simplicial abelian group m ↦ z_{equi}^{i}(X, m) and its associated chain complex, respectively.

By ^{\underline{zequizi}} 17.2, z_{equi}^{i}(X, m) is a subgroup of z^{i}(X, m) and z_{equi}^{i}(X, •) is a simplicial subgroup of z^{i}(X, •).

Example 18.2. The inclusion z_{equi}^{i}(X, ∗) ⊂ z^{i}(X, ∗) will not be a quasiisomorphism in general. Indeed, if i > d then z_{equi}^{i}(X, m) = 0 while z^{i}(X, m) is not generally zero. For example, consider X = Spec k. If i > 0 we have z_{equi}^{i}(Spec k, ∗) = 0. In contrast, z^{i}(Spec k, i) is the group of points on Δ^{i} which do not lie on any proper face. We will see in ^{\underline{thm1}} 19.7 that H_{i}z^{i}(Spec k, ∗) = H^{i,i}(Spec k) = K_{i}^{M}(k).

equiqiso Theorem 18.3. (Suslin ^{\underline{higher}} [HiCG, 2.1]) Let X be an equidimensional affine scheme of finite type over k, then the inclusion map:

z_{equi}^{i}(X, ∗) ^{⊂} z^{i}(X, ∗)

is a quasi-isomorphism for i ≤ dim X.

affhcg Corollary 18.4. Let X be an affine variety, then for all i ≥ 0

CH^{i}(X, m) = H_{m}(z_{equi}^{i}(X × 𝔸^{i}, ∗)).

In particular, CH^{i}(Spec k, m) = H_{m}(z_{equi}^{i}(𝔸^{i}, ∗)).

Proof. This is an immediate corollary of ^{\underline{equiqiso}} 18.3, definition ^{\underline{hcgdef}} 17.1 and the homotopy invariance of the higher Chow groups; see page ^{\underline{hcghi}} 111.

We need lemmas ^{\underline{lemma1}} 18.7, ^{\underline{lemma2}} 18.13 and ^{\underline{lemma2prime}} 18.13 to prove theorem ^{\underline{equiqiso}} 18.3. All of their proofs rely on a technical theorem ^{\underline{lemma4}} 18A.1, which will be proven in the appendix.

We begin by introducing some auxiliary notions. Let X be a scheme over S.

skel Definition 18.5. An N−skeletal map φ over X, relative to X → S, is a collection {φ_{n} : X × Δ^{n} → X × Δ^{n}}_{n=0}^{N} of S-morphisms, such that φ_{0} is the identity 1_{X} and for every face map ∂_{j} : Δ^{n−1} → Δ^{n} with n ≤ N the following diagram commutes.

[figure: X × Δ^{n−1}; φ_{n−1}; X × Δ^{n−1}; X × Δ^{n}; 1_{X} × ∂_{j}; φ_{n}; X × Δ^{n}; 1_{X} × ∂_{j}]

Note that φ_{N} determines φ_{n} for all n < N. When S = X, we shall just call φ an n-skeletal map over X.

The condition that an (N − 1)-skeletal map over X can be extended to an N-skeletal map is a form of the homotopy extension property, and follows from the Chinese Remainder Theorem when X is affine.

For example, a 1-skeletal map over X = Spec R (relative to S = X) is determined by a polynomial f ∈ R[t] such that f(0) = 0 and f(1) = 1; φ_{1} is Spec of the R-algebra map R[t] → R[t] sending t to f.

skelzi Definition 18.6. Given an N-skeletal map φ over X and n ≤ N, we define φz^{i}(X, n) to be the subgroup of z^{i}(X, n) generated by all V in X × Δ^{n} such that φ_{n}^{∗}(V) is defined and is in z^{i}(X, n). If n > N we set φz^{i}(X, n) = 0. In other words, φz^{i}(X, n) is the group of cycles in X × 𝔸^{n} which intersect all the faces properly and whose pullbacks along φ_{n} intersect all the faces properly.

By definition ^{\underline{skel}} 18.5 we know that the face map ∂_{j} : z^{i}(X, n) → z^{i}(X, n − 1) sends φz^{i}(X, n) to φz^{i}(X, n − 1). Thus φz^{i}(X, ∗) is a chain subcomplex of z^{i}(X, ∗). Moreover it follows from ^{\underline{skel}} 18.5 that the φ_{n}^{∗} assemble to define a chain map φ^{∗} : φz^{i}(X, ∗) → z^{i}(X, ∗).

Similarly, we can define φz_{equi}^{i}(X, n) to be the subgroup of z_{equi}^{i}(X, n) generated by all V such that φ_{n}^{∗}(V) is defined and is in z_{equi}^{i}(X, n). The same argument shows that φz_{equi}^{i}(X, ∗) is a subcomplex of z_{equi}^{i}(X, n) and that the φ_{n} form a chain map φ^{∗} : φz_{equi}^{i}(X, ∗) → z_{equi}^{i}(X, ∗).

lemma1 Lemma 18.7. (See ^{\underline{higher}} [HiCG, 2.8]) Let C_{∗} be a finitely generated subcomplex in z^{i}(X, ∗) with i ≤ dim X. Choose N so that C_{n} = 0 for n > N. Then there is an N-skeletal map φ over X such that C_{∗} ⊆ φz^{i}(X, ∗), and the chain map φ^{∗} : φz^{i}(X, ∗) → z^{i}(X, ∗) satisfies

φ^{∗}C_{∗} ⊆ z_{equi}^{i}(X, ∗).

[figure: 0; φz^{i}(X, 1); φz^{i}(X, 0); 0; z^{i}(X, 2); z^{i}(X, 1); i; ∩; φ_{1}^{∗}; z^{i}(X, 0); =; 0.]

Figure 18.1: A 1-skeletal map φ and its chain map φ^{∗}.

Example 18.8. If N = 1, and α ∈ k − {0, 1}, the subvariety V = X × {α} of X × 𝔸^{1} is in z^{1}(X, 1) but not z_{equi}^{1}(X, 1). If X = Spec R, fix r ∈ R and let φ_{1} : X × 𝔸^{1} → X × 𝔸^{1} be the 1-skeletal map defined by the R-algebra map R[t] → R[t] sending t to f(t) = t + r(t^{2} − t). The condition that φ_{1}^{∗}(V) is in z_{equi}^{1}(X, 1), i.e., dominant and equidimensional over Δ^{1}, is equivalent to the condition that the map r : X → 𝔸^{1} is equidimensional, i.e., that r − β be nonzero in the domain R for all β ∈ k. Indeed, the fibre of φ_{1}^{−1}(V) over t ≠ 0, 1 is supported on R/(r − (α − t/t^{2} − t)), and is empty if t = 0, 1. Such r always exists when dim X ≥ 1.

Proof of ^{\underline{lemma1}} 18.7. Suppose that C_{n} is generated by \{V_{n}^{k}\} ⊆ z^{i}(X, n). Set d = dim X − i and note that d ≥ 0 since i ≤ dim X. Then V_{n} = ∪V_{n}^{k} is closed in X × Δ^{n} of dimension n + d.

We proceed by induction on N. Since N is finite, we may assume that the ∂_{j}(V_{n}^{k}) are supported in V_{n−1}. Inductively, we may suppose that we have constructed an (N −1)-skeletal map {φ_{n}} such that the fibers of the projections φ_{n}^{−1}(V_{n}) → Δ^{n} have dimension ≤ d. Let ∂Δ^{N} be the union of the faces Δ^{N}. The compatibility granted by definition ^{\underline{skel}} 18.5 implies that these maps fit together to form a map from X×∂Δ^{N} to itself such that the fibers of φ^{−1}(X×∂Δ^{N})∩V_{N} → ∂Δ^{N} have dimension ≤ d. By Generic Equidimensionality ^{\underline{lemma4}} 18A.1, this map extends to a N-skeletal map φ_{N} : X × Δ^{N} → X × Δ^{N} over X such that the fibers of φ_{N}^{−1}(V_{N}) → 𝔸^{N} have dimension ≤ d. Because each component W of φ^{−1}(V_{n}^{k}) satisfies the inequality dim W ≤ n + d = dim V_{n}^{k}, each cycle φ_{n}^{∗}(V_{n}^{k}) is defined and lies in z_{equi}^{i}(X, n). Since C_{n} is generated by the V_{n}^{k}, it lies in φz^{i}(X, n) and satisfies φ^{∗}(C_{n}) ⊂ z_{equi}^{i}(X, n).

skelh Definition 18.9. Let φ^{0} and φ^{1} be N-skeletal maps over X. An N−skeletal homotopy Φ between φ^{0} and φ^{1} is an N-skeletal map {Φ_{n} : X × Δ^{n} × 𝔸^{1} → X × Δ^{n} × 𝔸^{1}}_{n=0}^{N} over X × 𝔸^{1} relative to the projection X × 𝔸^{1} → X, which is compatible with the φ^{j} in the sense that the following diagram commutes for every n.

[figure: X × Δ^{n}; i_{0}; X × Δ^{n} × 𝔸^{1}; i_{1}; X × Δ^{n}; X × Δ^{n}; φ_{n}^{0}; i_{0}; X × Δ^{n} × 𝔸^{1}; Φ_{n}; i_{1}; X × Δ^{n}; φ_{n}^{1}]

The subgroup Φz^{i}(X, n) of z^{i}(X, n) is defined to be the subgroup generated by all V in X × Δ^{n} such that (φ^{0})^{∗}(V), (φ^{1})^{∗}(V) and Φ^{∗}(V × 𝔸^{1}) are all defined. As in definition ^{\underline{skelzi}} 18.6, Φz^{i}(X, ∗) is a subcomplex of z^{i}(X, ∗). In fact, Φz^{i} is in (φ^{0}z^{i}) ∩ (φ^{1}z^{i}).

skel01 Lemma 18.10. If Φ is an N-skeletal homotopy between φ^{0} and φ^{1}, then the maps (φ^{0})^{∗} and (φ^{1})^{∗} from Φz^{i}(X, ∗) to z^{i}(X, ∗) are chain homotopic.

Proof. For 0 ≤ j ≤ n, let h_{j} denote the composite

[figure: X × Δ^{n+1}; 1_{X} × θ_{j}; X × Δ^{n} × 𝔸^{1}; Φ_{n}; X × Δ^{n} × 𝔸^{1}; pr; X × Δ^{n},]

where the isomorphisms θ_{i} : Δ^{n+1} → Δ^{n} × 𝔸^{1} were defined in ^{\underline{theta}} 2.15, and pr is the projection. That is, for V in Φz^{i}(X, n) we define

h_{j}^{∗}[V] = (1_{X} × θ_{j})^{∗}Φ_{n}^{∗}[V × 𝔸^{1}] ∈ z^{i}(X, n + 1).

The h_{j}^{∗} form a simplicial homotopy (see ^{\underline{w}}[Wei94, 8.3.11]) from ∂_{0}h_{0} = φ^{1} to ∂_{n+1}h_{n} = φ^{0}. Hence their alternating sum h = ∑(−1)^{j}h_{j}^{∗} satisfies h∂ + ∂h = (φ^{1})^{∗} − (φ^{0})^{∗}. (This is illustrated in figure ^{\underline{iphiche}} 18.2 when N = 2.)

[figure: 0; Φz^{i}(X, 2); ∂; Φz^{i}(X, 1); ∂; Φz^{i}(X, 0)]

[figure: h; h; 0]

[figure: z^{i}(X, 3); ∂; z^{i}(X, 2); φ_{2}^{1}; ∩; φ_{2}^{0}; ∂; z^{i}(X, 1); φ_{1}^{1}; ∩; φ_{1}^{0}; ∂; z^{i}(X, 0); φ_{0}^{1}; ∩; φ_{0}^{0} = 1_{X}]

Figure 18.2: The chain homotopy between i and φ^{∗} when N = 2.  iphiche

lemma1h Proposition 18.11. Let φ be an N-skeletal map, and {V_{n}^{k}} a finite set of varieties in φz^{i}(X, n), n ≤ N. Then there exists an N-skeletal homotopy Φ between φ and the identity map, such that each Φ^{∗}(V_{n}^{k} × 𝔸^{1}) is defined and lies in z_{equi}^{i}(X × 𝔸^{1}, n).

Proof. Set d = dim(X)−i. As in the proof of ^{\underline{lemma1}} 18.7, we construct Φ_{n} by induction on n. Inductively, we are given an (N −1)-skeletal map Φ_{n} such that the fibers of the projections Φ^{−1}(V_{n}^{k} × 𝔸^{1}) → Δ^{n} × 𝔸^{1} have dimension ≤ d. Let ∂(Δ^{N} × 𝔸^{1}) denote the union of (∂Δ^{N}) × 𝔸^{1} and Δ^{N} × {0, 1}. The compatibility with the faces of Δ^{N} and with i_{0}, i_{1} granted by definition ^{\underline{skelh}} 18.9 implies that the Φ_{n} and φ_{N} fit together to form a map ∂Φ_{N} from X × ∂(Δ^{N} × 𝔸^{1}) to itself such that the fibers of

∂Φ_{N}^{−1}(X × ∂(Δ^{N} × 𝔸^{1}) ∩ V_{N}^{k} × 𝔸^{1}) → Δ^{N} × 𝔸^{1}

have dimension ≤ d. By Generic Equidimensionality ^{\underline{lemma4}} 18A.1, with 𝔸^{n} = Δ^{N} ×𝔸^{1}, this map extends to a map Φ_{N} from X × Δ^{N} × 𝔸^{1} to itself which extends ∂Φ_{N} (i.e., an N-skeletal homotopy from the identity to φ over X), such that the fibers of Φ_{N}^{−1}(V_{N}^{k} × 𝔸^{1}) → Δ^{N} × 𝔸^{1} have dimension ≤ d over all points of Δ^{N} × 𝔸^{1} not on ∂(Δ^{N} × 𝔸^{1}). This completes the inductive step, and shows that each cycle Φ_{n}^{∗}(V_{n}^{k} × 𝔸^{1}) is defined and lies in z_{equi}^{i}(X × 𝔸^{1}, n).

In order to simplify the proof of theorem ^{\underline{equiqiso}} 18.3 we need to introduce the “topological” notion of weak homotopy.

Definition 18.12. Two maps f, g : K → L of complexes of abelian groups are called weakly homotopic if for every finitely generated subcomplex C of K, the restrictions f|_{C} and g|_{C} are chain homotopic.

It is easy to check that weakly homotopic maps induce the same maps on homology. If K and L are bounded complexes of free abelian groups, this notion is equivalent to the usual notion of chain homotopy between maps. To see that this notion is weaker than chain homotopy, consider a pure subgroup A of B which is not a summand, such as ⊕_{1}^{∞}ℤ ⊂ ∏_{1}^{∞} ℤ. Then the canonical map from (A → B) to (A → 0) is weakly homotopic to zero but not chain contractible.

lemma2prime lemma2 Lemma 18.13. (See ^{\underline{higher}} [HiCG, 2.3 and 2.6]) Let φ be an N-skeletal map over X. Then the maps i and φ^{∗} are are weakly homotopic on φz^{i}:

φz^{i}(X, ∗) [figure: ⊂; i; φ^{∗}] z^{i}(X, ∗),

and also on φz_{equi}^{i}:

φz_{equi}^{i}(X, ∗) [figure: ⊂; i; φ^{∗}] z_{equi}^{i}(X, ∗).

Note that the following diagram commutes

[figure: φz_{equi}^{i}(X, −); ⊂; φz^{i}(X, −); z_{equi}^{i}(X, −); i; ∩; φ^{∗}; ⊂; z^{i}(X, −).; i; ∩; φ^{∗}]

Week 13 - February 2, 2000

Moreover if a ∈ φz^{i}(X, n) ∩ z_{equi}^{i}(X, n), and φ^{∗}a ∈ z_{equi}^{i}(X, n), then a ∈ φz_{equi}^{i}(X, n).

Proof of ^{\underline{equiqiso}} 18.3. We have to prove that the induced map on homology classes is an isomorphism:

H_{n}(z_{equi}^{i}(X, ∗)) → H_{n}(z^{i}(X, ∗))  (18.13.1) iso

First we prove surjectivity. Let a ∈ z^{i}(X, n) be such that d(a) = 0. Lemma \underline{lemma1} 18.7 provides an integer N and an N-skeletal map \{φ_{n}\} such that a ∈ φz^{i}(X, n) and φ^{∗}(a) ∈ z_{equi}^{i}(X, n). By ^{\underline{lemma2}} 18.13, a−φ^{∗}a is a boundary in z^{i}(X, n), i.e., a and φ^{∗}(a) represent the same class in homology. Hence the map ^{\underline{iso}} 18.13.1 is surjective.

For injectivity we need to consider a ∈ z_{equi}^{i}(X, n) so that d(a) = 0 and b ∈ z^{i}(X, n + 1) with d(b) = a. Apply lemma ^{\underline{lemma1}} 18.7 to b and a. We find an (n + 1)-skeletal map φ such that a, b ∈ φz^{i}(X, ∗) and φ^{∗}a, φ^{∗}b ∈ z_{equi}^{i}(X, ∗). But now we have:

φ^{∗}a = φ^{∗}(db) = d(φ^{∗}b) = 0.

From lemma ^{\underline{lemma2prime}} 18.13, a and φ^{∗}a = 0 represent the same class in the homology of z_{equi}^{i}(X, ∗). Therefore a is a boundary in z_{equi}^{i}(X, ∗). Hence the map (^{\underline{iso}} 18.13.1) is also injective.

Now we shall prove lemma ^{\underline{lemma2}} 18.13 using ^{\underline{lemma4}} 18A.1.

Proof of ^{\underline{lemma2}} 18.13. Consider a subcomplex C_{∗} ^{⊂} φz^{i}(X, ∗) generated by some closed irreducible subvarieties V_{n}^{k} so that ∂_{j}(V_{n}^{k}) is a linear combination of generators. It suffices to prove that the inclusion of C_{∗} into z^{i}(X, ∗) is homotopic to φ^{∗}. But this is just an application of ^{\underline{lemma1h}} 18.11 and ^{\underline{skel01}} 18.10.

Lemma 18.14. (See ^{\underline{higher}} [HiCG] 2.4) For any V and φ_{n} as before and any Δ^{m} → Δ^{n} × 𝔸^{1}, (id_{X} × θ)^{−1}(V × 𝔸^{1}) has dimension ≤ m + t. check

So θ^{∗}(V) is well-defined and we can verify that h is an homotopy.

Appendix 18A- Generic Equidimensionality

This appendix is devoted to a proof of the following Generic Equidimensionality Theorem, due to Suslin. (See ^{\underline{higher}} [HiCG] 1.1.)

lemma4 Theorem 18A.1. Let S be an affine scheme of finite type over a field. Let V be a closed subscheme of S×𝔸^{n}, Z an effective divisor of 𝔸^{n} and φ : S×Z → S×𝔸^{n} any morphism over S. For every t ≥ 0 so that dim V ≤ n + t, there exists a map Φ : S × 𝔸^{n} → S × 𝔸^{n} over S so that:

1. Φ|_{S×Z} = φ;

2. the fibers of the projection Φ^{−1}(V) → 𝔸^{n} have dimension ≤ t over the points of 𝔸^{n} − Z.

The S-morphism φ : S × Z → S × 𝔸^{n} is determined by its component φ′ : S ×Z → 𝔸^{n}. If S ⊂ 𝔸^{m}, we can extend φ′ to a morphism ψ′ : 𝔸^{m} ×Z → 𝔸^{n}. If we knew the theorem for 𝔸^{m}, there would exist an extension Ψ′ : 𝔸^{m}×𝔸^{n} → 𝔸^{n} of ψ′ such that, setting Ψ(X, Y) = (X, Ψ′(X, Y)), the fibers of Ψ^{−1}(V) → 𝔸^{n} over points of 𝔸^{n} − Z have dimension ≤ t, and the restriction Φ of Ψ to S × 𝔸^{n} would satisfy the conclusion of the theorem. Thus we may suppose that S = 𝔸^{m}.

Write 𝔸^{m} = Spec k[x_{1}, …, x_{m}] and 𝔸^{n} = Spec k[y_{1}, …, y_{n}]. If the divisor Z is defined by a polynomial h ∈ k[Y] then the component φ′ : 𝔸^{m} × Z → 𝔸^{n} of φ extends to f = (f_{1}, …, f_{n}) : 𝔸^{m} × 𝔸^{n} → 𝔸^{n} for polynomials f_{i} ∈ k[X, Y] defined up to a multiple of h. For each n-tuple F = (F_{1}, …, F_{n}) of homogeneous forms in k[X] of degree N, consider the maps

Φ_{F} : 𝔸^{m} × 𝔸^{n} → 𝔸^{n}
Φ_{F} (X, Y) = (f_{1}(X, Y) + h(Y)F_{1}(X), …, f_{n}(X, Y) + h(Y)F_{n}(X)).

By construction, the restriction of Φ_{F} to Z × S is φ′, i.e., property (1) holds. It suffices to show that if N >> 0 and the F_{i} are in general position then Φ(X, Y) = (X, Φ_{F} (X, Y)) has the desired property (2).

If I = (g_{1}, …, g_{s}) is the ideal of k[X, Y] defining V, then the ideal J of k[X, Y] defining Φ^{−1}(V) is generated by the polynomials

g_{j}(X, Φ_{F}) = g_{j}(x_{1}, …, x_{m}, Φ_{1}, Φ_{2}, …, Φ_{n}), Φ_{i} = f_{i}(X, Y) + h(Y)F_{i}(X).

If b is a k-point of 𝔸^{n}, the ideal J_{b} of k[X] defining the fiber over b is generated by the g_{j}(X, Φ_{F} (X, b)). We need to show that if b ∉ Z, then J_{b} has height ≥ m − t.

Example 18A.2. Suppose that m = 1 and t = 0. We may assume that dim V = n, and that V is defined by g(x, Y) = 0. Then Φ^{−1}(V) is defined by g(x, Φ_{F}), F_{i}(x) = a_{i}x^{N}, and the fiber over b ∈ 𝔸^{n} − Z is defined by

g(x, f_{1}(x, b) + h(b)a_{1}x^{N}, …) = 0.

Since b ∉ Z, h(b) ≠ 0. Hence the left side of this equation is a nonzero polynomial in k[x] for almost all choices of a_{1}, …, a_{n} when N >> 0. Hence the fiber over b is finite.

The same argument works more generally when t = m − 1; we may assume that V is defined by g = 0, and the fiber over b is defined by g(X, Φ_{F} (X, b)) = 0. In order to see that the left side is nonzero for almost all choices of F_{1}, …, F_{n} one just needs to analyze the leading form of g(X, Φ_{F}) with respect to X.

For any ring R we grade the polynomial ring R[X] = R[x_{1}, …, x_{m}] with all x_{i} in degree 1. Any polynomial of degree d is the sum f = F_{d} + … + F_{0} where F_{i} is a homogeneous form of degree i; F_{d} is called the leading form of f with respect to X. If I is an ideal in R[X] the leading forms of elements of I generate a homogeneous ideal I′ of R[X].

grobner Lemma 18A.3. Let R be a catenary Noetherian ring, I ⊂ R[X] an ideal, and I′ the ideal of leading forms in I with respect to X. Then ht(I′) = ht(I).

Proof. Let I_{h} ⊂ S = R[x_{0}, …, x_{m}] be the homogeneous ideal defining the closure V̄ of V (I) in ℙ_{R}^{m}. Then ht(I) = ht_{S}(I_{h}) = ht_{S}(I_{h}, x_{0}) − 1. But I′ = (I_{h}, x_{0})S/x_{0}S, so ht(I′) = ht_{S}(I_{h}, x_{0}) − 1.

Now the ring k[X, Y] is bigraded, with each x_{i} of bidegree (0, 1) and each y_{i} of bidegree (1, 0). Thus each polynomial can be written as a sum g = ∑ G_{ij}, where the G_{ij} have bidegree (i, j). Ordering the bidegrees lexicographically allows us to talk about the bidegree of g, namely the largest (p, q) with G_{pq} ≠ 0; this G_{pq} is the bi-homogeneous leading form of g.

Without loss of generality, we assume that the generators g_{1}, …, g_{s} of I have the following property: the bi-homogeneous leading forms G_{j}(X, Y) of g_{j} generate the ideal of the leading forms of I.

h1.6.1 Lemma 18A.4. If F_{1}, …, F_{n} are homogeneous forms in k[X] of degree N > max{deg_{X}(f_{i}), deg_{X}(g_{j})} then the ideal J′ of leading forms in J with respect to X contains forms h^{r}G_{j}(X, F_{1}, …, F_{n}), for r >> 0.

Proof. (See ^{\underline{higher}} [HiCG] 1.6.1.) Recall that J is generated by the g_{j}(X, Φ_{F}). For any choice of the N-forms F_{i} it is easy to see that deg_{X} g_{j}(X, Φ_{F}) = deg_{X} G_{j}(X, Φ_{F}) = N deg_{Y} G_{j} + deg_{X} G_{j}, and that the leading form in g_{j}(X, Φ_{F}) with respect to X is h^{deg_{Y} G_{j}}G_{j}(X, F_{1}, …, F_{n}). h1.7 Proposition 18A.5. Let T ⊂ 𝔸^{m} × 𝔸^{n} be a closed subscheme of dimension

≤ n + t, t ≥ 0. If k is infinite, then for any N ≥ 0 we can find forms F_{1}, …, F_{n} in k[X] of degree N so that W = {w ∈ 𝔸^{m} : (w, F_{1}(w), …, F_{n}(w)) ∈ T } has dimension at most t.

Proof. The vector space on n-tuples F = (F_{1}, …, F_{n}) of homogeneous forms of degree N in k[X] is finite-dimensional, say dimension D. We identify it with the set of k-rational points of the affine space 𝔸^{D}. Consider the evaluation map

η : 𝔸^{m} × 𝔸^{D} → 𝔸^{m+n}, η(w, F) = (w, F (w)).

If w ≠ 0, the fibers of η : w×𝔸^{D} → w×𝔸^{n} are isomorphic to 𝔸^{D−n}, because the linear homomorphism η(w, −) : 𝔸^{D} → 𝔸^{n} is surjective. By inspection, η^{−1}(0 × 𝔸^{n}) = 0 × 𝔸^{D}. It follows that η^{−1}(T) has dimension at most D + t.

Now consider the projection π : η^{−1}(T) → 𝔸^{D}. The theorem on dimension of the fibers ^{\underline{hart}} [Har77] III.9.6 implies that there is a nonempty U ⊂ 𝔸^{D} whose fibers have dimension ≤ t. Choosing a rational point in U, the corresponding homogeneous forms (F_{1}, …, F_{n}) satisfy dim{w ∈ 𝔸^{m} : (w, F (w)) ∈ T } ≤ t.

Remark 18A.6. The case N = 0 is easy to visualize, since D = n. There is an open subset U of 𝔸^{n} so that for each b ∈ U the fiber T ∩ (𝔸^{m} × b) of the projection T → 𝔸^{n} over b has dimension at most t.

If T is defined by bi-homogeneous polynomials, then W is defined by homogeneous polynomials. Suslin states ^{\underline{h1.7}} 18A.5 for the corresponding projective varieties in ^{\underline{higher}} [HiCG] 1.7.

We are now ready to complete the proof of theorem ^{\underline{lemma4}} 18A.1. By ^{\underline{grobner}} 18A.3, J_{b} ⊂ k[X] has the same height as the ideal J′_{b} of its leading forms. Suppose that N > max{deg_{X}(f_{i}), deg_{X}(g_{j})}. Since h(b) ≠ 0, J′_{b} contains all the G_{j}(X, F) by \underline{h1.6.1} 18A.4. Let T ⊂ 𝔸^{m+n} be the variety defined by the ideal of bi-homogeneous forms of I, i.e., the G_{j}(X, Y). Hence the variety W = {w ∈ 𝔸^{m} : (w, F (w)) ∈ T } is defined by the G_{j}(X, F). By two applications of ^{\underline{grobner}} 18A.3, dim T = dim V ≤ n + t. Thus dim W ≤ t by ^{\underline{h1.7}} 18A.5. But the height of J′_{b} is at least the height of the ideal generated by the G_{j}(X, F), i.e., the codimension of W, which is at least m − t.

Motivic cohomology and higher Chow groups

mchcg

With the preparation of the last three lectures, we are ready to prove the fundamental comparison theorem:

10.2 Theorem 19.1. Let X be a smooth separated scheme over a perfect field k, then for all m and i there is a natural isomorphism:

H^{n,i}(X, ℤ) ^{≅} CH^{i}(X, 2i − n).

Because CH^{i}(X, 0) is the classical Chow group CH^{i}(X) we obtain:

Corollary 19.2. H^{2i,i}(X, ℤ) ≅ CH^{i}(X)

It is clear from definition ^{\underline{hcgdef}} 17.1 that CH^{i}(X, m) = 0 for m < 0. We immediately deduce the:

secondvanish Vanishing Theorem 19.3. For every smooth variety X and any abelian group A, we have H^{n,i}(X, A) = 0 for n > 2i.

The proof of ^{\underline{10.2}} 19.1 will proceed in two stages. First we will show (in theorem \underline{FS} 19.8) that ℤ(i)[2i] is quasi-isomorphic to U ↦ z^{i}(U × 𝔸^{i}, ∗) as a complex of Zariski sheaves. Then we will show (in ^{\underline{zardesc}} 19.12) that the hypercohomology of z^{i}(− × 𝔸^{i}, ∗) is CH^{i}(−, ∗).

We saw in ^{\underline{thm2}} 16.7 that ℤ(i) is quasi-isomorphic to the Suslin-Friedlander motivic complex ℤ^{SF} (i). Recall from page ^{\underline{equipst}} 107 that the shift ℤ^{SF} (i)[2i] is the chain complex C_{∗}z_{equi}(𝔸^{i}, 0) associated to the simplicial abelian presheaf with transfers C_{•}z_{equi}(𝔸^{i}, 0), which sends X to m ↦ z_{equi}(𝔸^{i}, 0)(X × Δ^{m}). The following result generalizes example ^{\underline{zequizi}} 17.2.

equisubset Lemma 19.4. Let T be smooth of dimension d. Then for all X there is an embedding of simplicial abelian groups:

C_{•}z_{equi}(T, d − i)(X) ^{⊂} z^{i}(X × T, •).

In particular (for T = 𝔸^{i}), ℤ^{SF} (i)[2i](X) is a subcomplex of z^{i}(X × 𝔸^{i}, ∗).

Proof. The cycles in C_{m}z_{equi}(T, d − i)(X) are equidimensional over X × Δ^{m} at all points, while the ones in z^{i}(X × T, m) need only be equidimensional at the generic points of the faces of X × T × Δ^{m}. Hence the first group is contained in the second group of cycles. Moreover, the face maps of the two simplicial groups are compatible by ^{\underline{1a12}} 1A.12.

20.5 Example 19.5. The complex ℤ^{SF} (i)[2i](Y) is a subcomplex of z^{i}(Y × 𝔸^{i}, ∗)_{W} (see ^{\underline{levineprop}} 17.5) for every finite correspondence W from X to Y. Indeed, z_{equi}(𝔸^{i}, 0)(Y × Δ^{m}) lies in z^{i}(Y × 𝔸^{i}, m)_{W×𝔸^{i}} because every generating cycle is quasi-finite over Y × Δ^{m}.

In contrast, it is easy to see that z_{equi}(Y × 𝔸^{i}, dim Y)(Δ^{m}) need not lie in z^{i}(Y × 𝔸^{i}, m)_{W×𝔸^{i}}, by letting X be a point of Y.

For any schemes X and T, consider the simplicial presheaf on X:

U ↦ z^{i}(U × T, •).

This can be regarded as a simplicial sheaf on the flat site over X and hence on both the (small) étale site and the Zariski site of X as well. We will write z^{i}(−×T, ∗) for the associated complex of sheaves. The homology of z^{i}(−×T, ∗) has the more general structure of a presheaf with transfers by ^{\underline{hcgpst}} 17.20.

nut Proposition 19.6. The homology of the embedding in ^{\underline{equisubset}} 19.4 is a morphism of presheaves with transfers:

H_{m}C_{∗}z_{equi}(𝔸^{i}, 0)(−) → H_{m}z^{i}(− × 𝔸^{i}, ∗) = CH^{i}(− × 𝔸^{i}, n).  (19.6.1) hzequizi

Proof. The source and target are presheaves with transfers by ^{\underline{equipst}} 16.3 and ^{\underline{hcgpst}} 17.20, respectively. It suffices to show that their transfer maps are compatible.

Let W be an elementary correspondence from X to Y. We need to verify that ϕ_{W} and W^{∗} are compatible with the map (^{\underline{hzequizi}} 19.6.1). If W is the graph of a flat map from X to Y, then ϕ_{W} and W^{∗} are compatible because both are just the flat pull-back of cycles. Since W^{∗} is defined in ^{\underline{pstch}} 17.16 by passing to an affine vector bundle torsor Y′ → Y, a simple diagram chase (which we leave to the reader) shows that it suffices to prove the statement when Y is affine.

Let Y be affine. Since H_{n}z^{i}(Y × 𝔸^{i}, m)_{W} = H_{n}z^{i}(Y × 𝔸^{i}, m) by ^{\underline{levineprop}} 17.5, the result will follow once we show that the following diagram commutes.

z_{equi}(𝔸^{i}, 0)(Y × Δ^{m}) \overline{\begin{matrix}ϕ_{W} \\  \end{matrix}}z_{equi}(𝔸^{i}, 0)(X × Δ^{m})

[figure: z^{i}(Y × 𝔸^{i}, m)_{W}; ^{\underline{20.5}} 19.5; ∩; W^{∗}; z^{i}(X × 𝔸^{i}, m); ^{\underline{equisubset}} 19.4; ∩]

Let i, f and π, respectively, denote the products with 𝔸^{i}×Δ^{m} of the inclusion W ^{⊂} X × Y, and the canonical projections X × Y → Y and X × Y → X. The transfer map W^{∗} was defined as W^{∗}(Z) = π_{∗}((W × 𝔸^{i} × Δ^{m}) · f^{∗}Z) in ^{\underline{ch}} 17.6. According to ^{\underline{equipst}} 16.3, the transfer map on z_{equi}(𝔸^{i}, 0)(Y × Δ^{m}) is ϕ_{W} (Z) = (iπ)_{∗}(Z_{W ×Δ^{m}}), where the pull-back Z_{W ×Δ^{m}} was defined on page \underline{ui} 13. By ^{\underline{17a.9}} 17A.12, Z_{W ×Δ^{m}} = (fi)^{∗}(Z), so we have:

ϕ_{W} (Z) = (iπ)_{∗}(fi)^{∗}(Z) = π_{∗}i_{∗}(fi)^{∗}(Z).

By ^{\underline{17a.8}} 17A.11, i_{∗}(fi)^{∗}(Z) = (W × 𝔸^{i} × Δ^{m}) · f^{∗}Z and therefore for every Z in z_{equi}(𝔸^{i}, 0)(Y × Δ^{m}) we have:

ϕ_{W} (Z) = π_{∗}((W × 𝔸^{i} × Δ^{m}) · f^{∗}Z) = W^{∗}(Z).

thm1 Example 19.7. If E is a field over k, then the map of ^{\underline{nut}} 19.6 evaluated at Spec E is an isomorphism:

H_{m}C_{∗}z_{equi}(𝔸^{i}, 0)(Spec E) ^{≅} H_{m}z^{i}(Spec E × 𝔸^{i}, ∗).

This follows from Suslin’s theorem ^{\underline{equiqiso}} 18.3 with X = 𝔸_{E}^{i} , since we may identify z_{equi}(𝔸_{k}^{i}, 0)(Δ_{E}^{m}) and z_{equi}(𝔸_{E}^{i}, 0)(Δ^{m}) by ^{\underline{zequike}} 16.6.

This implies that theorem ^{\underline{10.2}} 19.1 is true when evaluated on fields. To see this, set S = Spec E and recall that ℍ^{m}(S, C^{∗}) = H^{m}C^{∗}(S) for any complex of sheaves C^{∗}. By ^{\underline{thm2}} 16.7, the above map fits into the sequence of isomorphisms:

H^{n,i}(S, ℤ) ≅ H^{n}ℤ(i)(S) ≅ H^{n}ℤ^{SF} (i)(S) =

H_{2i−n}C_{∗}z_{equi}(𝔸^{i}, 0)(S) ^{≅} H_{2i−n}z^{i}(𝔸_{E}^{i}, ∗) =

CH^{i}(𝔸_{E}^{i}, 2i − n) ≅ CH^{i}(S, 2i − n).

FS Theorem 19.8. The map ℤ^{SF} (i)[2i] = C_{∗}z_{equi}(𝔸^{i}, 0) → z^{i}(− × 𝔸^{i}, ∗) is a quasi-isomorphism of complexes of Zariski sheaves.

Proof. The induced homomorphisms on homology presheaves,

H_{m}C_{∗}z_{equi}(𝔸^{i}, 0) → H_{m}z^{i}(− × 𝔸^{i}, ∗)  (19.8.1) 19.8.1

are morphisms of presheaves with transfer by ^{\underline{nut}} 19.6. The left side is homotopy invariant by ^{\underline{aletter}} 2.17 and the right side is homotopy invariant because the higher Chow groups are homotopy invariant (see p.^{\underline{hcghi}} 111). By ^{\underline{thm1}} 19.7, this is an isomorphism for all fields. By ^{\underline{smsemi}} 11.2, the sheafification of the map (^{\underline{19.8.1}} 19.8.1) is an isomorphism. Hence C_{∗}z_{equi}(𝔸^{i}, 0) → z^{i}(− × 𝔸^{i}, ∗) is a quasi-isomorphism for the Zariski topology.

10.1/2 Corollary 19.9. For any smooth scheme X, the inclusion of ^{\underline{equisubset}} 19.4 induces an

isomorphism:

H^{n,i}(X, ℤ) ^{≅} ℍ^{n−2i}(X, z^{i}(− × 𝔸^{i}, ∗)).

Proof. By ^{\underline{thm2}} 16.7 and ^{\underline{FS}} 19.8, we have the sequence of isomorphisms:

H^{n,i}(X, ℤ) = ℍ^{n}(X, ℤ(i)) ≅ ℍ^{n}(X, ℤ^{SF} (i)) =

ℍ^{n−2i}(X, ℤ^{SF} (i)[2i]) ^{≅} ℍ^{n−2i}(X, z^{i}(− × 𝔸^{i}, ∗)).

Corollary ^{\underline{10.1/2}} 19.9 is the first half of the proof of ^{\underline{10.2}} 19.1. The rest of this lecture is dedicated to proving the second half, that ℍ^{−m}(X, z^{i}(−×𝔸^{i}, ∗)) ≅ CH^{i}(X, m). To do this, we shall use Bloch’s Localization Theorem (see p.^{\underline{hcghi}} 111) to reinterpret the higher Chow groups as the hypercohomology groups of a complex of sheaves.

A chain complex of presheaves C is said to satisfy Zariski descent on X if H^{∗}(C(U)) → ℍ^{∗}(U, C_{Zar}) is an isomorphism for every open U in X.

Definition 19.10. Let C be a complex of presheaves on X_{Zar} (the small Zariski site of X). We say that C has the (Zariski) Mayer-Vietoris property if for every U ⊂ X, and any open covering U = V_{1} ∪ V_{2}, the diagram

[figure: C(U); C(V_{1}); C(V_{2}); C(V_{1} ∩ V_{2})]

is homotopy cartesian (i.e., the total complex is an acyclic presheaf). This implies that there is a long exact sequence

⋯ → H^{i}(C(U)) → H^{i}(C(V_{1})) ⊕ H^{i}(C(V_{2})) → H^{i}(C(V_{1} ∩ V_{2})) → ⋯

For example, any chain complex of flasque sheaves has the Mayer-Vietoris property. This is an easy consequence of the fact that C(U) → C(V) is onto for each V ⊂ U.

The following result is proven in ^{\underline{bg}} [BG73].

p Theorem 19.11 (Brown-Gersten). Let C be a complex of presheaves on X with the Mayer-Vietoris property. Then C satisfies Zariski descent. That is, the maps H^{∗}(C(U)) → ℍ^{∗}(U, C_{Zar}) are all isomorphisms.

Our main application of the Brown-Gersten theorem is to prove that Bloch’s complexes satisfy Zariski descent.

zardesc Proposition 19.12. Let X be any scheme of finite type over a field. For any scheme T, each z^{i}(− × T) satisfies Zariski descent on X. That is, for all m and i, we have:

CH^{i}(X × T, m) ≅ ℍ^{−m}(X, z^{i}(− × T)).

In particular (for T = 𝔸^{i}),

CH^{i}(X, m) ^{≅} CH^{i}(X × 𝔸^{i}, m) ^{≅} ℍ^{−m}(X, z^{i}(− × 𝔸^{i})).

Proof. (Bloch ^{\underline{acahkt}} [Blo86, 3.4]) By ^{\underline{p}}19.11, we have to show that C(U) = z^{i}(U × T) has the Mayer-Vietoris property. For each cover {V_{1}, V_{2}} of each U we set V_{12} = V_{1} ∩ V_{2} and consider the diagram:

[figure: 0; C(U − V_{1}); C(U); C(V_{1}); coker_{1}; 0]

[figure: 0; C(V_{2} − V_{12}); =; C(V_{2}); C(V_{12}); coker_{2}; 0.]

By Bloch’s Localization Theorem, the cokernels are both acyclic. A diagram chase shows that the middle square is homotopy cartesian, i.e., the Mayer-Vietoris condition is satisfied.

We are now ready to prove the main result of this section, theorem ^{\underline{10.2}} 19.1.

Proof of ^{\underline{10.2}} 19.1. Using ^{\underline{10.1/2}} 19.9 and ^{\underline{zardesc}} 19.12, we define the map to be the compositions of isomorphisms:

H^{n,i}(X, ℤ) ≅ ℍ^{n}(X, ℤ(i)) ^{≅} ℍ^{n−2i}(X, z^{i}(−×𝔸^{i})) ≅ CH^{i}(X, 2i−n).

Zariski descent has also been used by Bloch and Levine to show that the higher Chow groups are functorial for morphisms between smooth schemes. We conclude this lecture by showing that their definition agrees with ours.

chfunct Definition 19.13. (Bloch-Levine) Let f be a morphism from X to Y. Natural maps f^{∗} : CH^{i}(Y, m) → CH^{i}(X, m) for all m and i are defined as follows. As in the proof of ^{\underline{levineprop}} 17.5, write z^{i}(Y, ∗)_{f} for z^{i}(Y, ∗)_{Γ_{f}}.

If U ⊂ Y is open, z^{i}(Y, ∗)_{f} restricts to z^{i}(U, ∗)_{f}, and z_{f}^{i} is a complex of sheaves. Since Y is locally affine, z_{f}^{i} ≃ z^{i} by ^{\underline{levineprop}} 17.5 and there is a map z_{f}^{i} → f_{∗}z^{i} of complexes of sheaves on Y. The map is now defined using Zariski descent \underline{zardesc} 19.12 as the composite:

CH^{i}(Y, m) ≅ ℍ^{−m}(Y, z^{i}) ≅ ℍ^{−m}(Y, z_{f}^{i}) ^{f^{∗}} ℍ^{−m}(X, z^{i}) ≅ CH^{i}(X, m).

hyperflat Example 19.14. If q : Y′ → Y is flat, then z_{q}^{i} = z^{i}, and the map q^{∗} defined in \underline{chfunct} 19.13 is just the flat pull-back of cycles map q^{∗}, described in ^{\underline{chaffine}} 17.11.

hyperfunct Lemma 19.15. If X ^{g} Y ^{f} Z are morphisms of smooth schemes, then the maps defined in ^{\underline{chfunct}} 19.13 satisfy (fg)^{∗} = g^{∗}f^{∗}.

Proof. If fg ∐ f : X ∐ Y → Z, we can restrict (fg)^{∗} and f^{∗} to the subgroup z^{i}(Z, m)_{fg ∐ f}. Since (fg)^{∗} = g^{∗}f^{∗} on cycles (see ^{\underline{serre}} [Ser65, V-30]), f^{∗} maps this subgroup into z^{i}(Y, m)_{g}. By construction, the diagram of groups

[figure: z^{i}(Z, m)_{fg ∐ f}; ⊂; z^{i}(Z, m)_{fg}; z^{i}(Y, m)_{g}; f^{∗}; g^{∗}; z^{i}(X, m); (fg)^{∗}]

commutes. Sheafifying and applying hypercohomology, ^{\underline{levineprop}} 17.5 and Zariski descent \underline{zardesc} 19.12 show that the composite

CH^{i}(Z, m) ≅ ℍ^{−m}(Z, z_{fg ∐ f}^{i}) ^{f^{∗}} ℍ^{−m}(Y, z_{g}^{i}) ^{g^{∗}} ℍ^{−m}(X, z^{i}) ≅ CH^{i}(X, m)

is just (fg)^{∗}, as required. hcgagree Proposition 19.16. The map f^{∗} : CH^{i}(Y, m) → CH^{i}(X, m) defined in ^{\underline{chfunct}} 19.13

agrees with the map f^{∗} = Γ_{f}^{∗} defined in ^{\underline{pstch}} 17.16.

Proof. Suppose first that X and Y are affine, and consider the commutative diagram

[figure: CH^{i}(Y, m); =; H^{−m}z^{i}(Y, ∗); ≅; H^{−m}z^{i}(Y, ∗)_{f}; H^{−m}z^{i}(X, ∗); =; CH^{i}(X, m); ℍ^{−m}(Y, z^{i}); ≅; ≅; ℍ^{−m}(Y, z_{f}^{i}); ℍ^{−m}(X, z^{i}).; ≅]

The arrows marked ‘ ≅′ are isomorphisms by ^{\underline{levineprop}} 17.5 and ^{\underline{zardesc}} 19.12. The top composite is the map of ^{\underline{chaffine}} 17.11, which by ^{\underline{affWagree}} 17.17 is the map Γ_{f}^{∗} of ^{\underline{pstch}} 17.16. The bottom composite is the map f^{∗} of ^{\underline{chfunct}} 19.13, proving that f^{∗} = Γ_{f}^{∗} in this case.

In the general case, ^{\underline{jouansurj}} 17.14 gives a diagram

[figure: X′; g; Y′; X; p; f; Y,; q]

where X′ → X and Y′ → Y are affine vector bundle torsors. By definition \underline{pstch} 17.16, Γ_{f}^{∗} is (p^{∗})^{−1}Γ_{g}^{∗}q^{∗}, where p^{∗} and q^{∗} are flat pull-back of cycles. By ^{\underline{hyperflat}} 19.14, these are the same as the maps p^{∗} and q^{∗} defined in ^{\underline{chfunct}} 19.13. Since Γ_{g}^{∗} = g^{∗} by the first part of the proof and g^{∗}q^{∗} = (qg)^{∗} = (pf)^{∗} = p^{∗}f^{∗} by ^{\underline{hyperfunct}} 19.15, we have:

Γ_{f}^{∗} =^{\underline{pstch} 17.16}(p^{∗})^{−1}Γ_{g}^{∗}q^{∗} = (p^{∗})^{−1}g^{∗}q^{∗} =^{\underline{hyperfunct} 19.15} (p^{∗})^{−1}p^{∗}f^{∗} = f^{∗}.

Covering morphisms of triples

leccov

The main goal of the rest of the lectures will be to prove that if F is a homotopy invariant presheaf with transfers, then the presheaf H_{Nis}^{n}(−, F) is homotopy invariant. This was stated in theorem ^{\underline{theorem''}} 13.7 and it was used in lectures 13-19. The remaining lectures depend upon lectures 11, 12, and the first part of 13 (^{\underline{nisshxfer}} 13.1–^{\underline{nisextcoho}} 13.5), but not on the material from ^{\underline{boot}} 13.6 to the end of lecture 19.

morstd3 Definition 20.1. Let T_{Y} = (Ȳ , Y_{∞}, Z_{Y}) and T_{X} = (X̄, X_{∞}, Z_{X}) be standard triples (as defined in ^{\underline{dfnstd3}} 11.5). For convenience, set Y = Ȳ −Y_{∞} and X = X̄ −X_{∞}. A covering morphism f : T_{Y} → T_{X} of standard triples is a finite morphism f : Ȳ → X̄ such that:

• f^{−1}(X_{∞}) ⊂ Y_{∞} (and hence f(Y) ⊂ X);

• f|_{Y} : Y → X is étale;

• f induces an isomorphism Z_{Y} ^{≅} Z_{X} with f^{−1}(Z_{X}) ∩ Y = Z_{Y}.

Note that f need not induce a finite morphism f : Y → X.

Y_{∞,1}  Ȳ Z_{Y} Y_{∞,2} Z^{X} X^{∞}  X̄
Figure 20.1: A covering morphism f : Ȳ → X̄  covfigure

Week 20 - April 5, 2000

Any covering morphism of standard triples induces an upper distinguished square Q = Q(X, Y, X − Z_{X}) (see ^{\underline{uds}} 12.5):

[figure: (Y − Z_{Y}); Y; (X − Z_{X}); X.; f]

We say that this upper distinguished square comes from the covering morphism of standard triples.

17.2 Example 20.2. Suppose that an affine X has a covering X = U ∪ V and a good compactification (X̄, X_{∞}) over some smooth S. Then the Zariski square

U ∩ V  \overline{}U
Q(X, U, V) :
[figure: V; X]

comes from a morphism of triples, provided that X̄ − (U ∩ V) lies in an affine open neighborhood in X̄.

Indeed, if Z = X − V then T = (X̄, X_{∞}, Z) is a standard triple and T′ = (X̄, X̄ − U, Z) is also a standard triple. The identity on X̄ induces a covering morphism T′ → T and the above square comes from this morphism.

Recall from ^{\underline{split3}} 11.11 that a splitting of a standard triple (X̄, X_{∞}, Z) over V ⊂ X is a trivialization of L_{Δ_{X}} on V ×_{S} Z.

pullsplit Lemma 20.3. Let f : T_{Y} → T_{X} be a covering morphism of standard triples. A splitting of T_{X} over V induces a splitting of T_{Y} over f^{−1}(V) ∩ Y.

Proof. Since T_{X} was split over V ⊆ X̄, we are given t : L_{ΔX}|_{V ×Z_{X}} ≅ O. We need a trivialization

f^{−1}(t) : L_{ΔY} |_{f^{−1}(V)×Z_{Y}} ≅ O.

Now (f × f)^{−1}(Δ_{X}) is the disjoint union of Δ_{Y} and some Q, so (f × f)^{∗}(L_{ΔX}) is L_{ΔY} ⊗ L_{Q}, where L_{Q} is the associated line bundle. Since f induces an isomorphism Z_{Y} → Z_{X}, Q is disjoint from Y ×_{S} Z_{Y}. Since L_{Q} has a canonical trivialization outside Q, we have L_{Q} ≅ O on Y ×_{S} Z_{Y}. Since (f × f)^{∗}(t) is a trivialization of L_{Δ_{Y}} ⊗ L_{Q} on (f × f)^{−1}(V ×_{S} Z_{X}), we may regard (f × f)^{∗}(t) as a trivialization of L_{ΔY} on (f^{−1}(V) ∩ Y) ×_{S} Z_{Y}.

Example 20.4. Let Ȳ → X̄ be a finite separable morphism of smooth projective curves, X_{∞} ⊂ X̄ a finite nonempty set containing the branch locus, and y ∈ Ȳ a k-rational point so that x = f(y) is not in X_{∞}. Set Y_{∞} = f^{−1}(X_{∞}) ∐ f^{−1}(x) − {y}. Then (Ȳ , Y_{∞}, {y}) → (X̄, X_{∞}, {x}) is a covering morphism of standard triples. If X = Spec A and P is the prime ideal of A defining x, then P B is prime in the coordinate ring B of Y. If a ∈ A then by \underline{xsplit} 11.13, lemma ^{\underline{pullsplit}} 20.3 states that if P [1/a] is principal, then so is P B[1/a]. mvdef Definition 20.5. Let Q be any commutative square of the form

[figure: B; i; Y; A; f; i; X.; f]

We write MV (Q) for the following chain complex in Cor_{k}:

MV (Q) : 0  B ^{(−f,i)} A ⊕ Y ^{(i,f)} X  0.

If F is a presheaf, then F (MV (Q)) is the complex of abelian groups:

0  F (X) ^{(i,f)} F (A) ⊕ F (Y) ^{(−f,i)} F (B)  0.

The general theorem below will involve an intricate set of data which we now describe. Let f : (Ȳ , Y_{∞}, Z_{Y}) → (X̄, X_{∞}, Z_{X}) be a covering morphism of standard triples. Let Q denote the square that comes from f. Let Q′ = (X′, Y′, A′) be another upper distinguished square with Y′ and X′ affine so that Q and Q′ are of the form:

[figure: B′; Y′; B; Y; Q′ :  Q :; A′; f′; i′; X′; f′; A; f; i; X.; f]  (20.5.1) morphdist

may 24 handout mainstd3 Theorem 20.6. Let j : Q′ → Q be a morphism of upper distinguished squares

of the form ^{\underline{morphdist}} 20.5.1 such that:

• Q comes from a covering morphism of standard triples;

• X′ → X is an open embedding, and (X̄, X_{∞}, Z_{X}) splits over X′;

• X′ and Y′ are affine.

Then for any homotopy invariant presheaf with transfers F, the map of complexes F (MV (Q)) → F (MV (Q′)) is chain homotopic to zero.

[figure: 0; F (X); (i, f); F (A) ⊕ F (Y); (−f, i); F (B); 0; 0; F (X′); j_{X}; (i′, f′); F (A′) ⊕ F (Y′); ( \begin{matrix}j_{A} \\ j_{Y} \end{matrix} ); (−f′, i′); F (B′); j_{B}; 0]

17.5 Lemma 20.7. There are maps λ_{A} : F (A) → F (X′) and λ_{B} : F (B) → F (Y′) so that the following diagram commutes:

[figure: F (X); i; F (A); f; F (B); i; F (Y)]

[figure: j_{X}; j_{Y}]

[figure: F (X′); ∃λ_{A}; f′; F (Y′).; ∃λ_{B}]

Proof. By ^{\underline{pullsplit}} 20.3, both triples T_{X} and T_{Y} split. Hence the maps in question exist and the outer triangles commute by ^{\underline{prp1''}} 11.15. The construction of the trivializations in the proof of ^{\underline{prp1''}} 11.15 shows that the middle square commutes.

Recall from ^{\underline{a1he}} 2.23 and ^{\underline{h0singa1he}} 7.2 that two elements of Cor(X, Y) are 𝔸^{1}-homotopic if they agree in the quotient H_{0}^{sing}(X × Y/X) of Cor(X, Y).

Aug8 Lemma 20.8. Let h be a rational function on X̄ ×_{S} Ȳ which is invertible in a neighborhood U of A′ ×_{S} Y_{∞} and A′ ×_{S} Z_{Y}, and equals 1 on A′ ×_{S} Y_{∞}. Then the Weil divisor D defined by h defines an element ψ of Cor(A′, B) such that the composition iψ ∈ Cor(A′, Y) is 𝔸^{1}-homotopic to zero.

Proof. As a divisor on the normal variety A′ ×_{S} Ȳ, we can write D = ∑ n_{i}D_{i} with each D_{i} integral and supported off of U. Since each D_{i} misses A′ ×_{S} Y_{∞}, it is quasi-finite over A′. Since D_{i} is proper over A′, and has the same dimension as A′, it is finite and surjective over A′. As such, each D_{i} and hence D defines an element of C_{0}(A′ ×_{S} B/A′) which is a subgroup of C_{0}(A′ ×B/A′) = Cor(A′, B). By construction (see ^{\underline{C0lift}} 7.15), the image of D in Pic(A′ ×_{S} Ȳ , A′ ×_{S} (Y_{∞} ∐ Z)) is given by (O, h), the trivial line bundle with trivialization 1 on A′ ×_{S} Y_{∞}, and h on A′ ×_{S} Z_{Y}. The composition with i : B → Y sends D to an element of C_{0}(A′ ×_{S} Y/A′) whose image in Pic(A′ ×_{S} Ȳ , A′ ×_{S} Y_{∞}) is the class of (O, h). By ^{\underline{relpic}} 7.16, this group is isomorphic to H_{0}^{sing}(A′ ×_{S} B/A′). But in this group (O, h) = (O, 1) is the zero element. This implies that the image is zero in H_{0}^{sing}(A′ × B/A′).

We say that a diagram in Cor_{k} is homotopy commutative if every pair of composites f, g : X → Y with the same source and target are 𝔸^{1}-homotopic. Any homotopy invariant presheaf with transfers identifies 𝔸^{1}-homotopic maps, and hence converts a homotopy commutative diagram into a commutative diagram.

For example, one could rephrase ^{\underline{17.5}} 20.7 as giving maps λ_{A} ∈ Cor(X′, A) and λ_{B} ∈ Cor(Y′, B), well-defined up to 𝔸^{1}-homotopy, such that a certain diagram is homotopy commutative. storto Lemma 20.9. Let j : Q′ → Q be as in the statement of ^{\underline{mainstd3}} 20.6. Then there is a

map ψ in Cor_{k}(A′, B) fitting into a homotopy commutative diagram:

B′ \overline{\begin{matrix}λ_{B} ∘ i′ − j_{B} \\  \end{matrix}}B

.............. ψ

[figure: A′; f′; λ_{A} ∘ i′ − j_{A}; A.; f]

Moreover the composition A′ ^{ψ} B ^{i} Y is 𝔸^{1}-homotopic to 0.

Corollary 20.10. For every homotopy invariant presheaf with transfers F we have a commutative diagram:

F (A)  \overline{f} F (B)

.............. ψ

[figure: F (A′); i′ ∘ λ_{A} − j_{A}; f′; F (B′),; i′ ∘ λ_{B} − j_{B}]

and the composite F (Y) ^{i} F (B) ^{ψ} F (A′) is zero.

Proof of ^{\underline{storto}} 20.9. In order to streamline notation, we write × for ×_{S} and Z for Z_{Y}.

Let L_{ΔX′} be the line bundle on X′ × X̄ corresponding to the graph ΔX′ of X′ ^{⊂} X̄, and L_{ΔY′} for the line bundle on Y′ × Ȳ corresponding to the graph ΔY′ of Y′ ^{⊂} Ȳ. In between these, we have the line bundle M on X′ × Ȳ, obtained by pulling back L_{ΔX′}.

Since these three line bundles come from effective divisors, they have canonical global sections. We will write s_{X} for the canonical global section of L_{ΔX′} on X′ × X̄, s_{M} for M on X′ × Ȳ, and s_{Y} for L_{ΔY′} on Y′ × Ȳ. Each global section determines a section on X′ ×Z_{X}, X′ ×Z, and Y′ ×Z, respectively. Since A′ ⊆ X′ − Z_{X} and B′ ⊆ Y′ − Z, the restrictions of s_{X}, s_{M}, s_{Y} also determine trivializations in each case, of L_{ΔX′} on A′ × Z_{X}, of M on A′ × Z, and of L_{ΔY′} on B′ × Z.

Because Z ≅ Z_{X}, the inclusion of X′ × Z_{X} in X′ × X̄ lifts to X′ × Ȳ, and we may identify the pullbacks of L_{ΔX′} and M to X′ × Z, together with their respective trivializations s_{X} and s_{M} on A′ × Z.

Since the standard triple (X̄, X_{∞}, Z_{X}) splits over X′, we are given a fixed trivialization t_{X} of L_{ΔX′} on X′ × Z_{X}. As with s_{X}, we may identify t_{X} with a trivialization t_{M} of M on X′ × Z. By ^{\underline{pullsplit}} 20.3, t_{X} also induces a trivialization t_{Y} of L_{ΔY} on Y′ × Z. Since Z_{X} lives in an affine neighborhood U_{X} in X̄, we extend t_{X} to X′ × U_{X} and we fix this particular extension. Pulling back, the same is true for t_{M} and t_{Y} and we fix those two extensions too.

Because t_{X}, t_{M}, t_{Y} are trivializations, there are regular functions r_{X}, r_{M}, r_{Y} so that:

s_{X} = r_{X}t_{X} on X′ × Z_{X}; s_{M} = r_{M}t_{M} on X′ × Z; s_{Y} = r_{Y} t_{Y} on Y′ × Z.

Because s_{X} is a trivialization on A′ ×Z_{X}, r_{X} is invertible on A′ ×Z_{X}. Similarly, r_{M} is invertible on A′ × Z, and r_{Y} is invertible on B′ × Z. (See figure ^{\underline{coov}} 20.2.)

A′ × Z_{Y}^{A′ × Y_{∞,1}}^{A′ × Y_{∞,2}}  ^{1}^{r_{M}}^{1} A′ × Ȳ  A′ × X̄^{A′ × Z_{X}}  ^{r_{X}}^{1}^{A′ × X_{∞}}
Figure 20.2: The covering morphism f : Ȳ → X̄ over A′  coov

Because (Ȳ , Y_{∞}, Z) is a standard triple, there is an affine open neighborhood U of Y_{∞} ∐ Z in Ȳ. Hence X′ × U is an affine open neighborhood of X′ × Z and X′ × Y_{∞} in X′ × Ȳ. Since Z and Y_{∞} are disjoint, the Chinese Remainder Theorem yields a regular function h on X′ × U which equals 1 on X′ × Y_{∞} and equals r_{M} on X′ × Z. Let D ⊂ X′ × Ȳ denote the the principal divisor corresponding to h. By lemma ^{\underline{Aug8}} 20.8, the divisor −D defines an element ψ of Cor(A′, B) such that the composition iψ ∈ Cor(A′, Y) is homotopically trivial. defined ψ as

−D to clean up signs By ^{\underline{C0lift}} 7.15, the map Cor(A′, B) → Pic(A′ ×Ȳ , A′ ×(Y_{∞} ∐ Z)) sends ψ to the class of (O_{A′×Ȳ} , 1_{∞} ∐ r_{M}^{−1}).

It remains to verify that the diagram in ^{\underline{storto}} 20.9 is homotopy commutative.

We first interpret the horizontal maps in ^{\underline{storto}} 20.9. By the construction of λ_{A} and λ_{B} in ^{\underline{prp1''}} 11.15 and ^{\underline{17.5}} 20.7, the compositions λ_{A} ∘ i′ ∈ Cor(A′, A) and λ_{B} ∘ i′ ∈ Cor(B′, B) represent the classes of (L_{ΔA′}, s_{∞} ∐ t_{X}) and (L_{ΔB′}, s_{∞} ∐ t_{Y}) in Pic(A′ × X̄, A′ × (X_{∞} ∐ Z)) and Pic(B′ × Ȳ , B′ × (Y_{∞} ∐ Z)), respectively. On the other hand, the inclusions j_{A} and j_{B} represent the classes of (L_{ΔA′}, s_{∞} ∐ s_{X}) and (L_{ΔB′}, s_{∞} ∐ s_{Y}), respectively. It follows that the differences j_{A} − λ_{A} ∘ i′ ∈ Cor(A′, A) and j_{B} − λ_{B} ∘ i′ ∈ Cor(B′, B) represent the classes of (O_{A′×X̄}, 1_{∞} ∐ r_{X}) and (O_{B′×Ȳ} , 1_{∞} ∐ r_{Y}), respectively. (Cf. exercise \underline{C0rep} 11.16.)

The composition ψf′ ∈ Cor(B′, B) represents (O_{B′×Ȳ} , f^{∗}h^{−1}). Since f^{∗}h is a rational function on B′ × Ȳ which is 1 on B′ × Y_{∞} and r_{Y} on B′ × Z, we have ψf′ = λ_{B} ∘ i′ − j_{B} in Pic(B′ × Ȳ , B′ × (Y_{∞} ∐ Z)).

Now the composition fψ ∈ Cor(A′, A) represents the push-forward of ψ along H_{0}(A′ × B/A′) → H_{0}(A′ × A/A′). By ^{\underline{invrelpic}} 7.24, this represents the class of (O_{A′×X̄}, f_{∗}(1_{∞} ∐ r_{M}^{−1})). By definition ^{\underline{covf}} 7.22, the norm of h is a rational function which extends the trivialization f_{∗}(1_{∞} ∐ r_{M}) to an affine neighborhood. Since h is identically 1 on f^{−1}(X_{∞}) ⊂ Y_{∞}, N(h) = 1 on A′ × X_{∞} by ^{\underline{normszz}} 7.23. We will show that N(h) = r_{X} on A′ × Z_{X} in lemma ^{\underline{l20.10}} 20.11 below. Hence fψ = λ_{A}i′ − j_{A} in Cor(A′, A), as desired.

l20.10 Lemma 20.11. Let f : U → V be a finite map with U and V normal. Suppose that Z ⊂ V and Z′ ⊂ U are reduced closed subschemes such that the induced map Z′ → Z is an isomorphism, and U → V is étale in a neighborhood of Z′.

If h ∈ O^{∗}(U) is 1 on f^{−1}(Z) − Z′, then N(h)|_{Z} and h|_{Z′} are identified by Z′ ≅ Z.

Proof. Suppose first that f has a section s : V → U sending Z to Z′. Then U ≅ s(U) ∐ U′ and h is 1 on f^{−1}(Z) ∩ U′. In this case, the assertion follows from the componentwise calculation of the norm N(h), together with ^{\underline{normszz}} 7.23.

In the general case, let U′ ⊂ U be a neighborhood of Z′ which is étale over V, and let h′ ∈ O^{∗}(U′ ×_{V} U) be the pullback of h. The graph Z′′ ⊂ U′ ×_{V} U of Z′ → Z is isomorphic to Z′, and U′ ×_{V} U′ is an étale neighborhood of Z′′ in U′×_{V} U. By construction, h′ is 1 on U′×_{V} (f^{−1}(Z)−Z′′) and U′×_{V} U → U′ has a canonical section sending Z′ to Z′′; in this case we have shown that N(h′)|_{Z′} is identified with h|_{Z′}. Since norms commute with base change, we can identify N(h) with N(h′) under O^{∗}(V) ⊆ O^{∗}(U′). This proves the lemma.

Proof of ^{\underline{mainstd3}} 20.6. We have maps s_{1} = (λ_{A}, 0) : F (A) ⊕ F (Y) → F (X′) and s_{2} = (ψ, λ_{B}) : F (B) → F (A′) ⊕ F (Y′). In order for these maps to form a chain homotopy from j to zero, we must have sd + ds = j. This amounts to six equations, three of which come from the commutativity of the trapezoid in \underline{17.5} 20.7. The other three, which involve ψ are: ψi ≃ 0, j_{A} ≃ i′λ_{A} − ψf and j_{B} ≃ i′λ_{B} − f′ψ. These are provided by ^{\underline{storto}} 20.9.

step2.1.5 Corollary 20.12. Let Q = Q(X, Y, A) be an upper distinguished square of smooth schemes coming from a covering morphism of standard triples and S = {x_{1}, …, x_{n}} a finite set of points in X.

Then there exists an affine neighborhood X′ of S in X such that:

• The induced square Q′ = Q(X′, Y′, A′) is upper distinguished, where A′ = X′ ∩ A and Y′ = f^{−1}(X′);

• For any homotopy invariant presheaf with transfers F, the map F (MV (Q)) → F (MV (Q′)) is chain homotopic to zero.

[figure: 0; F (X); F (A) ⊕ F (Y); F (B); 0; 0; F (X′); F (A′) ⊕ F (Y′); F (B′); 0] Proof. By ^{\underline{lemmuzzo}} 11.14, S has an affine neighborhood X′ over which the triple (X̄, X_{∞}, Z) splits. We may shrink X′ in order to assume that it is disjoint from f(Y_{∞}). Hence Y′ = f^{−1}(X′) is finite over X′, and hence affine, as well as being contained in Y. The induced square Q′ is upper distinguished because it is obtained from Q by base change. Thus the hypotheses of theorem ^{\underline{mainstd3}} 20.6 are satisfied for Q′ → Q.

Zariski sheaves with transfers

With the technical results of the last lecture in hand, we are ready to prove the following results.

azarhi Theorem 21.1. Let F be a homotopy invariant presheaf with transfers. Then the Zariski sheaf F_{Zar} is homotopy invariant.

step2.2 Theorem 21.2. Let F be a homotopy invariant presheaf with transfers. Then F_{Zar} = F_{Nis}.

Combining ^{\underline{azarhi}} 21.1 and ^{\underline{step2.2}} 21.2, we obtain theorem ^{\underline{n0}} 21.3 below, which is the case n = 0 of theorem ^{\underline{theorem''}} 13.7. This theorem does not require k to be perfect.

n0 Theorem 21.3. If F is a homotopy invariant presheaf with transfers, then the Nisnevich sheaf F_{Nis} is homotopy invariant.

We will prove theorems ^{\underline{azarhi}} 21.1 and ^{\underline{step2.2}} 21.2 in order, using a sequence of lemmas. We make the running assumption that F is a homotopy invariant presheaf with transfers. The Mayer-Vietoris sequence F (MV (Q)) associated to a commutative square Q is defined in ^{\underline{mvdef}} 20.5.

step1.1 Lemma 21.4. Let U be an open subset of 𝔸^{1} and U = U_{1} ∪U_{2} be a Zariski covering of U. Then the complex F (MV (Q)) is split exact, where Q = Q(U, U_{1}, U_{2}).

F (MV (Q)) : 0  F (U)  F (U_{1}) ⊕ F (U_{2})  F (U_{1} ∩ U_{2})  0

In particular, F is a Zariski sheaf on 𝔸^{1}.

\underline{week21.tex}

Proof. Setting Y_{∞} = ℙ^{1} − U, Y′_{∞} = ℙ^{1} − U_{1} and Z = U − U_{2}, the identity of ℙ^{1} is a covering morphism (ℙ^{1}, Y′_{∞}, Z) → (ℙ^{1}, Y_{∞}, Z) of standard triples as in example ^{\underline{17.2}} 20.2. Both triples are split over U itself by ^{\underline{xsplit}} 11.13, so by theorem ^{\underline{mainstd3}} 20.6 with Q′ = Q, the complex F (MV (Q)) is chain contractible, i.e., split exact. step1.2 Lemma 21.5. If F is a homotopy invariant Zariski sheaf with transfers, and

U is an open subset of 𝔸^{1}, then H_{Zar}^{n}(U, F) = 0 for n > 0.

Proof. If U = \{U_{1}, …, U_{n}\} is a finite cover of U, it follows from ^{\underline{step1.1}} 21.4 and induction on n that the following sequence is exact.

0 → F (U) → ⊕_{i}F (U_{i}) → ⊕_{i,j}F (U_{i} ∩ U_{j}) → … → F (∩_{i}U_{i}) → 0

Hence the Čech cohomology of F satisfies Ȟ^{i}(U, F) = 0 for i > 0. But then H^{1}(U, F) = Ȟ^{1}(U, F) = 0 by ^{\underline{hart}} [Har77, Ex III.4.4]. Since dim U = 1, we must also have H^{i}(U, F) = 0 for i > 1 (see ^{\underline{hart}} [Har77, III.2.7]).

Exercise 21.6. Show that ^{\underline{step1.1}} 21.4 and ^{\underline{step1.2}} 21.5 fail for F = O_{X}^{∗} if 𝔸^{1} is replaced by an affine elliptic curve.

step1.2nis Lemma 21.7. If F is a homotopy invariant Nisnevich sheaf with transfers, and

U is an open subset of 𝔸^{1}, then H_{Nis}^{n}(U, F) = 0 for n > 0. ^{\underline{step1.2nis}} 21.7 is used to

prove ^{\underline{21.2}} 23.2

Proof. Since dim U = 1, we have H_{Nis}^{n}(U, F) = 0 for n > 1. By ^{\underline{milne}} [Mil80, III.2.10], H_{Nis}^{1}(U, F) = Ȟ^{1}(U, F). Therefore we only need to show that Ȟ^{1}(U, F) = 0.

Since F takes disjoint unions to direct sums, the Čech cohomology can be computed using covering families V → X, instead of the more general {V_{i} → X}. By ^{\underline{niscurve}} 12.6, any such cover of U has a refinement U = \{A, V \}, where A ⊂ U is dense open, V → U is étale, and the square Q = Q(U, V, A) is upper distinguished (see ^{\underline{uds}} 12.5). Embed V in a smooth projective curve V̄ finite over ℙ^{1}, and set V_{∞} = V̄ −V. By construction (see ^{\underline{morstd3}} 20.1), Q comes from the covering morphism of standard triples (V̄ , V_{∞}, Z) → (ℙ^{1}, U_{∞}, Z), where U_{∞} = ℙ^{1} −U and Z = U −A. Since (ℙ^{1}, U_{∞}, Z) splits over U by ^{\underline{xsplit}} 11.13, theorem ^{\underline{mainstd3}} 20.6 with Q′ = Q implies that the complex F (MV (Q)) is split exact. That is Ȟ^{1}(U, F) = 0. Passing to the limit over all such covers yields Ȟ^{1}(U, F) = 0.

step1.3 Lemma 21.8. Let F be a homotopy invariant presheaf with transfers. If X is smooth and U ⊂ X is dense open, then F_{Zar}(X) → F_{Zar}(U) is injective.

Proof. As F_{Zar} is a sheaf it suffices to verify this locally. Let f ∈ F_{Zar}(X) be a nonzero section which vanishes in F_{Zar}(U). Pick a point x ∈ X so that f is nonzero in the stalk F_{x} = F (Spec O_{X,x}). By shrinking X around x we may assume that f ∈ F (X). By shrinking U, we may assume that f vanishes in F (U) and hence in F (V) for V = Spec(O_{X,x}) ∩ U. By ^{\underline{prp1}} 11.1, f is nonzero in F (V), and this is a contradiction.

Proof of ^{\underline{azarhi}} 21.1. We have to prove that i^{∗} : F_{Zar}(X×𝔸^{1}) → F_{Zar}(X) is an isomorphism, where i : X → X ×𝔸^{1}. It is enough to prove that i^{∗} is injective. We may assume that X is connected and therefore irreducible. Let γ : Spec K X be the generic point. We get a diagram:

F_{Zar}(X × 𝔸^{1}) \overline{i^{∗}} F_{Zar}(X)

[figure: F_{Zar}(Spec K × 𝔸^{1}); (γ × 1)^{∗}; ≅; F_{Zar}(Spec K); γ^{∗}]

where the vertical maps are injective by ^{\underline{step1.3}} 21.8. The bottom map is an isomorphism by ^{\underline{step1.1}} 21.4 since we may regard F as a homotopy invariant presheaf with transfers over the field K by ^{\underline{trkl}} 2.9:

F_{Zar}(𝔸_{K}^{1}) = F (𝔸_{K}^{1}) ^{≅} F (Spec K) = F_{Zar}(Spec K).

Thus i^{∗} is injective.

Let s_{Zar}(F) be the separated presheaf (with respect to the Zariski topology) associated to the presheaf F. It is defined by the formula:

s_{Zar}(F)(X) = F (X)/F_{0}(X), F_{0}(X) = colim_{\begin{matrix}covers \\ \{U_{i}→X\} \end{matrix}} ker F (X) → ∏ F (U_{i}).

used in both \underline{pro} 21.12, ^{\underline{azartr}} 21.15 l.2 Lemma 21.9. s_{Zar}(F) is a homotopy invariant presheaf with transfers.

Proof. The homotopy invariance of s_{Zar}F is immediate from the fact that homotopy invariance is preserved by quotient presheaves. The existence of transfers is more difficult. Let Z ⊂ S × X be an elementary correspondence from S to X. We must show that the corresponding transfer F (X) → F (S) sends F_{0}(X) to F_{0}(S), i.e., that the image of F_{0}(X) vanishes at each stalk F (Spec O_{S,s}). It suffices to suppose S local, so that Z is semilocal. Hence there is a semilocal subscheme X′ of X with Z ⊂ S × X′. But by ^{\underline{prp1}} 11.1, F (X′) injects into F (U) for each dense U ⊂ X′, so F_{0}(X′) = 0. Hence F_{0}(X) → F (S) is zero, because it factors through F_{0}(X′) = 0.

For the next few lemmas, S will be the semilocal scheme of a smooth quasiprojective variety X at a finite set of points. Since any finite set of points lies in an affine neighborhood, we may even assume that X is affine. Clearly, S is the intersection of the filtered family of its affine open neighborhoods X_{α} in X.

l.3 Lemma 21.10. Suppose that F is a homotopy invariant presheaf with transfers. Then for any open covering S = U_{0} ∪ V there is an open U ⊂ U_{0} such that S = U ∪ V and the sequence F (MV (Q)) is exact, where Q = Q(S, U, V):

0 → F (S) → F (U) ⊕ F (V) → F (U ∩ V) → 0.

Proof. We may assume that S is connected, since we can work separately with each component. By assumption, there are open Ũ_{0}, Ṽ in X such that U_{0} = S ∩ Ũ_{0}, V = S ∩ Ṽ. Since Ũ_{0} is open in X, there is an affine open Ũ contained in Ũ_{0} which contains the finite set of closed points of U_{0}. Setting U = S ∩ Ũ, we have S = U ∪ V. We will show that F (MV (Q)) is exact for the square Q = Q(S, U, V).

We first suppose that k is an infinite field. For each α, set U_{α} = X_{α} ∩ Ũ and V_{α} = X_{α} ∩ Ṽ. The canonical map from Q to the square Q_{α} = Q(X_{α}, U_{α}, V_{α}) induces a morphism of Mayer-Vietoris sequences, F (MV (Q_{α})) → F (MV (Q)). It suffices to show that these morphisms are chain homotopic to zero, because F (MV (Q)) is the direct limit of the F (MV (Q_{α})).

Let Z ⊂ X denote the union of X − (Ũ ∩ Ṽ) and the closed points of S. For each X_{α}, we know by ^{\underline{teoremuzzo}} 11.17 that there is an affine neighborhood X′_{α} of S in X_{α} and a standard triple T_{α} = (X̄_{α}, X_{∞,α}, Z_{α}) with X′_{α} ≅ X̄_{α} − X_{∞,α} and Z_{α} = X_{α} ∩ Z. Set U′_{α} = X′_{α} ∩ Ũ and V′_{α} = X′_{α} ∩ Ṽ. Since X̄_{α} − (U′_{α} ∩ V′_{α}) lies in X_{∞,α} ∪ Z_{α}, it lies in an affine open subset of X̄_{α} (by definition ^{\underline{dfnstd3}} 11.5). By \underline{17.2} 20.2, the Zariski square Q′_{α} = Q(X′_{α}, U′_{α}, V′_{α}) comes from a covering morphism of triples T′_{α} → T_{α}.

By ^{\underline{lemmuzzo}} 11.14, the triple T_{α} is split over an affine neighborhood X′′_{α} of S in X′_{α}. Set U′′_{α} = X′′_{α} ∩ Ũ and V′′_{α} = X′′_{α} ∩ Ṽ, and form the square Q′′_{α} = Q(X′′_{α}, U′′_{α}, V′′_{α}). Since X′′_{α} and Ũ are affine, so is U′′_{α}. By theorem ^{\underline{mainstd3}} 20.6, the ^{\underline{mainstd3}} 20.6 needs U′′_{α} morphism F (MV (Q′_{α})) → F (MV (Q′′_{α})) is chain homotopic to zero. Since affine F (MV (Q_{α})) → F (MV (Q)) factors through this morphism, it too is chain homotopic to zero.

[figure: 0; F (X_{α}); F (U_{α}) ⊕ F (V_{α}); F (U_{α} ∩ V_{α}); 0]

[figure: 0; F (X′_{α}); F (U′_{α}) ⊕ F (V′_{α}); F (U′_{α} ∩ V′_{α}); 0]

[figure: 0; F (X′′_{α}); F (U′′_{α}) ⊕ F (V′′_{α}); F (U′′_{α} ∩ V′′_{α}); 0]

[figure: 0; F (S); F (U) ⊕ F (V); F (U ∩ V); 0.]

If k is finite, exactness follows by a transfer argument. Any element a in the homology of F (MV (Q)) must vanish when we pass to Q ⊗_{k} k′ for any infinite algebraic extension k′ of k. Since a must vanish for some finite subextension k′_{0}, a has exponent [k′_{0} : k]. Since [k′_{0} : k] can be chose to be a power of any prime, we conclude that a = 0.

Lemma ^{\underline{l.3}} 21.10 corrects ^{\underline{preth}} [CohTh, 4.23], which omitted the passage from U_{0} to U.

2211 Corollary 21.11. Let S′ and S′′ be semilocal schemes of a smooth quasiprojective scheme X at finite sets of points, and set S = S′ ∪ S′′. Then the Mayer-Vietoris sequence F (MV (Q)) is exact, where Q = Q(S, S′, S′′):

0 → F (S) → F (S′) ⊕ F (S′′) → F (S′ ∩ S′′) → 0.

Proof. Write S′ as the intersection of opens U_{α} ⊂ S and S′′ as the intersection of opens V_{β} ⊂ S. The sequence F (MV (Q)) is the direct limit of the sequences F (MV (Q_{αβ})), where Q_{αβ} = Q(S, U_{α}, V_{β}). By ^{\underline{l.3}} 21.10, there are U_{αβ} ⊂ U_{α} such that the sequences F (MV (Q(S, U_{αβ}, V_{β}))) are exact. Hence the morphisms from F (MV (Q_{αβ})) to F (MV (Q)) are zero on homology. Passing to the direct limit, we see that the homology of F (MV (Q)) is zero, i.e., it is exact.

Note that the sequence 0 → F(S) → F(S′) ⊕ F(S′′) → F(S′ ∩ S′′) is always exact when F is a Zariski sheaf on S. This is because it is the direct limit of the exact sequences 0 → F(S) → F(U_{α}) ⊕ F(V_{β}) → F(U_{α} ∩ V_{β}) associated to the family of open covers {U_{α}, V_{β}} of S with S′ ⊂ U_{α} and S′′ ⊂ V_{β}.

pro Lemma 21.12. Let S be the semilocal scheme of a smooth quasi-projective scheme X at a finite set of points. Then F_{Zar}(S) = F (S).

Proof. By ^{\underline{prp1}} 11.1, F (S) = (s_{Zar}F)(S). Since s_{Zar}F is a homotopy invariant presheaf with transfers by ^{\underline{l.2}} 21.9, we may replace F by s_{Zar}F and assume that F is separated. We now proceed by induction on the number of the closed points of S. Let S′ be the local scheme at a closed point x of S, and S′′ the semilocal scheme at the remaining points. Consider the following commutative diagram.

[figure: 0; F (S); F (S′) ⊕ F (S′′); F (S′ ∩ S′′); 0; F_{Zar}(S); F_{Zar}(S′) ⊕ F_{Zar}(S′′); =; F_{Zar}(S′ ∩ S′′); into]

The top row is exact by ^{\underline{2211}} 21.11, and we have noted that the bottom row is exact because F_{Zar} is a Zariski sheaf. The right vertical map is an injection because F is separated. The middle vertical map is the identity by induction. A diagram chase shows that the left vertical map is an isomorphism, as desired.

We need an analogue of lemma ^{\underline{etcov}} 6.16 for the Zariski topology, showing that we can lift finite correspondences to open covers under mild conditions.

topcov Lemma 21.13. Let W be a closed subset of X × Y, x ∈ X a point and V ⊂ Y an open subset such that p^{−1}(x) ⊂ {x} × V, where p : W → X is the projection. Then there is a neighborhood U of x such that W ×_{X} U is contained in U × V.

Proof. The subset Z = W − W ∩ (X × V) is closed, and x ∉ p(Z). Because p is a closed map, p(Z) is closed and U = X − p(Z) is an open neighborhood of x. By construction, W ×_{X} U is contained in U × V.

topcovf Corollary 21.14. Let W ∈ Cor(X, Y) have support W and let p : W → X be the projection. If x ∈ X and V ⊂ Y are such that p^{−1}(x) ⊂ {x} × V, then there is a neighborhood U of x and a canonical W_{U} ∈ Cor(U, V) such that the following diagram commutes.

[figure: U; W_{U}; V; X; W; Y]

Proof. Writing W = ∑ n_{i}[W_{i}], we may apply lemma ^{\underline{topcov}} 21.13 to each W_{i}. Since W_{i} is finite over X, W_{i}×_{X} U is finite over U, so W_{U} = ∑ n_{i}[W_{i}×_{X} U] is the required finite correspondence. It is canonical because if U′ ⊂ U, the composition of U′ ⊂ U with W_{U} is W_{U′} = ∑ n_{i}[W_{i} ×_{X} U′]. azartr Theorem 21.15. Let F be a homotopy invariant presheaf with transfers. Then

the Zariski sheaf F_{Zar} has a unique structure of presheaf with transfers such that F → F_{Zar} is a morphism of presheaves with transfers. advertised in

lectures 6 and 13 (pp.41, 91)

Proof. By ^{\underline{l.2}} 21.9 we may assume that F is separated, i.e., that F (V) ⊆ F_{Zar}(V) for every V. We may also assume that X and Y are irreducible without loss of generality.

We begin with a general construction, starting with an element f ∈ F_{Zar}(Y) and a finite correspondence W from X to Y. Fix a point x ∈ X. Since p : W → X is finite, the image of p^{−1}(x) under the natural map W → Y consists of only finitely many points; let S denote the semilocal scheme of Y at these points. Since F (S) = F_{Zar}(S) by ^{\underline{pro}} 21.12, there is an open V_{x} ⊂ Y such that f_{x} = f|_{V_{x}} ∈ F_{Zar}(V_{x}) lies in the subgroup F (V_{x}) ⊆ F_{Zar}(V_{x}). By ^{\underline{topcovf}} 21.14, there is a neighborhood U_{x} of x such that W restricts to a finite correspondence W_{x} from U_{x} to V_{x}. Let W^{∗}(f)_{x} denote the image of f_{x} under W_{x}^{∗} : F (V_{x}) → F (U_{x}) ⊆ F_{Zar}(U_{x}).

Uniqueness. Suppose that F → F_{Zar} is a morphism of presheaves with transfers. Given W ∈ Cor(X, Y) and f ∈ F_{Zar}(Y), it suffices to show that W^{∗}(f) ∈ F_{Zar}(X) is uniquely defined in some neighborhood of any point x. The construction above shows that the image of W^{∗}(f) in F_{Zar}(U_{x}) must equal W^{∗}(f)_{x}, which is defined using only the sheaf structure on F_{Zar} and the transfer structure on F.

Existence. Fix W ∈ Cor(X, Y) and f ∈ F_{Zar}(Y). In the construction above, we produced a neighborhood U_{x} of every point x ∈ X, an open set V_{x} in Y so that f_{x} = f|_{V_{x}} belongs to the subgroup F (V_{x}) of F_{Zar}(V_{x}), and considered the image W^{∗}(f)_{x} = W_{x}^{∗}(f_{x}) of f_{x} in F (U_{x}) ⊆ F_{Zar}(U_{x}).

We first claim that the W^{∗}(f)_{x} agree on the intersections U_{xx′} = U_{x} ∩ U_{x′}. The element W^{∗}(f) ∈ F_{Zar}(X) will then be given by the sheaf axiom (see figure \underline{ffzar} 21.1).

F_{Zar}(Y)  F_{Zar}(X)

[figure: ∏ F_{Zar}(V_{x}); ⊃; ∏ F (V_{x}); W_{U_{x}}^{∗}; ∏ F (U_{x}); ⊂; ∏ F_{Zar}(U_{x})]

[figure: ∏ F_{Zar}(V_{xx′}); into; ∏ F (V_{xx′}); W_{xx′}^{∗}; ∏ F (U_{xx′}); ∏ F_{Zar}(U_{xx′})]

Figure 21.1: The transfer map for F_{Zar}  ffzar

Pick two points x, x′ ∈ X and set U_{xx′} = U_{x} ∩ U_{x′}, V_{xx′} = V_{x} ∩ V_{x′}. Since W ×_{X} U_{x} lies in U_{x} × V_{x} for all x (by ^{\underline{topcov}} 21.13), it follows that W ×_{X} U_{xx′} lies in U_{xx′} × (V_{x} ∩ V_{x′}). Hence there is a finite correspondence W_{xx′} from U_{xx′} lifting both W_{x} and W_{x′} in the sense of ^{\underline{topcovf}} 21.14. That is, the middle square commutes in figure ^{\underline{ffzar}} 21.1.

Fix x ∈ X and choose V ⊂ Y, f_{V} ∈ F (V) and U_{x} as above. Because F is separated we have F (V) ⊂ F_{Zar}(V), so the element f_{V} ∈ F (V) is well defined. Given a dense V_{0} ⊂ V, the map F (V) → F (V_{0}) sends f_{V} to f_{V_{0}}, because F_{Zar}(V) ⊂ F_{Zar}(V_{0}) by ^{\underline{step1.3}} 21.8. Given U_{0} ⊂ U_{x}, the proof of ^{\underline{topcovf}} 21.14 shows that the the canonical lift W_{U_{0}} ∈ Cor(U_{0}, V) is the composition of the inclusion U_{0} ⊂ U with the canonical lift W_{U} ∈ Cor(U, V). Hence F_{Zar}(U_{x}) → F_{Zar}(U_{0}) sends the element W^{∗}(f)_{x} to the image of f_{V_{0}} under F (V_{0}) → F (U_{0}) ⊂ F_{Zar}(U_{0}).

It is now easy to check using ^{\underline{step1.3}} 21.8 that the maps W^{∗} are additive and give F_{Zar} the structure of a presheaf with transfers.

week22 starts here Proof of ^{\underline{step2.2}} 21.2. We have to prove that F_{Zar} = F_{Nis}. Let F and F′′ denote the kernel and cokernel presheaves of F → F_{Nis}, respectively. By ^{\underline{nisshxfer}} 13.1, they are presheaves with transfer whose associated Nisnevich sheaf is zero. Since sheafification is exact, it suffices to show that F′_{Zar} = F′′_{Zar} = 0. That is, we may assume that F_{Nis} = 0.

By ^{\underline{azarhi}} 21.1 and ^{\underline{azartr}} 21.15, F_{Zar} is also a homotopy invariant presheaf with transfers. Since F_{Nis} = (F_{Zar})_{Nis}, we may assume that F = F_{Zar}, i.e., that F is a Zariski sheaf. Therefore it suffices to show that F (S) = 0 for every local scheme S of a smooth variety X.

By ^{\underline{step2.1}} 12.7, it suffices to show that, for any upper distinguished square

B  \overline{i} Y
Q :
[figure: A; f; i; X; f]

as in ^{\underline{uds}} 12.5, the square F (Q ×_{X} S) is a pull-back.

Shrinking X around x, we may suppose by ^{\underline{teoremuzzo}} 11.17 that X is affine and fits into a standard triple (X̄, X_{∞}, Z) with A = X −Z. Shrinking Y around the finite set f^{−1}(x), we may also suppose by ^{\underline{teoremuzzo}} 11.17 that Y is affine, and fits into a standard triple so that Q comes from a finite morphism of standard triples in the sense of ^{\underline{morstd3}} 20.1. Hence ^{\underline{step2.1.5}} 20.12 implies that the map F (MV (Q)) → F (MV (Q ×_{X} S)) is chain homotopic to zero.

[figure: 0; F (X); F (A) ⊕ F (Y); F (B); 0; 0; F (S); F (A ∩ S) ⊕ F (Y ×_{X} S); F (B ×_{X} S); 0]

Taking the limit over smaller and smaller neighborhoods X of x, we see that F (MV (Q×_{X} S)) is exact. But then F (Q×_{X} S) is a pullback square, as claimed.

Contractions

Let F be a homotopy invariant presheaf. Define a new presheaf F_{−1} (known as the contraction of F in the literature) by the formula:

F_{−1}(X) = coker (F (X × 𝔸^{1}) → F (X × (𝔸^{1} − 0))) .

For r > 1 we define F_{−r} to be (F_{1−r})_{−1}.

Since the inclusion t = 1 : X ^{⊂} X × (𝔸^{1} − 0) ⊂ X × 𝔸^{1} is split by the projection X × 𝔸^{1} → X, we have a canonical decomposition F (X × (𝔸^{1} − 0)) ≅ F (X) ⊕ F (X)_{−1}. Hence, F_{−1} is also homotopy invariant, and if F is a sheaf then so is F_{−1}. Here are some examples of this construction.

Zi-1 Example 22.1. If F = O_{X}^{∗} then F_{−1} = ℤ, because O^{∗}(X×(𝔸^{1}−0)) = O^{∗}(X)×

\{t^{n}\} for every integral X. By ^{\underline{zreg}} 4.1, there is a quasi-isomorphism ℤ(1)_{−1} ≃ ℤ[−1].

More generally, the higher Chow groups CH^{i}(−, n) are homotopy invariant (see p. ^{\underline{hcghi}} 111) and their contractions are given by the formula:

CH^{i}(X, n)_{−1} ≅ CH^{i−1}(X, n − 1).  (22.1.1) CH-1

This follows from the the Localization Sequence (see p. ^{\underline{hcghi}} 111):

CH^{i−1}(X, n) ^{(t=0)_{∗}} CH^{i}(X×𝔸^{1}, n) → CH^{i}(X×(𝔸^{1}−0), n) → CH^{i−1}(X, n−1),

which is split as above by the pullback along t = 1 (using ^{\underline{chfunct}} 19.13).

Theorem ^{\underline{10.2}} 19.1 allows us to rewrite the formula in (^{\underline{CH-1}} 22.1.1) as:

H^{m,i}(X, ℤ)_{−1} = ℍ_{Zar}^{m}(X, ℤ(i))_{−1} ≅ ℍ_{Zar}^{m−1}(X, ℤ(i − 1)) = H^{m−1,i−1}(X, ℤ).

This yields the formula ℤ(i)_{−1} ≃ ℤ(i − 1)[−1] in the derived category, and in DM.

Example 22.2. We will see in the next lecture (in ^{\underline{restatement}} 23.1 and ^{\underline{fundthm}} 23.8) if F is a homotopy invariant Zariski sheaf with transfers then H^{n}(−, F) is homotopy invariant and H_{Zar}^{n}(−, F)_{−1} ≅ H_{Zar}^{n}(−, F_{−1}).

used in ^{\underline{locconXZ}} 22.14

loccons-1 Example 22.3. Suppose that 1/n ∈ k, and let M be a locally constant ntorsion sheaf, such as μ_{n}. The argument of ^{\underline{Zi-1}} 22.1 applied to étale cohomology, shows that

H_{ét}^{m}(X, M ⊗ μ_{n})_{−1} ≅ H_{ét}^{m−1}(X, M).

used in ^{\underline{pstHsupp}} 23.9

Hsupp Exercise 22.4. Let U be the standard covering of X × (𝔸^{n} − 0) by U_{1} = X × (𝔸^{1} − 0) × 𝔸^{n−1}, ..., U_{n} = X × 𝔸^{n−1} × (𝔸^{1} − 0). If F is homotopy invariant and n ≥ 2, show that Ȟ^{0}(U, F) ≅ F (X), Ȟ^{n−1}(U, F) ≅ F_{−n}(X), and that Ȟ^{r}(U, F) = 0 for all other r.

Now suppose that F is a Zariski sheaf, and that its cohomology groups are also homotopy invariant. Show that, for all m and n > 0, the cohomology with supports satisfies:

H_{X×{0}}^{m}(X × 𝔸^{n}, F) ≅ H^{m−n}(X, F)_{−n}.

Hint: Use the Čech spectral sequence Ȟ^{p}(U, H^{q}F) ⇒ H^{p+q}(X × (𝔸^{n} − 0), F).

nis-1 Proposition 22.5. Let F be a homotopy invariant presheaf with transfers. Then (F_{Nis})_{−1} ≅ (F_{−1})_{Nis}. used in ^{\underline{p24.4}} 23.3

Proof. By ^{\underline{nisshxfer}} 13.1 and ^{\underline{n0}} 21.3, F_{Nis} is a homotopy invariant sheaf with transfers. By inspection, the natural map (F_{−1})_{Nis} → (F_{Nis})_{−1} is a morphism of presheaves with transfers. By ^{\underline{smsemi}} 11.2 (applied to the kernel and cokernel), it suffices to show that F_{−1}(S) = (F_{Nis})_{−1}(S) when S = Spec E for a field E. The left side is F (𝔸_{E}^{1} − 0)/F (𝔸_{E}^{1}) by definition, while the right side equals F_{Nis}(𝔸_{E}^{1} − 0)/F_{Nis}(𝔸_{E}^{1}). These are equal by ^{\underline{step1.1}} 21.4 and ^{\underline{step2.2}} 21.2.

In the rest of this lecture, we will compare F_{−1} to various sheaves F_{(Y,Z)}, which we now define.

defcok Definition 22.6. Given a closed embedding i : Z ^{⊂} Y, and a presheaf F, we define a Nisnevich sheaf F_{(Y,Z)} on Z as follows. Let K = K_{(Y,Z)} denote the presheaf cokernel of F → j_{∗}j^{∗}F, where j : V ^{⊂} Y is the complement of Z. That is, K(U) is the cokernel of F (U) → F (U ×_{Y} V) for all U. We set F_{(Y,Z)} = (i^{∗}K)_{Nis}.

Since sheafification is exact, there is a canonical exact sequence of sheaves

F_{Nis} → (j_{∗}j^{∗}F)_{Nis} → i_{∗}F_{(Y,Z)} → 0.  (22.6.1) defcok1

VV stopped reading here on 7/15/02 excisepair Example 22.7. If Z = {z} is a closed point on Y, then the value at Z of F_{(Y,Z)}

is the cohomology with supports, H_{Z}^{1}(Y, F_{Nis}). Indeed, if S is the Hensel local scheme of Y at Z then F_{(Y,Z)}(Z) is the cokernel of F_{Nis}(S) → F_{Nis}(S − Z, F), i.e., H_{Z}^{1}(S, F_{Nis}). But this equals H_{Z}^{1}(Y, F_{Nis}) by excision ^{\underline{hart}} [Har77, Ex.III.2.3]. Similarly, we have H^{n}(−, F)_{(Y,Z)} ≅ H_{Z}^{n+1}(Y, F) for n > 0. This follows from excision and the exact sequence

H^{n−1}(S, F) → H^{n−1}(U, F) → H_{z}^{n}(S, F) → 0.

ex23.2 Example 22.8. Fix a Nisnevich sheaf F and consider the presheaf H^{n}(−, F). We claim that if n > 0 then

H^{n}(−, F)_{(Y,Z)} = i^{∗}R^{n}j_{∗}(F).

Indeed, in ^{\underline{defcok1}} 22.6.1 we have H^{n}(−, F)_{Nis} = 0, and R^{n}j_{∗}(F) is the sheaf on Y associated to the presheaf j_{∗}j^{∗}H^{n}(−, F) = j_{∗}H^{n}(−, F |_{V}). Hence

i_{∗}H^{n}(−, F)_{(Y,Z)} ≅ (j_{∗}j^{∗}H^{n}(−, F))_{Nis} = R^{n}j_{∗}(F).

Now apply i^{∗} and observe that i^{∗}i_{∗} is the identity.

21.4 Example 22.9. Let i : S ^{⊂} S × 𝔸^{1} be the embedding i(s) = (s, 0), with complement S × (𝔸^{1} − 0). By definition, F_{−1}(U) = K(U × 𝔸^{1}) where the cokernel presheaf K is defined in ^{\underline{defcok}} 22.6. The adjunction yields a natural map from K(U × 𝔸^{1}) to i_{∗}i^{∗}K(U × 𝔸^{1}) = i^{∗}K(U). That is, we have a natural morphism of sheaves on S:

(F_{−1})_{Nis} → F_{(S×𝔸^{1},S×0)}.

22.4 Proposition 22.10. Let F be a homotopy invariant presheaf with transfers. Then (F_{−1})_{Nis}|_{S} ≅ F_{(S×𝔸^{1},S×0)} for all smooth S.

Proof. We need to compare F_{−1} and j_{∗}j^{∗}F/F at a sufficiently small neighborhood of any point s of any smooth affine S. We will use the standard triple T = (ℙ_{S}^{1} → S, S × \{∞\}, S × \{0\}), which is split over S × 𝔸^{1} by ^{\underline{P1split}} 11.12. For each affine neighborhood U of S × 0 in S × 𝔸^{1}, set T_{U} = (ℙ_{S}^{1}, ℙ_{S}^{1} − U, S × {0}).

We claim that by shrinking S we can make T_{U} into a standard triple. At issue is whether or not (ℙ_{S}^{1} − U) ∪ (S × {0}) lies in an affine open subscheme of ℙ_{S}^{1}. Since the fiber U_{s} over s is open in ℙ_{s}^{1}, there is an affine open V ⊂ ℙ_{k}^{1} so that s ×_{k} V contains both {0} and the finite set ℙ_{s}^{1} − U_{s}. Hence the complements of U and S × V in ℙ_{S}^{1} intersect in a closed subset, disjoint from the fiber ℙ_{s}^{1}. Since ℙ_{S}^{1} is proper over S, we may shrink S about s (keeping S affine) to assume that the complements are disjoint. Hence the affine S × V contains the complement ℙ_{S}^{1} − U as well as S × {0}, as claimed.

Now the identity on ℙ_{S}^{1} is a finite morphism of standard triples T_{U} → T in the sense of ^{\underline{morstd3}} 20.1 by ^{\underline{17.2}} 20.2. Setting U_{0} = U − (S × \{0\}), the square Q coming from this is:

[figure: U_{0}; U; S × (𝔸^{1} − 0); j; S × 𝔸^{1}]

By the standard triples theorem ^{\underline{mainstd3}} 20.6 applied to Q′ = Q, the complex F (MV (Q)) is split exact:

0 → F (S × 𝔸^{1}) → F (S × (𝔸^{1} − 0)) ⊕ F (U) → F (U_{0}) → 0.

\underline{Week 23 - May 24, 2000}

Since F is homotopy invariant, this implies that F (U) → F (U_{0}) is injective and that F_{−1}(S) ≅ F (U_{0})/F (U). Since j : S × (𝔸^{1} − 0) ↪ S × 𝔸^{1} has j_{∗}j^{∗}F (U) = F (U_{0}), the right side is j_{∗}j^{∗}F/F (U). Passing to the limit over U and S, we get the statement.

etpair Lemma 22.11. Let f : Y → X be an étale morphism and Z a closed subscheme of X such that f^{−1}(Z) → Z is an isomorphism. Then for every presheaf F:

F_{(X,Z)} ^{≅} F_{(Y,f^{−1}(Z))}.

Proof. Since this is to be an isomorphism of Nisnevich sheaves, we may assume that X is Hensel local, and that Z is not empty. Then Y is Hensel semilocal; the assumption that f^{−1}(Z) ≅ Z implies that Y is local and in fact Y ≅ X. In this case the two sides are the same, namely F (X−Z)/F (X) ≅ F (Y −Z)/F (X).

Lemma ^{\underline{etpair}} 22.11 uses the Nisnevich topology in a critical way. For the Zariski topology, the corresponding result requires F to be a homotopy invariant presheaf with transfers, and may be proven along the same lines as ^{\underline{22.4}} 22.10; see \underline{preth} [CohTh, 4.13].

week23.tex starts here

p23.5 Theorem 22.12. Let i : Z → X be a closed embedding of smooth schemes of codimension 1, and F a homotopy invariant presheaf with transfers. Then there exists a covering X = ∪U_{α} and isomorphisms on each U_{α} ∩ Z:

F_{(U_{α},U_{α}∩Z)} ≅ (F_{−1})_{Nis}.

That is, for each α there is an exact sequence of Nisnevich sheaves on U_{α}:

0 → F_{α} → j_{α}_{∗}j_{α}^{∗}F_{α} → i_{∗}(F_{−1})_{Nis} → 0.

Here F_{α} = (F |_{U_{α}})_{Nis} and j_{α} denotes the inclusion U_{α} ∩ (X − Z) ^{⊂} U_{α}. Moreover, for every smooth T we also have isomorphisms on (U_{α} ∩ Z) × T:

F_{(U_{α}×T,(U_{α}∩Z)×T)} ≅ (F_{−1})_{Nis}.

Proof. We have to show that every smooth pair (X, Z) of codimension one is locally like (S ×𝔸^{1}, S ×0). If dim(Z) = d then, by shrinking X about any point (and writing X instead of U), we may find an étale map f : X → 𝔸^{d+1} such that Z ≅ f^{−1}(𝔸^{d}).

[figure: Z; ⊂; i; X; 𝔸^{d}; f; ⊂; 𝔸^{d+1}≅ 𝔸^{d} × 𝔸^{1}; f]

By construction, Z ×𝔸^{1} is étale over 𝔸^{d} ×𝔸^{1}. Form the pullback X′ = X ×_{𝔸^{d+1}} Z×𝔸^{1} and note that both X′ → X and X′ → Z×𝔸^{1} are étale with Z′ = Z×_{𝔸^{d}}Z lying above Z and Z ×0, respectively. Since Z′ → Z is étale and has a canonical section Δ, we can write Z′ = Δ(Z) ∐ W. Setting X′′ = X − W, both X′′ → X and X′′ → Z × 𝔸^{1} are étale, with Δ(Z) the inverse image of Z and Z × 0, respectively. Applying lemma ^{\underline{etpair}} 22.11 twice and then ^{\underline{22.4}} 22.10, we obtain the required isomorphisms of Nisnevich sheaves on Z:

F_{(X,Z)} ^{≅} F_{(X′′,Δ(Z))} ^{≅} F_{(Z×𝔸^{1},Z×0)} ≅ (F_{−1})_{Nis}.

To see that the sequence of sheaves is exact, we only need to observe that F_{α} injects into j_{∗}j^{∗}F_{α} by lemma ^{\underline{step1.3}} 21.8, since F_{α} = (F |_{U_{α}})_{Zar} by ^{\underline{step2.2}} 21.2.

In order to prove the final assertion, it suffices to replace Z, X and 𝔸^{d} with Z × T, X × T and 𝔸^{d} × T in the above argument.

\underline{preth} [CohTh, 4.14]

Porism 22.13. The same proof shows that if Z → X is a closed embedding of smooth schemes of codimension r, then locally F_{(X,Z)} ≅ F_{(Z×𝔸^{r},Z×0)}.

locconXZ Example 22.14. Let M be a locally constant n-torsion étale sheaf and consider F (X) = H^{1}(X, M ⊗ μ_{n}). By ^{\underline{loccons-1}} 22.3, (F_{−1})_{Nis} ≅ M. By ^{\underline{milne}} [Mil80, p. 243], we also have F_{(X,Z)} ≅ M. In this case, the isomorphisms F_{(X,Z)} ≅ (F_{−1})_{Nis} of ^{\underline{p23.5}} 22.12 hold for any cover of X.

Homotopy Invariance of Cohomology

lechic

We finally have all the tools to prove ^{\underline{theorem''}} 13.7 which we restate here for the convenience of the reader.

p.119 ^{\underline{orange}} [VSF00]

restatement Theorem 23.1. Let k be a perfect field and F a homotopy invariant presheaf with transfers. Then H_{Nis}^{n}(−, F_{Nis}) is a homotopy invariant presheaf (with transfers) for every n.

Proof. It suffices to prove that the H_{Nis}^{n}(−, F_{Nis}) are homotopy invariant, since we already know that they are presheaves with transfers from ^{\underline{nistr}} 13.4. We shall proceed by induction on n. The case n = 0 was completed in Theorem ^{\underline{n0}} 21.3, so we know that F_{Nis} is homotopy invariant. Hence, we may assume that F = F_{Nis}.

Consider X × 𝔸^{1} ^{π} X. Since π_{∗}F (U) = F (U × 𝔸^{1}) ≅ F (U), we have π_{∗}F = F. By induction we know that R^{q}π_{∗}F = 0 for 0 < q < n. By theorem \underline{21.2} 23.2 below, R^{n}π_{∗}F = 0 as well. Hence the Leray spectral sequence

H_{Nis}^{p}(X, R^{q}π_{∗}F) ⇒ H_{Nis}^{p+q}(X × 𝔸^{1}, F)

collapses enough to yield H_{Nis}^{n}(X, F) ≅ H_{Nis}^{n}(X ×𝔸^{1}, F). That is, the presheaf H_{Nis}^{n}(−, F) is homotopy invariant.

We have thus reduced the proof of ^{\underline{restatement}} 23.1 to the following theorem. Recall from ^{\underline{ega4}} [EGA IV, 17.5] that the Hensel local scheme Spec(R) of a smooth variety at some point is formally smooth, i.e., geometrically regular.

21.2 Theorem 23.2. Let k be a perfect field, and F a homotopy invariant Nisnevich sheaf with transfers such that R^{q}π_{∗}F = 0 for 0 < q < n. If S is a formally smooth Hensel local scheme over k, then H_{Nis}^{n}(S × 𝔸^{1}, F) = 0.

The requirement that k be perfect comes from the following fact (see \underline{ega0} [EGA0_{IV}, 19.6.4]): if k is perfect, every regular local k-algebra is formally smooth over k.

Proof. We will proceed by induction on d = dim(S). If d = 0 then S = Spec(K) for some field K; in this case, H_{Nis}^{n}(S × 𝔸^{1}, F) = H_{Nis}^{n}(𝔸_{K}^{1}, F) = 0 by ^{\underline{step1.2nis}} 21.7. Here we have used exercise ^{\underline{trkl}} 2.9 to regard F as a homotopy invariant presheaf with transfers over K.

If dim(S) > 0, and U is any proper open subscheme, then R^{q}π_{∗}F |_{U} = 0 for 0 < q ≤ n, by induction on d. Thus the canonical map π|_{U}^{∗} : H_{Nis}^{n}(U, F) → H_{Nis}^{n}(U ×𝔸^{1}, F) is an isomorphism, and its inverse is induced by the restriction s|_{U} of the zero section s : S → S × 𝔸^{1} to U. From the diagram

[figure: H_{Nis}^{n}(S × 𝔸^{1}, F); j^{∗}; H_{Nis}^{n}(U × 𝔸^{1}, F); 0 =H_{Nis}^{n}(S, F); s^{∗}; H_{Nis}^{n}(U, F); s|_{U}^{∗} ≅]

we see that the top map j^{∗} is zero for all such U.

Now S = Spec(R) for a regular local ring (R, 𝔪); choose r ∈ 𝔪 − 𝔪^{2} and set Z = Spec(R/r), U = S −Z. Because Z is regular and k is perfect, Z is formally smooth over k. For this choice, the map j^{∗} is an injection by proposition ^{\underline{p24.4}} 23.3 below. Hence the source H_{Nis}^{n}(S × 𝔸^{1}, F) of j^{∗} must be zero.

We use k is perfect but maybe there’s a workaround. p24.4 Proposition 23.3. Let k be a perfect field and S the Hensel local scheme of a

smooth scheme X at some point. Let U be the complement of a smooth divisor Z on S. Under the inductive assumption that R^{q}π_{∗}F = 0 for all 0 < q < n, the following map is a monomorphism:

H_{Nis}^{n}(S × 𝔸^{1}, F) → H_{Nis}^{n}(U × 𝔸^{1}, F).

Proof. Let i and j denote the inclusions of Z × 𝔸^{1} and U × 𝔸^{1} into S × 𝔸^{1} respectively. Regarding F as a sheaf on S × 𝔸^{1}, the map in question factors as:

H_{Nis}^{n}(S × 𝔸^{1}, F) ^{τ} H_{Nis}^{n}(S × 𝔸^{1}, j_{∗}j^{∗}F) ^{η} H_{Nis}^{n}(U × 𝔸^{1}, j^{∗}F).

We first show that the right-hand map η is injective. This will follow from ^{\underline{l24.5}} 23.4 below, once we have shown that R^{q}j_{∗}F = 0 for 0 < q < n. The inductive assumption implies that H^{q}(F) is a homotopy invariant presheaf with transfers. Since q > 0 we have H^{q}(F)_{Nis} = 0. Now see from ^{\underline{nis-1}} 22.5 that (H^{q}(F)_{−1})_{Nis} ≅ (H^{q}(F)_{Nis})_{−1} = 0. By ^{\underline{ex23.2}} 22.8 and ^{\underline{p23.5}} 22.12 (with T = 𝔸^{1}) we have addendum to

\underline{p23.5} 22.12 used

R^{q}j_{∗}F ≅ i_{∗}H^{q}(F)_{(S×𝔸^{1},Z×𝔸^{1})} ≅ i_{∗}(H^{q}(F)_{−1})_{Nis} = 0.

We now prove that the left-hand map τ is injective as well. Since F is a homotopy invariant presheaf with transfers, F injects into j_{∗}j^{∗}F by lemma ^{\underline{step1.3}} 21.8. By ^{\underline{defcok}} 22.6, there is a short exact sequence of Nisnevich sheaves on S × 𝔸^{1}:

0 → F → j_{∗}j^{∗}F → i_{∗}F_{(S×𝔸^{1},Z×𝔸^{1})} → 0.

Since S is local, theorem ^{\underline{p23.5}} 22.12 (with T = 𝔸^{1}) implies that F_{(S×𝔸^{1},Z×𝔸^{1})} ≅ F_{−1} on Z × 𝔸^{1}. Consider the associated long exact sequence in cohomology.

H^{n−1}(S × 𝔸^{1}, j_{∗}j^{∗}F) → H^{n−1}(Z × 𝔸^{1}, F_{−1}) ^{∂}

H^{n}(S × 𝔸^{1}, F) → H^{n}(S × 𝔸^{1}, j_{∗}j^{∗}F) → H^{n}(Z × 𝔸^{1}, F_{−1})

It suffices to show that the map H^{n−1}(S ×𝔸^{1}, j_{∗}j^{∗}F) → H^{n−1}(Z ×𝔸^{1}, F_{−1}) is onto. If n > 1, this follows from the homotopy invariance of F_{−1} and the fact that Z is Hensel local:

H^{n−1}(Z × 𝔸^{1}, F_{−1}) ≅ H^{n−1}(Z, F_{−1}) = 0.

If n = 1, we argue as follows. Since F and F_{−1} are homotopy invariant, the two left horizontal maps are isomorphisms in the commutative diagram:

[figure: F (U); ≅; F (U × 𝔸^{1}); =; H^{0}(S × 𝔸^{1}, j_{∗}j^{∗}F); F_{−1}(Z); ^{\underline{p23.5}} 22.12 onto; ≅; F_{−1}(Z × 𝔸^{1}); =; H^{0}(Z × 𝔸^{1}, F_{−1}).]

The left vertical map is onto by ^{\underline{p23.5}} 22.12, because S is local. It follows that the right vertical map is onto, as desired.

l24.5 Lemma 23.4. Let G be any sheaf on U×𝔸^{1} such that R^{q}j_{∗}G = 0 for 0 < q < n. Then the canonical map H^{n}(X × 𝔸^{1}, j_{∗}G) → H^{n}(U × 𝔸^{1}, G) is an injection.

Proof. Consider the Leray spectral sequence

H^{p}(X × 𝔸^{1}, R^{q}j_{∗}G) ⟹ H^{p+q}(U × 𝔸^{1}, G).

Using the assumption on the vanishing of the R^{q}j_{∗}G, it is easy to see that there is a short exact sequence:

0 → H^{n}(X × 𝔸^{1}, j_{∗}G) → H^{n}(U × 𝔸^{1}, G) → H^{0}(X × 𝔸^{1}, R^{n}j_{∗}G).

We have now completed the proof of homotopy invariance of the cohomology sheaves, which was promised in lecture 13 (as theorem ^{\underline{theorem''}} 13.7).

For the rest of this lecture, we fix a homotopy invariant Zariski sheaf with transfers F over a perfect field k. Because we have proven theorem ^{\underline{theorem''}} 13.7, we may use proposition ^{\underline{znis}} 13.8, which says that H_{Zar}^{∗}(X, F) ≅ H_{Nis}^{∗}(X, F). We will sometimes supress the subscript and just write H^{∗}(X, F).

znisz Corollary 23.5. If S is a smooth semi-local scheme over k and F is a homotopy invariant sheaf with transfers, then for all n > 0:

• H^{n}(S, F) = 0;

• H^{n}(S × T, F) = 0 for every open subset T of 𝔸_{k}^{1}.

Proof. (Cf. ^{\underline{znis}} 13.8.) By ^{\underline{restatement}} 23.1, each H^{n}(−, F) is a homotopy invariant presheaf with transfers. If E is the field of fractions of S, then H^{n}(Spec E, F) = 0 for n > 0 because dim E = 0. By ^{\underline{prp1}} 11.1, this implies that H^{n}(S, F) = 0.

Now H^{n}(X) = H^{n}(X × T, F) is also a homotopy invariant presheaf with transfers by ^{\underline{restatement}} 23.1, and H^{n}(S) injects into H^{n}(Spec E) = H^{n}(Spec(E) × T, F) by ^{\underline{prp1}} 11.1. By ^{\underline{trkl}} 2.9 and ^{\underline{step1.2nis}} 21.7, this group vanishes for n > 0.

Fdvr Example 23.6. Let (R, 𝔪) be a discrete valuation ring containing k, with field of fractions E and residue field K = R/𝔪. Setting S = Spec R and Z = Spec K, theorem ^{\underline{p23.5}} 22.12 yields F_{(S,Z)} ≅ F_{−1} and an exact sequence of Nisnevich sheaves on S, 0 → F → j_{∗}F → i_{∗}F_{−1} → 0. Since H_{Nis}^{1}(S, F) = 0 by ^{\underline{znisz}} 23.5, the Nisnevich cohomology sequence yields the exact sequence:

0 → F (Spec R) → F (Spec E) → F_{−1}(Spec K) → 0.

More generally, if R is a semilocal principal ideal domain with maximal ideals 𝔪_{i}, the same argument (using ^{\underline{znisz}} 23.5) yields an exact sequence:

0 → F (Spec R) → F (Spec E) → ⊕_{i}F_{−1}(Spec R/𝔪_{i}) → 0.

Exercise 23.7. If X is a smooth curve over k, show that F_{−1}(x) ≅ H_{x}^{1}(X, F) for every closed point x ∈ X. Conclude that there is an exact sequence

0 → F (X) → F (Spec k(X)) → ⨁_{x∈X} F_{−1}(x) → H_{Zar}^{1}(X, F) → 0.

fundthm Proposition 23.8. Let k be a perfect field and F a homotopy invariant Zariski sheaf with transfers. Then H^{n}(−, F)_{−1} ≅ H^{n}(−, F_{−1}) for all smooth X. That is, there is a natural isomorphism:

H_{Zar}^{n}(X × (𝔸^{1} − {0}), F) ≅ H_{Zar}^{n}(X, F) ⊕ H_{Zar}^{n}(X, F_{−1}).

Proof. Write T for 𝔸^{1} − {0} and consider the projection π : X × T → X. Let S be the local scheme at a point x of X. The stalk of R^{q}π_{∗}F at x is H^{q}(S × T, F), which vanishes for q > 0 by ^{\underline{znisz}} 23.5. Therefore the Leray spectral sequence degenerates to yield H^{n}(X×T, F) ≅ H^{n}(X, π_{∗}F). But π_{∗}F ≅ F ⊕F_{−1} by the definition of F_{−1}.

compare to \underline{preth} [CohTh, 4.35] pstHsupp Example 23.9. Let F be a homotopy invariant Zariski sheaf with transfers.

Combining proposition ^{\underline{fundthm}} 23.8 with ^{\underline{restatement}} 23.1 and ^{\underline{Hsupp}} 22.4, we get the formula:

H_{Z×{0}}^{n}(Z × 𝔸^{r}, F) ≅ H^{n−r}(Z, F_{−r}).

If Z = Spec(K) for a field K, this shows that H_{{0}}^{n}(𝔸_{K}^{r}, F) vanishes for n ≠ r, while the value of H_{{0}}^{r}(𝔸_{K}^{r}, F) at Spec(K) is F_{−r}(Spec(K)).

Pdual Lemma 23.10. Let S be a d-dimensional regular local scheme over a perfect field k. If F is a homotopy invariant sheaf with transfers and Z is the closed point of S, then H_{Z}^{n}(S, F) vanishes for n ≠ d, while H_{Z}^{d}(S, F) ≅ F_{−d}(Z).

Proof. Since the case d = 0 is trivial, and d = 1 is given in example ^{\underline{Fdvr}} 23.6, we may assume that d > 1. Write U for S − Z. Since F (S) injects into F (U) by \underline{prp1} 11.1, H_{Z}^{0}(S, F) = 0. For n > 0, we may use H^{n−1}(−, F), which is a homotopy invariant presheaf with transfers by ^{\underline{restatement}} 23.1. By ^{\underline{etpair}} 22.11 and two applications of ^{\underline{excisepair}} 22.7, we have

H_{Z}^{n}(S, F) ≅ H^{n−1}(−, F)_{(S,Z)} ≅ H^{n−1}(−, F)_{(Z×𝔸^{d},Z×0)} ≅ H_{Z×0}^{n}(Z × 𝔸^{d}, F).

By ^{\underline{pstHsupp}} 23.9, this group vanishes for n ≠ d, and equals F_{−d}(Z) if n = d.

If z is a point of X with closure Z, and A is an abelian group, let (i_{z})_{∗}(A) denote the constant sheaf A on Z, extended to a sheaf on X.

VV checked this 7/15/02 gersten Theorem 23.11. Let X be smooth over k, and F a homotopy invariant Zariski

sheaf with transfers. Then there is a canonical exact sequence of Zariski sheaves on X:

0 → F → ∐_{codim z=0} (i_{z})_{∗}(F) → ∐_{codim z=1} (i_{z})_{∗}(F_{−1}) → … → ∐_{codim z=r} (i_{z})_{∗}(F_{−r}) → …

Proof. It suffices to assume that X is local with generic point x_{0} and closed point x_{d}, and construct the exact sequence

0 → F (S) → F (x_{0}) → ∐_{codim z=1} (F_{−1}(z)) → … → ∐_{codim z=r} (F_{−d}(z)) → … → F (x_{d}) → 0.

When dim(X) = 1 this is ^{\underline{Fdvr}} 23.6, so we may assume that d = dim(X) > 1. For any r ≤ d, let H^{n}(X^{r}, F) denote the direct limit of the groups H^{n}(X − T, F) with codim(T) > r. For any Zariski sheaf F, and r > 0, the direct limit (over T and all Z of codimension r) of the long exact cohomology sequences H_{Z}^{∗}(X − T, F) → H^{∗}(X − T, F) → H^{∗}(X − Z − T, F) yields an exact sequence

0 → ∐_{{codim z \atop =r}}H_{z}^{0}(X_{z}, F) → F (X^{r}) → F (X^{r−1}) → ∐_{{codim z \atop =r}}H_{z}^{1}(X_{z}, F) → H^{1}(X^{r}, F) …

Each X_{z} is an r-dimensional local scheme. Hence the groups H_{z}^{n}(X_{z}, F) vanish except for n = r by ^{\underline{Pdual}} 23.10, and H_{z}^{r}(X_{z}, F) ≅ F_{−r}(z). For r > 0 this yields:

F (X) ≅ F (X^{d−1}) ≅ ⋯ F (X^{r}) ≅ ⋯ ≅ F (X^{1});
0 = H^{r}(X, F) ≅ H^{r}(X^{d−1}, F) ≅ ⋯ ≅ H^{r}(X^{r+1}, F);

and (since X^{0} is a point)

0 = H^{r}(X^{0}, F) ≅ H^{r}(X^{1}, F) ≅ ⋯ ≅ H^{r}(X^{r−1}, F).

Using these, we get exact sequences:

0 → F (X) → F (x_{0}) → ∐_{codim z=1} H_{z}^{1}(X_{z}, F) → H^{1}(X^{1}, F) → 0;

and (for 0 < r ≤ d)

0 → H^{r−1}(X^{r−1}, F) → ∐_{codim z=r} H_{z}^{r}(X_{z}, F) → H^{r}(X^{r}, F) → 0.

Splicing these together (and using ^{\underline{Pdual}} 23.10) yields the required exact sequence.

Remark 23.12. Since the sheaves (i_{z})_{∗}(F_{−r}) are flasque, theorem ^{\underline{gersten}} 23.11 gives a flasque resolution of the sheaf F. Taking global sections yields a chain complex which computes the cohomology groups H^{n}(X, F). This shows that the coniveau spectral sequence

E_{1}^{p,q} = ⨁_{codim x=p} H_{z}^{p+q}(X, F) ⟹ H^{p+q}(X, F)

degenerates, with E_{2}^{p,0} = H^{p}(X, F) and E_{2}^{p,q} = 0 for q ≠ 0.

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