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''}} ??. 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}} ?? 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''}} ??: 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}} ??) 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}} ??) 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}} ??) that Bloch’s higher Chow groups are presheaves with transfers over any field. The final ingredient is a result of Suslin (^{\underline{equiqiso}} ??) 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''}} ??. Using technical results from lecture 20, we first prove (in ^{\underline{n0}} ??) 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 6

1A The category Cor_{S} . . . . . . . . . . . . . . . . . . . . . . . . . 10 2 Presheaves with transfers 14 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

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}} ??), 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.

pstsing Exercise 2.10. Let X be a (non-smooth) scheme of finite type over k. For each smooth U, define ℤ_{tr}(X)(U) to be the group Cor(U, X) of ^{\underline{elcor}} 1.1. Show that the composition ∘ defined after ^{\underline{finpush}} 1.4 makes ℤ_{tr}(X) into a presheaf with transfers.

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 2.11. 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 2.12. 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 2.13. 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 2.14. 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 2.15. 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 2.16. 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 2.17. 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}} 2.16 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}} 2.15 and ^{\underline{2.11}} 2.17, we obtain

aletter Corollary 2.18. 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 2.19. (^{\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 2.20. 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 2.21. 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} 2.17, the identity map id = i_{1}^{∗}(C_{∗}h) is chain homotopic to 0 = i_{0}^{∗}(C_{∗}h).

cstaro Example 2.22. The paradigm for lemma ^{\underline{thm6}} 2.21 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}} 2.21.

Here is a second application of ^{\underline{thm6}} 2.21. Note that the projection p : X×𝔸^{1} → X induces a map ℤ_{tr}(X × 𝔸^{1}) → ℤ_{tr}(X).

217half Corollary 2.23. 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} 2.21 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 2.24. 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 2.25. 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}} 2.23, 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 2.26. 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}} ??, ^{\underline{lecnishxfer}} ??, and ^{\underline{lecmot}} ??;

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

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}} ?? from the derived category of PST(k). By ^{\underline{217half}} 2.23, 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}} 2.25.

kfunctpst Exercise 2.27. If k ⊂ F is a finite separable field extension, exercise ^{\underline{kfunct}} 1.12 implies that there are adjoint functors i^{∗} : PST(k) → PST(F), i_{∗} : PST(F) → PST(k). Show that there is a natural transformation π : i^{∗}i_{∗}M → M whose composition with the canonical map M → i^{∗}i_{∗}M is multiplication by [F : k].

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}} 2.12, 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}} ?? 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}} 2.12.

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}} ?? 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}} 2.13, (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}} 2.17, 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 a maximal prime-to-l extension 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}} 2.12 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}} ?? 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}} ??) 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}} ??). 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}} ??). 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}} 2.26) 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}} 2.20, because C_{0}(X/S) = ℤ_{tr}(X)(Spec k). Also by ^{\underline{h0ztr}} 2.20, 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}} ?? 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 quasiaffine 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} ??.

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]).

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