Cohomological theory of presheaves with transfers.

Vladimir Voevodsky

Contents

1 Introduction 1

2 Divisors with compact support on relative curves. 3

3 Pretheories. 9 3.1 Categories of pretheories. . . . . . . . . . . . . . . . . . . . . . 9 3.2 Homotopy invariant pretheories. . . . . . . . . . . . . . . . . . 11 3.3 Elementary properties of transfer maps. . . . . . . . . . . . . . 15 3.4 Examples of pretheories. . . . . . . . . . . . . . . . . . . . . . 20

4 Zariski sheaves associated with pretheories. 21 4.1 Technical lemmas. . . . . . . . . . . . . . . . . . . . . . . . . . 21 4.2 Pretheories in a neighborhood of a smooth subscheme. . . . . . 27 4.3 Pretheories on curves over a field. . . . . . . . . . . . . . . . . 31 4.4 Pretheories on semi-local schemes. . . . . . . . . . . . . . . . . 32 4.5 Zariski cohomology with coefficients in pretheories. . . . . . . . 36 4.6 The Gersten resolution. . . . . . . . . . . . . . . . . . . . . . . 41

5 Sheaves in Nisnevich and etale topologies associated with pretheories. 43 5.1 Pretheoretical structures on Nisnevich cohomology. . . . . . . . 43 5.2 Cohomology of affine space and the comparison theorem. … 45 5.3 Applications to Suslin homology. . . . . . . . . . . . . . . . . . 47 5.4 Cohomology of blowups. . . . . . . . . . . . . . . . . . . . . . 50 5.5 Sheaves in the etale topology associated with pretheories. … 53

1 Introduction

Let k be a field and Sm/k be the category of smooth schemes over k. In this paper we study contravariant functors from the category Sm/k to additive categories equipped with transfer maps. More precisely we consider contravariant functors F : (Sm/k)^{op} → A together with a family of morphisms ϕ_{X/S}(Z) : F (X) → F (S) given for any smooth curve X → S over a smooth scheme S over k and a relative divisor Z on X over S which is finite over S. If these maps satisfy some natural properties (see definition 3.1) such a collection of data is called a pretheory over k. Some examples of pretheories are given in §3.4.

A pretheory F is called homotopy invariant if for any smooth scheme X one has F (X × A^{1}) = F (X). For any pretheory F with values in an abelian category we define a family of homotopy invariant pretheories \underline{h}_{i}(F). Applying this construction to particular pretheories F one can get algebraic cycle homology and bivariant morphic homology of Friedlander and Gabber (see [1]) or algebraic singular homology and algebraic Lawson homology defined in [7].

In the first section we prove some elementary facts about divisors with proper support on relative curves (see also [7]). In Section 3 we define pretheories and prove some of their basic properties.

In the third section we study local behavior of homotopy invariant pretheories. In most cases we assume that our pretheory takes values in the category of abelian groups and study the sheaf F_{Zar} in the Zariski topology on Sm/k associated with the presheaf F. The main theorem (Theorem 4.27) of this section states that if F is a homotopy invariant pretheory over a perfect field k with values in the category of abelian groups then the functors U → H_{Zar}^{i}(U, F_{Zar}) have canonical structures of homotopy invariant pretheories.

As an application we show in §4.6 that for any such pretheory F the sheaf F_{Zar} has the Gersten resolution.

In the next section we consider the sheaves in the Nisnevich topology (see [5]) associated with pretheories with values in the category of abelian groups. It turns out that the Nisnevich topology has several serious advantages over the Zariski topology in the context of pretheories. In particular for any reasonable (see Definition 5.1), not necessarily homotopy invariant, pretheory F with values in the category of abelian groups the functors U → H_{Nis}^{i}(U, F_{Nis}) have canonical structures of pretheories (Theorem 5.3). On the other hand, using the results of Section 4 we show that for any homotopy invariant pretheory F over a perfect field with values in the category of abelian groups one has canonical isomorphisms

H_{Nis}^{i}(−, F_{Nis}) ≅ H_{Zar}^{i}(−, F_{Zar})

for all i ≥ 0 (Theorem 5.7). It allows us to reformulate all the results of the previous section for the Nisnevich topology. In the last paragraph we apply our technique to prove the existence of a long exact sequence of Mayer-Vietoris type for algebraic singular homology (Theorem 5.17). Though apparently we are only interested in open coverings the use of the Nisnevich topology in the proof is crucial.

In paragraph 5.4 we consider the behavior of homotopy invariant pretheories with respect to blow-ups. Proposition 5.21 has important applications to algebraic cycle cohomology (see [2]).

In the last paragraph we consider the etale topology. The first result is the rigidity Theorem 5.25 (which is just a version of the rigidity theorem proven in [7]). It says that the sheaf in the etale topology on Sm/k associated with a homotopy invariant pretheory with values in the category of torsion abelian groups of torsion prime to characteristic of the field k is locally constant. For rational coefficients we get in the case of a perfect field k isomorphisms (Theorem 5.28):

H_{et}^{i}(−, F_{et}) ⊗ Q ≅ H_{Zar}^{i}(−, F_{Zar}) ⊗ Q.

Most results of this paper are formulated for smooth schemes over a perfect field. This restriction on the base scheme is important. In fact almost all the results of this paper are wrong even for smooth schemes over the spectrum of a discrete valuation ring of characteristic zero.

The idea to consider presheaves with transfer (instead of qfh-sheaves) which allows us to obtain the “correct” answers with integral coefficients is due completely to my conversations with A.Suslin. I am aslo very grateful to D. Kazhdan for his patient interest in my work and to Yu. Tschinkel who told me about the existence of the Nisnevich topology.

2 Divisors with compact support on relative curves.

Let S be a smooth scheme. A curve p : X → S over S is an equi-dimensional morphism of finite type of relative dimension one. The morphism ∅ → S will also be considered as a curve over S.

For any curve X → S over a smooth scheme S denote by c_{equi}(X/S, 0) the free abelian group generated by closed integral subschemes of X which are finite over S and surjective over an irreducible component of S.

Let Z be any closed subscheme of X which is finite over S. Consider the cycle Cycl_{X}(Z) of X in Z. It follows immediately from our definition that it belongs to c_{equi}(X/S, 0) if and only if any irreducible component of Z dominates an irreducible component of S. In particular Cycl_{X}(Z) belongs to c_{equi}(X/S, 0) for any closed subscheme Z in X which is finite and flat over S.

As was shown in [8] for any morphism of smooth schemes f : S′ → S one can define a homomorphism

cycl(f) : c_{equi}(X/S, 0) → c_{equi}(X ×_{S} S′/S′, 0)

such that the following conditions hold:

1. For a composable pair of morphisms S′′ →^{g} S′ →^{f} S one has

cycl(f ∘ g) = cycl(g) ∘ cycl(f)

2. For any dominant morphism f : S′ → S of smooth schemes and an integral closed subscheme Z in c_{equi}(X/S, 0) one has

cycl(f)(Z) = Cycl_{X×_{S}S′}(Z ×_{S} S′)

3. For any morphism of smooth schemes f : S′ → S and a closed subscheme Z in X which is finite and flat over S one has:

cycl(f)(Cycl_{X}(Z)) = Cycl_{X×_{S}S′}(Z ×_{S} S′).

It can be shown easily (using Platification Theorem, see [6] or [8, Th. 2.2.2]) that homomorphisms cycl(f) are uniquely determined by the conditions (1)-(3) given above.

Let g : X_{1} → X_{2} be a morphism of curves over a smooth scheme S and Z be an integral closed subscheme in X_{1} which belongs to c_{equi}(X_{1}/S, 0). Since it is proper over S its image g(Z) in X_{2} is a closed subscheme which belongs to c_{equi}(X_{2}/S, 0). The morphism Z → g(Z) is a finite surjective morphism of integral schemes. Denote its degree (i.e. the degree of the corresponding function fields extension) by n(Z, g).

We define the direct image homomorphism

g_{∗} : c_{equi}(X_{1}/S, 0) → c_{equi}(X_{2}/S, 0)

setting

g_{∗}(Z) = n(Z, g)g(Z).

The following proposition is a particular case of [8, Prop. 3.6.2].

Proposition 2.1 Let X_{1} → S, X_{2} → S be a pair of curves over a smooth scheme S, g : X_{1} → X_{2} be a morphism over S and f : S′ → S be a morphism of smooth schemes. Then the following diagram commutes:

c_{equi}(X_{1}/S, 0) →^{cycl(f)} c_{equi}(X_{1} ×_{S} S′/S′, 0)
g_{∗} ↓  ↓ g_{∗}
c_{equi}(X_{2}/S, 0) →^{cycl(f)} c_{equi}(X_{2} ×_{S} S′/S′, 0).

The following technique provides us with a method of construction of elements in the groups c_{equi}(X/S, 0) which will be used extensively in this paper.

Let D be a Cartier divisor on X. Denote by L_{D} the corresponding line bundle on X. There is a canonical trivialization t_{D} : O_{U} → (L_{D})_{|U} of L_{D} on U = X −D. If D is effective this trivialization extends to a canonical section s_{D} : O_{X} → L_{D} of L_{D} on X. Suppose that X is normal and connected. Then for any line bundle L on X, open subset U of X and a trivialization t of L on U there exists a unique Cartier divisor D(L, U, t) and an isomorphism L ≅ L_{D(L,U,t)} such that t = t_{L_{D}} (see [4, 21.1.4]). Note that D(L, U, t) considered as a closed subset of X lies in X − U.

Let p̄ : X̄ → S be a proper normal curve over a smooth scheme S and X ⊂ X̄ be an open subscheme in X̄ which is quasi-affine and smooth over S. Denote by X_{∞} the reduced scheme X̄ − X.

Proposition 2.2 Let L be a line bundle on X̄, U be an open neighborhood of X_{∞} in X̄ and t : O_{U} → L_{|U} be a trivialization of L on U. Consider the Cartier divisor D(L, U, t) definied by the triple (L, U, t). Then Cycl_{X}(D(L, U, t)) belongs to c_{equi}(X/S, 0).

Proof: Let

Cycl_{X}(D(L, U, t)) = ∑ n_{i}Z_{i}

where Z_{i} are integral closed subschemes of X. Since p̄ : X̄ → S is proper and X is quasi-affine over S the open subset U is dense in all fibers of p̄ and X̄ − U is finite over S. Since D(L, U, t) as a closed subset in X̄ belongs to X̄ − U the closed subschemes Z_{i} are finite over S. On the other hand each of Z_{i} is of pure codimension one in X and being finite over S it must dominate an irreducible component of S.

Lemma 2.3 Under the assumption of Proposition 2.2 consider a morphism of smooth schemes f : S′ → S. Then for any triple (L, U, t) as above we have:

cycl(f)(Cycl_{X}(D(L, U, t))) = Cycl_{X×_{S}S′}(D(f_{X}^{∗}(L), f_{X}^{−1}(U), f_{X}^{∗}(t)))

where f_{X} is the projection X ×_{S} S′ → X.

Proof: It follows from the fact that D(L, U, t) is flat over S by the third property of base change homomorphisms.

Let Z_{0}, Z_{1} be two elements of c_{equi}(X/S, 0). We say that they are equivalent if there is an element Z in S × A^{1} such that

cycl(i_{0})(Z) = Z_{0}
cycl(i_{1})(Z) = Z_{1}

where i_{0}, i_{1} : S → S × A^{1} are the closed embeddings which correspond to the points 0 and 1 of A^{1} respectively. We denote the group of equivalence classes of elements of c_{equi}(X/S, 0) with respect to this equivalence realtion by h_{0}(X/S). It follows immediately from the definition that homomorphisms cycl(f) give us homomorphisms h_{0}(X/S) → h_{0}(X ×_{S} S′/S′) and by Proposition 2.1 the groups h_{0}(X/S) are covariantly functorial with respect to morphisms X_{1} → X_{2} of curves over S.

Definition 2.4 Let p : X → S be a curve over a smooth scheme S. A good compactification of X is a pair (p̄: X̄ → S, j : X → X̄) where X̄ is a proper curve over S and j is an open embedding over S such that the closed subset X_{∞} = X̄ − X in X̄ has an open neighborhood which is affine over S.

Note that if (p̄ : X̄ → S, j : X → X̄) is a good compactification of a curve p : X → S and f : S′ → S is a morphism of smooth schemes then (p̄′ : X̄ ×_{S} S′ → S′, j ×_{S} Id_{S′} : X ×_{S} S′ → X̄ ×_{S} S′) is a good compactification of the curve p′ : X ×_{S} S′ → S′.

The following lemma shows that if a normal curve over a smooth scheme has a good compactification then we may always choose it in such a way that the corresponding proper curve is also normal.

Lemma 2.5 Let p : X → S be a normal curve over a smooth scheme S and (p̄ : X̄ → S, j : X → X̄) be a good compactification of X. Let further X̄′ be the normalization of X̄. Then the open embedding j : X → X̄ defines an open embedding j′ : X → X̄′ and the pair (p̄′ : X̄′ → S, j′ : X → X̄′) is a good compactification of X.

Proof: Since the morphism X̄′ → X̄ is finite it is also affine which implies that the preimage of the affine open neighborhood of X_{∞} in X̄′ is an affine open neighborhood of X′_{∞} = X̄′ − X in X̄′.

Proposition 2.6 Let p : X → S be a smooth quasi-affine curve over a smooth scheme S and (p̄: X̄ → S, j : X → X̄) be a good compactification of X. Let X_{∞} be the reduced scheme X̄ − X.

Let further L be a line bundle on X̄ and s : O_{X_{∞}} → L_{|X_{∞}} be a trivialization of L over X_{∞}. Then the following statements hold.

1. For any two extensions s̃_{1}, s̃_{2} of the trivialization s to an open neighborhood U of X_{∞} the cycles Cycl_{X}(D(L, U, s̃_{1})), Cycl_{X}(D(L, U, s̃_{2})) give the same element in h_{0}(X/S).

2. If S is affine there exists an open neighborhood U of X_{∞} in X̄ and an extension s̃ : O_{U} → L_{|U} of the trivialization s to a trivialization of L on U.

Proof: The second statement follows trivially from the fact that X_{∞} has an affine open neighborhood in X̄.

To prove the first one consider the curve

X̄ × A^{1} → S × A^{1}

over S × A^{1} and let L′ be the pull-back of L with respect to the projection X̄ × A^{1} → X̄. The trivializations s̃_{1}, s̃_{2} define trivializations s̃′_{1}, s̃′_{2} of L′ on U×A^{1} which coincide on X_{∞}×A^{1}. Consider the section ts̃_{1}′+(1−t)s̃′_{2} of L′ on U ×A^{1} where t is the regular function given by the projection U ×A^{1} → A^{1}. Its restriction to X_{∞}×A^{1} is equal to the trivialization induced by s and hence it definies a trivialization h of L′ on an open neighborhood V of X_{∞} × A^{1} in X̄ ×A^{1}. The divisor D(L′, V, h) associated with h gives us by Proposition 2.2 an element Z in c_{equi}(X × A^{1}/S × A^{1}, 0). Consider the closed embeddings i_{0}, i_{1} : S → S×A^{1} which correspond to the points 0 and 1 on A^{1} respectively. Lemma 2.3 implies that

cycl(i_{0})(Z) = Cycl_{X}(D(L, U, s̃_{2}))

and

cycl(i_{1})(Z) = Cycl_{X}(D(L, U, s̃_{1})).

Let p : X → S be a smooth quasi-affine curve over an affine smooth scheme S and (p̄ : X̄ → S, j : X → X̄) be a good compactification of X. Denote by P ic(X̄, X_{∞}) the set of isomorphism classes of pairs of the form (L, s) where L is a line bundle on X̄ and s is a trivialization of L on X_{∞}. Tensor multiplication of line bundles and their sections gives a structure of abelian group on P ic(X̄, X_{∞}). Proposition 2.6 implies that there is a well defined homomorphism

P ic(X̄, X_{∞}) → h_{0}(X/S).

Lemma 2.3 implies immediately that this homomorphism commutes in the obvious sense with base change homomorphisms cycl(−).

Note that if we consider pairs of the form (OX̄, s) we get a homomorphism

O^{∗}(X_{∞}) → h_{0}(X/S).

Remark 2.7 It can be shown (see [7]) that for a smooth quasi-affine curve X → S over a smooth affine scheme S and a good compactification of X the homomorphism P ic(X̄, X_{∞}) → h_{0}(X/S) is in fact an isomorphism. We will not use this fact in the present paper.

3 Pretheories.

3.1 Categories of pretheories.

Let k be a field. We denote by Sm/k the category of smooth schemes of finite type over k.

Definition 3.1 A pretheory (F, ϕ, A) over a field k is the following collection of data.

1. An additive category A.

2. A contravariant functor F : Sm/k → A.

3. For any object S of Sm/k and any smooth curve p : X → S over S a homomorphism of abelian groups

ϕ_{X/S} : c_{equi}(X/S, 0) → Hom(F (X), F (S)).

These data should satisfy the following conditions.

1. For any object S of Sm/k, any smooth curve p : X → S over S and any S-point i : S → X of X one has ϕ_{X/S}(i(S)) = F (i).

2. Let f : S_{1} → S_{2} be a morphism of smooth schemes of finite type over k and p : X_{2} → S_{2} be a smooth curve over S_{2}. Consider the Cartesian square

\begin{matrix}X_{1} & →^{g} & X_{1} \\ ↓ & & ↓ \\ S_{1} & →^{f} & S_{2}. \end{matrix}

Then for any Z in c_{equi}(X_{2}/S_{2}, 0) one has

F (f) ∘ ϕ_{X_{2}/S_{2}}(Z) = ϕ_{X_{1}/S_{1}}(f^{∗}(Z)) ∘ F (g).

3. For any pair X, Y of objects of Sm/k the canonical morphism

F (X ∐ Y) → F (X) ⊕ F (Y)

is an isomorphism.

Definition 3.2 Let (F_{1}, ϕ_{1}, A), (F_{2}, ϕ_{2}, A) be a pair of pretheories with values in an additive category A. A morphism of pretheories

T : (F_{1}, ϕ_{1}, A) → (F_{2}, ϕ_{2}, A)

is a morphism T : F_{1} → F_{2} of presheaves on the category Sm/k such that for any smooth scheme S over k, any smooth curve p : X → S over S and any element Z ∈ c_{equi}(X/S, 0) the following diagram commutes:

F_{1}(X) →^{ϕ_{1}(Z)} F_{1}(S)
T_{X} ↓  ↓ T_{S}
F_{2}(X) →^{ϕ_{2}(Z)} F_{2}(S).

Denote by P reth(k, A) the category of pretheories over k with values in an additive category A. It is obviously an additive category. Moreover if A is an abelian category one has the following obvious proposition.

Proposition 3.3 Let k be a field and A be an abelian category. Then the category P reth(k, A) is abelian and the forgetfull functor

P reth(k, A) → P reshv(Sm/k, A)

to the category of A-valued presheaves on Sm/k is exact.

Let (F, ϕ, A) be a pretheory over a field k and U be an object of Sm/k. We define a pretheory (F_{U}, ϕ_{U}, A) as follows:

1. The presheaf F_{U} takes a smooth scheme U of finite type over k to the object F (U ×_{k} X).

2. Let p : X → S be a smooth curve over a smooth scheme S of finite type over k and Z be an element of c_{equi}(X/S, 0). Then we define (ϕ_{U})_{X/S}(Z) to be the homomorphism ϕ(p^{∗}(Z)) where p is the projection S×U → S.

One can easily see that (F_{U}, ϕ_{U}, A) is indeed a pretheory and this construction gives us a functor

(Sm/k)^{op} × P reth(k, A) → P reth(k, A).

For any pretheory (F, ϕ, A) with values in an abelian category A we denote by (F_{−1}, ϕ_{−1}, A) the pretheory

coker((F_{A^{1}}, ϕ_{A^{1}}, A) → (F_{A^{1}−{0}}, ϕ_{A^{1}−{0}}, A)).

Consider a separable field extension k ⊂ L and let f : Spec(L) → Spec(k) be the corresponding morphism of schemes. Then for any pretheory (F, ϕ, Ab) over k where Ab is the category of abelian groups the inverse image p^{∗}(F) of the presheaf F has a canonical structure of a pretheory over L. It gives us an exact functor p^{∗} : P reth(k, Ab) → P reth(L, Ab). If U is a smooth scheme over k then one has a canonical isomorphism:

p^{∗}((F_{U}, ϕ_{U}, Ab)) ≅ p^{∗}((F, ϕ, Ab))_{U_{L}}.

In particular there is a canonical isomorphism

p^{∗}((F, ϕ, Ab)_{−1}) = p^{∗}((F, ϕ, Ab))_{−1}.

3.2 Homotopy invariant pretheories.

Definition 3.4 Let k be a field and (F, ϕ, A) be a pretheory over k. It is called homotopy invariant if for any object X of Sm/k the projection X × A^{1} → X induces an isomorphism F (X) → F (X × A^{1}).

For any field k and an additive category A denote by HIP reth(k, A) the full subcategory in the category P reth(k, A) which consists of homotopy invariant pretheories. In the following proposition we summarize basic properties of homotopy invariant pretheories which follow directly from our definitions.

Proposition 3.5 Let k be a field and A be an abelian category. Then the category HIP reth(k, A) is abelian and the inclusion functor HIP reth(k, A) → P reth(k, A) is exact. Moreover any subpretheory of a homotopy invariant pretheory is a homotopy invariant pretheory and any extension of homotopy invariant pretheories is a homotopy invariant pretheory.

Let F be a presheaf on the category Sm/k with values in an abelian category. Let Δ^{•} = (Δ^{n}, ∂_{i}^{n}, s_{i}^{n}) be the standard cosimplicial object in Sm/k (see [2, §4]). Denote by \underline{C}_{∗}(F) the complex of presheaves such that \underline{C}_{n}(F)(U) = F (U × Δ^{n}) and the differencial is given by the alternated sums of morphisms induced by ∂_{i}^{n}. Denote by \underline{h}_{i}(F) the cohomology presheaves H^{−i}(\underline{C}_{∗}(F)) of this complex.

If (F, ϕ, A) is a pretheory with values in an abelian category A then \underline{C}_{∗}(F) is a complex of pretheories (or equivalently a pretheory with values in the category of complexes over A) and in particular the presheaves \underline{h}_{i}(F) have canonical structures of pretheories.

Proposition 3.6 For any pretheory (F, ϕ, A) over k where A is an abelian category the pretheories \underline{h}_{i}(F, ϕ, A) are homotopy invariant.

Proof: Since a pretheory is homotopy invariant if and only if the underlying presheaf is it is sufficient to show that for any presheaf of abelian groups F on Sm/k the presheaves \underline{h}_{i}(F) are homotopy invariant. Let U be a smooth scheme over k. We have to show that the morphisms \underline{h}_{i}(F)(U) → \underline{h}_{i}(F)(U × A^{1}) induced by the projection U × A^{1} → U are isomorphisms.

Denote by i_{0}, i_{1} : U → U × A^{1} the closed embeddings Id_{U} × {0} and Id_{U} × {1} respectively. Let us show first that the morphisms

i_{0}^{∗} : \underline{h}_{i}(F)(U × A^{1}) → \underline{h}_{i}(F)(U)
i_{1}^{∗} : \underline{h}_{i}(F)(U × A^{1}) → \underline{h}_{i}(F)(U)

coincide. It is sufficient to prove that the corresponding morphisms of complexes of abelian groups \underline{C}_{∗}(F)(U × A^{1}) → \underline{C}_{∗}(F)(U) are homotopic.

Define homomorphisms

s_{n} : F (U × A^{1} × Δ^{n}) → F (U × Δ^{n+1})

by the formula

s_{n} = ∑_{i=0}^{n}(−1)^{i}(Id_{U} × ψ_{i})^{∗}

where ψ_{i} : Δ^{n+1} → Δ^{n} × A^{1} is the linear isomorphism taking v_{j} to v_{j} × 0 if j ≤ i or to v_{j−1} × 1 if j > i (here v_{j} = (0, …, 1, …, 0) is the j-th vertex of Δ^{n+1} (resp. Δ^{n})). A straightforward computation shows that sd + ds = i_{1}^{∗} − i_{0}^{∗}.

Consider now the morphism

Id_{U} × μ : U × A^{1} × A^{1} → U × A^{1}

where μ : A^{1} × A^{1} → A^{1} is given by multiplication of functions. Applying the previous result to the embeddings

i_{0}, i_{1} : U × A^{1} → (U × A^{1}) × A^{1}

we conclude that the homomorphism

\underline{h}_{i}(F)(U × A^{1}) → \underline{h}_{i}(F)(U × A^{1})

induced by the composition

U × A^{1} →^{pr_{1}} U →^{i_{0}} U × A^{1}

is the identity homomorphism which implies immediately the assertion of the proposition.

Note that we have a canonical morphism:

(F, ϕ, A) → \underline{h}_{0}(F, ϕ, A).

The following proposition follows easily from the definition of \underline{h}_{0}(−).

Proposition 3.7 For any pretheory (F, ϕ, A) and any homotopy invariant pretheory (G, ψ, A) the homomorphism

Hom(\underline{h}_{0}(F, ϕ, A), (G, ψ, A)) → Hom((F, ϕ, A), (G, ψ, A))

induced by the projection

(F, ϕ, A) → \underline{h}_{0}(F, ϕ, A).

is an isomorphism.

Thus \underline{h}_{0}(−) is a functor P reth(k, A) → HIP reth(k, A) which is left adjoint to the inclusion functor. In particular it is right exact. The functors \underline{h}_{i}(−) are in some sense the left derived functors of \underline{h}_{0}(−). More precisely one has:

Proposition 3.8 Let

0 → (F_{1}, ϕ_{1}, A) → (F_{2}, ϕ_{2}, A) → (F_{3}, ϕ_{3}, A) → 0

be a short exact sequence of pretheories with values in an abelian category A. Then there is a canonical long exact sequence of homotopy invariant pretheories of the form:

… → \underline{h}_{i}(F_{1}, ϕ_{1}, A) → \underline{h}_{i}(F_{2}, ϕ_{2}, A) → \underline{h}_{i}(F_{3}, ϕ_{3}, A) → \underline{h}_{i−1}(F_{1}, ϕ_{1}, A) → …

Proof: It follows immediately from the obvious fact that F → \underline{C}_{∗}(F) is an exact functor from the category of presheaves to the category of complexes of presheaves.

Unfortunately one can not apply usual techniques from homological algebra to define \underline{h}_{i}(−) as the left derived functors of \underline{h}_{0} because even if A is the category of abelian groups there is not enough pretheories (F, ϕ, A) such that \underline{h}_{i}(F, ϕ, A) = 0 for all i > 0.

We will often use below the following simple results.

Proposition 3.9 Let (F, ϕ, A) be a pretheory such that A is an abelian category and U be a smooth scheme of finite type over k. Then there are canonical isomorphisms of pretheories

(\underline{h}_{i}(F, ϕ, A))_{U} ≅ \underline{h}_{i}((F, ϕ, A)_{U})

in particular if (F, ϕ, A) is a homotopy invariant pretheory then (F, ϕ, A)_{U} is also homotopy invariant.

Proposition 3.10 Let k ⊂ L be a separable extension of fields and f : Spec(L) → Spec(k) be the corresponding morphism. Then the inverse image functor f^{∗} : P reth(k, Ab) → P reth(L, Ab) takes homotopy invariant pretheories to homotopy invariant pretheories and commutes with the corresponding functors \underline{h}_{i}.

Proposition 3.11 Let (F, ϕ, A) be a pretheory over k. It is homotopy invariant if and only if for any smooth scheme S of finite type over k and any smooth curve X → S over S the morphism ϕ_{X/S} can be factored through the canonical epimorphism c_{equi}(X/S, 0) → h_{0}(X/S) (see the definition of h_{0}(−) after Lemma 2.3).

Proof: The “only if” part is obvious. Suppose that (F, ϕ, A) is a pretheory such that the morphisms ϕ_{X/S} can be factored through h_{0}(X/S). Let us show that it is homotopy invariant.

Let i : X → X × A^{1} be the embedding which corresponds to the point {0} of A^{1}. It is obviously sufficient to show that the following composition is equal to the identity morphism:

F (X × A^{1}) →^{F (i)} F (X) →^{F (pr_{1})} F (X × A^{1}).

Consider the curve (X × A^{1}) × A^{1} over X × A^{1} and let

Z ∈ h_{0}((X × A^{1}) × A^{1}/X × A^{1})

be the element which corresponds to the divisor (X × A^{1}) × {0}. Then by the properties (1) and (2) of pretheories our composition is equal to the composition:

F (X × A^{1}) →^{F (pr_{1}×Id_{A^{1}})} F ((X × A^{1}) × A^{1}) →^{ϕ(Z)} F (X × A^{1}).

The element Z coincides in h_{0}((X × A^{1}) × A^{1}/X × A^{1}) with the element represented by the graph of the morphism

Id_{X} × Δ : X × A^{1} → X × A^{1} × A^{1}

and hence by the property (1) of pretheories we have ϕ(Z) = F (Id_{X} × Δ). Therefore our composition is equal to

F ((pr_{1} × Id_{A^{1}}) ∘ (Id_{X} × Δ)) = F (Id_{X×A^{1}}) = Id_{F (X×A^{1})}.

Proposition is proven.

3.3 Elementary properties of transfer maps.

Proposition 3.12 Let k be a field and (F, ϕ, A) be a homotopy invariant pretheory over k. Let S be an object of Sm/k, j : U → X be an open embedding of smooth curves over S and a be an element of c_{equi}(U/S, 0). Then one has

ϕ_{X/S}(j_{∗}(a)) = ϕ_{U/S}(a) ∘ F (j).

Proof: Let Z be the closed subset X − U of X. Consider the smooth curve W = X × A^{1} − Z × {0} over S × A^{1}. Let p : S × A^{1} → S be the canonical projection. Then there exists a unique element ã in c_{equi}(W/S × A^{1}, 0) such that its image in c_{equi}(X × A^{1}/S × /A^{1}, 0) equals cycl(p)(j_{∗}(a)). Let i_{0}, i_{1} : S → S × A^{1} be the morphisms which correspond to the points {0} and {1} of A^{1} respectively. Then the fiber of W over i_{0} is canonically isomorphic to U and the fiber of W over i_{1} is canonically isomorphic to X and with respect to these isomorphisms we have

cycl(i_{0})(ã) = a
cycl(i_{1})(ã) = j_{∗}(a).

Let q : W → X be the obvious projection. Consider the following diagram:

F (X) →^{ϕ_{X/S}(j_{∗}(a))}  F (S)
Id ↗  ↑  ↑ F (i_{1})
F (X) →^{F (q)} F (W) →^{ϕ_{W/S×A^{1}}(ã)} F (S × A^{1})
F (j) ↘  ↓  ↓ F (i_{0})
F (U) →^{ϕ_{U/S}(a)}  F (S)

It is commutative by the property (2) of pretheories. Since F is homotopy invariant F (i_{0}), F (i_{1}) are isomorphisms and we have

ϕ_{U/S}(a) ∘ F (j) = ϕ_{X/S}(j_{∗}(a)).

The proposition is proven.

Let k ⊂ E_{1} ⊂ E_{2} be finite separable extensions of k and f : Spec(E_{2}) → Spec(E_{1}) be the corresponding morphism of smooth schemes over k. For any pretheory (F, ϕ, A) over k we define a morphism

F_{∗}(f) : F (Spec(E_{2})) → F (Spec(E_{1}))

as follows. Consider the smooth curve A_{E_{2}}^{1} over Spec(E_{1}). The point {0} of A^{1} gives as an element in c_{equi}(A_{E_{2}}^{1}/Spec(E_{1}), 0) and we set F_{∗}(f) to be the composition

F (Spec(E_{2})) →^{F (p)} F (A_{E_{2}}^{1}) →^{ϕ_{A{1 \atop E_{2}} /Spec(E_{1})}(\{0\})} F (Spec(E_{1}))

where p : A_{E_{2}}^{1} → Spec(E_{2}) is the canonical projection.

Proposition 3.13 Let C be a smooth curve over k and x be a closed point of C. Suppose that the residue field k_{x} of x is separable over k and consider the following diagram of smooth schemes over k

Spec(k_{x}) →^{i}  C
p_{0}  ↘ ↓ p
Spec(k)

Then for any homotopy invariant pretheory (F, ϕ, A) over k the following diagram commutes

F (C)  →^{F (i)} F (Spec(k_{x}))
ϕ_{C/Spec(k)}(x)  ↓  ↙  F_{∗}(p_{0})
F (Spec(k))

Proof: Since k_{x} is separable over k the embedding Spec(k_{x}) → C is a smooth pair over k and hence there exists an open neighborhood U of x in C and an etale morphism f : U → A_{k}^{1} such that f^{−1}({0}) = x. By Proposition 3.12 we may suppose that U = C.

Consider a Cartesian square of the form

X
f̃ ↙  ↘ pr
A_{k}^{2}  C
g ↘  ↙ f
A_{k}^{1}

where g is the morphism given by multiplication of functions (i.e g(u, v) = uv).

The composition

X →^{f̃} A_{k}^{2} →^{pr_{1}} A_{k}^{1}

defines a smooth curve over A_{k}^{1} (here pr_{1} is the projection (u, v) ↦ u). Consider the reduced closed subscheme Z = f̃^{−1}({v = 0}) in X. One can easily see that it is integral and belongs to c_{equi}(X/A_{k}^{1}, 0). The fibers of X/A_{k}^{1} over the points {0} and {1} of A_{k}^{1} are isomorphic to A_{k_{x}}^{1} and C respectively and the intersections of these fibers with Z are the points {0} (on A_{k_{x}}^{1}) and x (on C).

Let a be an element of F (C). Consider the element ϕ_{X/A_{k}^{1}}(Z)(F (pr)(a)). By the property (2) of pretheories its restriction to the fiber over {0} is equal to F_{∗}(p_{0})(F (i)(a)) and its restriction to the fiber over {1} is equal to ϕ_{S/Spec(k)}(x)(a). Therefore these two elements coincide since F is homotopy invariant.

Proposition 3.14 Let k ⊂ E_{1} ⊂ E_{2} be finite separable extensions of a field k and d = [E_{2} : E_{1}]. Consider the diagram

Spec(E_{2}) →^{f} Spec(E_{1})
p_{2}  ↘ ↓ p_{1}
Spec(k)

Then for any homotopy invariant pretheory (F, ϕ, A) over k one has

F_{∗}(p_{2}) ∘ F (f) = d F_{∗}(p_{1}).

Proof: Consider the smooth curve A_{E_{1}}^{1} over Spec(k). Since E_{2} is separable over k it has a primitive element and hence there exists a closed point x on A_{E_{1}}^{1} such that the field of functions of x is isomorphic to E_{2}. Let us identify Spec(E_{1}) with the point {0} of A_{E_{1}}^{1}. Then we have two morphisms f_{1}, f_{0} : Spec(E_{2}) → A_{E_{1}}^{1} where f_{0} corresponds to the morphism f with respect to our identification and f_{1} corresponds to the point x. Since the compositions of f_{1} and f_{0} with the canonical projection A_{E_{1}}^{1} → Spec(E_{1}) coincide these two morphisms are homotopic over Spec(k) and hence F (f_{0}) = F (f_{1}) since F is homotopy invariant.

On the other hand as an element in h_{0}(A_{E_{1}}^{1}/Spec(k)) the point x is equal to d{0}. Since F is homotopy invariant we conclude by Proposition 3.13 that

dF_{∗}(p_{1}) = dϕ_{A_{E_{1}}^{1}/Spec(k)}({0}) = ϕ_{A_{E_{1}}^{1}/Spec(k)}(x) =
= F_{∗}(p_{2}) ∘ F (f_{1}) = F_{∗}(p_{2}) ∘ F (f).

The proposition is proven.

Corollary 3.15 Let k ⊂ E_{1} ⊂ E_{2} be finite separable extensions of a field k, d = [E_{2} : E_{1}] and f : Spec(E_{2}) → Spec(E_{1}) be the corresponding morphism. Then for any homotopy invariant pretheory (F, ϕ, A) over k one has

F_{∗}(f)F (f) = d Id_{F (Spec(E_{1}))}.

Lemma 3.16 Let k be a field and C be a smooth curve over k. Then the group h_{0}(C/Spec(k)) is generated by closed points x of C whose residue fields k_{x} are separable over k.

Proof: Let z be a closed point of C such that the field k_{z} is not separable over k. Let further C̄ be a smooth compactification of C. Denote by x_{1}, …, x_{k} the closed points C̄ − C. To show that the element of h_{0}(C/Spec(k)) which corresponds to z is a sum of elements which correspond to closed points of C with separable residue fields it is sufficient to find a rational function f on C̄ such that the following conditions hold:

1. f = 1 at the points x_{1}, …, x_{k}.

2. The divisor D(f) of f is of the form ∑ q_{i} − z − ∑ p_{j} where q_{i}, p_{j} are closed points of C such that the fields k_{q_{i}}, k_{p_{j}} are separable extensions of k.

Note first that since C is smooth over k there is infinitely many closed points on C with separable residue fields. Therefore there exists a finite set p_{1}, …, p_{n} of such points and a meromorphic function g on C̄ such that g(x_{i}) = 0 for all i = 1, …, k and g has poles in the points z, p_{1}, …, p_{n} with multiplicities [k_{z} : k], [k_{p_{1}}, k], …, [k_{p_{k}} : k] respectively.

Consider the function g as a morphism C̄ → P_{k}^{1}. It is smooth in the points p_{1}, …, p_{k} and hence it is smooth everywhere but in a finite set of closed points s_{1}, …, s_{t} (note that since the extension k_{z}/k is not separable and g(z) = ∞ is a point with separable residue field on P^{1} we have z ∈ {s_{1}, …, s_{t}}). We may assume that k is infinite (since finite fields are perfect). Then there is a k-point y on P_{k}^{1} such that y ≠ 0, ∞ and g is smooth over y. We set f = (y − g)/y. One can easily see that this function satisfies the conditions (1)-(2) from above.

Proposition 3.17 Let f : X_{1} → X_{2} be a morphism of smooth curves over a field k and (F, ϕ, A) be a homotopy invariant pretheory over k. Then the following diagram commutes

c_{equi}(X_{1}/Spec(k), 0) →^{ϕ_{X_{1}/Spec(k)}} Hom(F (X_{1}), F (Spec(k)))
f_{∗} ↓  ↓
c_{equi}(X_{2}/Spec(k), 0) →^{ϕ_{X_{2}/Spec(k)}} Hom(F (X_{2}), F (Spec(k))).

Proof: It follows immediately from Propositions 3.13, 3.14 and Lemma 3.16.

Definition 3.18 Let k be a field and (F, ϕ, A) be a pretheory over k. It is said to be of homological type if for any smooth scheme S over k, any morphism f : X_{1} → X_{2} of smooth curves over S and any element a in c_{equi}(X_{1}/S, 0) one has

ϕ_{X_{2}/S}(f_{∗}(a)) = ϕ_{X_{1}/S}(a) ∘ F (f).

3.4 Examples of pretheories.

Etale cohomology. For any field k consider the functor H_{et}(−, F) from the category Sm/k to the derived category of Gal(k̄/k)-modules which takes a smooth scheme X to the object Rp_{∗}(p^{∗}(F)) where F is a sheaf on the small etale site Spec(k)_{et} and p : X_{et} → Spec(k)_{et} is the morphism of sites defined by the structural morphism of X. This functor has a canonical structure of a pretheory. If F is a sheaf of Z[1/char(k)]-modules this pretheory is homotopy invariant. A fortiori, for any i ≥ 0 the functor from Sm/k to the category of abelian groups which takes X to the group H_{et}^{i}(X, F) has a canonical structure of pretheory which is homotopy invariant if F is a sheaf of Z[1/char(k)]-modules.

If S is the shift functor on the derived category of Gal(k̄/k)-modules then for any locally constant n-torsion sheaf F such that n is prime to characteristic of k we have (by Kunnet formula for the etale cohomology) a canonical isomorphism

(H_{et}(−, F))_{−1} ≅ S^{−1} ∘ H_{et}(−, F ⊗ μ_{n}^{−1}).

In particular

(H_{et}^{i}(−, F))_{−1} ≅ H_{et}^{i−1}(−, F ⊗ μ_{n}^{−1})

where μ_{n} is the sheaf of n-th roots of unit.

Algebraic K-theory. For any field k consider the functor K_{i} from Sm/k to the catgeory of abelian groups which takes a smooth scheme X to the Quillen’s K-group K_{i}(X). Since for any smooth scheme S over k and any smooth curve p : X → S over S an element of c_{equi}(X/S, 0) is a formal linear combination of closed subschemes in X which are flat and finite over S one can define transfer homomorphisms

ϕ_{X/S} : c_{equi}(X/S, 0) → Hom(K_{i}(X), K_{i}(S)).

Unfortunately, as was shown by Mark Walker these homomorphisms do not satisfy in general the second axiom of the definition of pretheory. It can be proven though that if we consider the separated in the Zariski topology presheaves K̃_{i}(−) which correspond to K_{i}(−) we get a pretheory which is clearly homotopy invariant.

Note also that we have canonical isomorphisms (K̃_{i})_{−1} ≅ K̃_{i−1}.

Algebraic De Ram cohomology. Let k be a field of characteristic zero. One can define a structure of pretheory on the functor which takes a smooth scheme p : X → Spec(k) over k to the object Rp_{∗}(Ω^{•}) of the derived category of k-vector spaces. In particular we get structures of pretheories on the functors X ↦ H_{DR}^{k}(X).

qfh-sheaves. Let F be a qfh-sheaf of abelian groups on the category Sch/k of schemes of finite type over a field k (see [10], [7]). Then one can easily see that the restriction of F to the category Sm/k has a canonical structure of a pretheory and the same holds for all functors of the form X ↦ H_{qfh}^{i}(X, K) where K is a complex of sheaves in the qfh-topology.

4 Zariski sheaves associated with pretheories.

4.1 Technical lemmas.

Definition 4.1 Let S be a smooth scheme. A standard triple

(p̄: X̄ → S, X_{∞}, Z)

over S is the following collection of data:

1. a proper normal curve p̄: X̄ → S over S,

2. a pair of reduced closed subschemes Z, X_{∞} in X.

These data should satisfy the following conditions:

1. the closed subset Z ∪X_{∞} has an open neighborhood in X̄ which is affine over S,

2. Z ∩ X_{∞} = ∅,

3. the scheme X = X̄ − X_{∞} is quasi-affine and smooth over S.

One can easily see that if (p̄ : X̄ → S, X_{∞}, Z) is a standard triple then for any closed reduced subscheme Z′ of Z the triple (p̄: X̄ → S, X_{∞}, Z′) is also a standard triple.

Lemma 4.2 Let S be a smooth scheme and (p̄: X̄ → S, X_{∞}, Z) be a standard triple over S. Denote the scheme X̄ − X_{∞} by X and let p : X → S be the corresponding morphism.

Let f : X′ → X be an etale morphism. Denote by W_{f} the closed subspace in X which consists of points x such that f is not finite over x and let Z_{0} be a closed subspace in Z. Suppose that the following conditions hold:

1. W_{f} ⊂ Z

2. the morphism f^{−1}(Z_{0}) → Z_{0} is finite.

Denote by f̄: X̄′ → X̄ the finite morphism with normal X̄′ associated with f. Then the triple

(p̄ ∘ f̄: X̄′ → S, X̄′ − X′, f^{−1}(Z_{0}))

is a standard triple over S.

Proof: Note first that since the morphism f^{−1}(Z_{0}) → Z_{0} → S is finite the subspace f^{−1}(Z_{0}) is closed in X̄′. On the other hand f^{−1}(Z_{0}) lies in X′ and hence f^{−1}(Z_{0}) ∩ (X̄′ − X′) = ∅.

The scheme X′ → X → S is smooth and quasi-affine over S since etale morphisms are smooth and quasi-affine. It is sufficient to verify that f^{−1}(Z_{0}) ∪ (X̄′ − X′) has an affine open neighborhood in X̄′. Since W_{f} ⊂ Z the image f̄(f^{−1}(Z_{0})∪(X̄′ −X′)) of this subscheme lies in X_{∞} ∪Z and hence has an affine open neighborhood since f̄ is affine morphism.

Lemma 4.3 Let S = Spec(O) be a smooth affine semi-local scheme over a field k, p̄: X̄ → S be a projective curve over S and Z be a closed subscheme in X̄. Then Z has an affine open neighborhood in X̄ if and only if the fibers of Z over the closed points of S are finite.

Proof: The “only if” part is obvious. Suppose that the fibers of Z over the closed points of S are finite and let z_{1}, …, z_{n} be the corresponding points of X. Since X is projective over S and S is affine there exists a quasi-projective scheme X̃ over k and an affine morphism X → X̃. Our lemma follows now from the fact that any finite set of points in a quasi-projective scheme has an affine open neighborhood.

Let S be a smooth affine scheme and T = (p̄: X̄ → S, X_{∞}, Z) be a standard triple over S. Denote by p : X → S the scheme X̄ − X_{∞} over S. Consider the smooth curve X ×_{S} X → X over X. The diagonal Δ : X → X ×_{S} X gives is a relative Cartier divisor on X ×_{S} X over X. Denote by L_{Δ} be the corresponding line bundle.

Definition 4.4 A splitting of T over an open subset U of X is a trivialization of the restriction of L_{Δ} to U ×_{S} Z.

Lemma 4.5 Let k be a field, S be a smooth scheme over k, T = ((p̄: X̄ → S, X_{∞}, Z) a standard triple over S and j : U → X be an affine open subset in X such that T splits over U. Then for any homotopy invariant pretheory (F, ϕ, A) over k there is a morphism F (X − Z) → F (U) such that the composition F (X) → F (X − Z) → F (U) coincides with the morphism F (j) : F (X) → F (U).

Proof: Consider the line bundle M on U ×_{S} X̄ which corresponds to the obvious U-point of U ×_{S} X̄ and let s be its section which defines our divisor. Its restriction to U ×_{S} X_{∞} gives a trivialization s̃ of M there. For a splitting t of T over U let t̃ be the trivialization of M on U ×_{S} (X_{∞} ∐ Z) equal to s̃ on U ×_{S} X_{∞} and to t on U ×_{S} Z.

Suppose now that U is affine. Then by Proposition 2.6 the pair (M, t̃) defines an element A_{t} in h_{0}(U ×_{S} (X − Z)/U) such that the direct image of A_{t} to h_{0}(U ×_{S} X) coincides with the class of tautological U-point of U ×_{S} X. In particular if (F, ϕ, A) is a homotopy invariant pretheory over k then any splitting t of T over an affine open subset U gives us a morphism F (X − Z) → F (U) and the composition F (X) → F (X − Z) → F (U) coincides by Proposition 3.12 with the morphism F (j).

Lemma 4.6 Let k be a field, S be a smooth scheme over k, T = ((p̄: X̄ → S, X_{∞}, Z) be a standard triple over S and U be an affine open subset in X such that T splits over U. Suppose further that Z = Z_{1} ∐ Z_{2} where Z_{1}, Z_{2} are reduced closed subschemes of X such that Z_{1} ∩ Z_{2} = ∅.

Then for any homotopy invariant pretheory (F, ϕ, A) over k the canonical morphism of complexes

0 → F (X) →  F (X−Z_{1})⊕F (X−Z_{2})  →  F (X−Z) → 0
↓  ↓  ↓
0 → F (U) → F ((X−Z_{1})∩U)⊕F ((X−Z_{2})∩U) → F ((X−Z)∩U) → 0

is homotopic to zero.

Proof: We have to define morphisms

\begin{matrix}f_{i} : F (X − Z) → F ((X − Z_{i}) ∩ U) \\ g_{i} : F (X − Z_{i}) → F (U) \end{matrix}

which satisfy the usual axioms of a homotopy. As in the proof of Lemma 4.5 we will first construct some elements

\begin{matrix}A_{i} ∈ h_{0}(((X − Z_{i}) ∩ U) ×_{S} (X − Z)/((X − Z_{i}) ∩ U)) \\ B_{i} ∈ h_{0}(U ×_{S} (X − Z_{i})/U) \end{matrix}

and then will use the transfer maps with respect to these elements to define the morphisms we need.

Consider the line bundle M on U ×_{S} X̄/U which corresponds to the obvious U-point of U ×_{S} X̄ and let s be its section which defines our divisor. Its restriction to U ×_{S} X_{∞} gives a trivialization s_{∞} of M there. By our assumption the restriction of M to the closed subscheme U ×_{S} Z is trivial. Since U is affine there is an affine open neighborhood V of the subscheme U ×_{S} (X_{∞} ∐ Z) in U ×_{S} X̄ and a trivialization t of M on V whose restriction to U ×_{S} X_{∞} is equal to s_{∞}.

Suppose for simplicity of the notations that i = 1. Consider the smooth curve ((X − Z_{1}) ∩ U) ×_{S} (X − Z) over (X − Z_{1}) ∩ U. Consider now the restriction of M to the subscheme ((X − Z_{1}) ∩ U) ×_{S} (X_{∞} ∐ Z). Since the corresponding divisor on ((X − Z_{1}) ∩ U) ×_{S} X̄ does not intersect ((X − Z_{1}) ∩ U) ×_{S} (X_{∞} ∐ Z_{1}) we have the canonical trivialization s_{1} of M there. Consider now the trivialization r of M on ((X −Z_{1})∩U)×_{S} (X_{∞} ∐ Z) which is equal to s_{1} on ((X − Z_{1}) ∩ U) ×_{S} (X_{∞} ∐ Z_{1}) and to the restriction of t on ((X − Z_{1}) ∩ U) ×_{S} Z_{2}.

We claim that r can be extended to an open neighborhood V_{0} of ((X − Z_{1}) ∩ U) ×_{S} (X_{∞} ∐ Z) in ((X − Z_{1}) ∩ U) ×_{S} X̄ (of course the only problem is that if codim(Z_{1}) ≥ 1 the scheme (X −Z_{1})∩U is not affine). Since V is affine there exists a regular function h on V whose restriction to U ×_{S} (X_{∞} ∐ Z_{2}) is equal to one and whose restriction to U ×_{S} Z_{1} is equal to s/t. Let Y be the divisor of h. Then the open subset V_{0} = ((X − Z_{1}) ∩ U) ×_{S} X̄ ∩ (V − Y) is an open neighborhood of ((X − Z_{1}) ∩ U) ×_{S} (X_{∞} ∐ Z). Consider now the trivialization th of M on V_{0}. It is obviously an extension of the form we need.

We define now A_{1} to be the element which corresponds to r by Proposition 2.6(a). The definition of A_{2} is similar.

Let us now construct the elements B_{i}. Since U is affine it is sufficient to define a trivialization of M on U ×_{S} (X_{∞} ∐ Z_{i}). We set it to be equal to the restriction of t.

Denote by C the element in h_{0}(U ×_{S} (X − Z)/U) which corresponds to the trivialization t of M.

We got morphisms:

C̃ : F (X − Z) → F (U)
Ã_{i} : F (X − Z) → F ((X − Z_{i}) ∩ U)
B̃_{i} : F (X − Z_{i}) → F (U).

Let us define the morphism g_{2} to be equal to the morphism B̃_{2}, the morphism g_{1} to be equal to zero, the morphism f_{1} to be equal to Ã_{1} and the morphism f_{2} to be equal to the difference δ − Ã_{2} where δ is the composition

F (X − Z) →^{C̃} F (U) → F ((X − Z_{2}) ∩ U).

One can verify easily now using Proposition 3.12 that we indeed constructed a homotopy of our morphism to the zero morphism.

Proposition 4.7 Let S be a smooth scheme over a field k and T_{i} = (p̄_{i} : X̄_{i} → S, X_{∞,i}, Z_{i}), i = 1, 2 be two standard triples over S. Denote the schemes X̄_{i} − X_{∞,i} by X_{i}.

Let f̄: X̄_{1} → X̄_{2} be a finite morphism such that the following conditions hold.

1. f̄^{−1}(X_{∞,2}) ⊂ X_{∞,1}.

2. Let f : X_{1} → X_{2} be the morphism defined by f̄ then f^{−1}(Z_{2}) = Z_{1} and the morphism f^{−1}(Z_{2}) → Z_{2} is an isomorphism.

Let (F, ϕ, A) be a homotopy invariant pretheory over k of homological type such that A is an abelian category and V be an open affine subset of X_{2} such that T_{2} splits over V. Then there is a morphism

ψ : F (X_{1} − Z_{1}) → F (V − V ∩ Z_{2})

satisfying the following conditions.

1. The composition

F (X_{1}) → F (X_{1} − Z_{1}) → F (V − V ∩ Z_{2})

is zero.

2. The following diagram is commutative:

F (X_{2} − Z_{2})
↓  ↘
F (X_{1} − Z_{1}) →  F (V − V ∩ Z_{2})/F (V)
↘  ↓
F (f^{−1}(V) − f^{−1}(V) ∩ Z_{1})/F (f^{−1}(V))

Proof: Let M be the line bundle on X_{2} ×_{S} X̄_{1} which is the preimage of the diagonal line bundle on X_{2}×_{S} X̄_{2} with respect to the morphism Id×_{S} f̄and let s be a section of M which defines the corresponding divisor X̃_{2} → X_{2} ×_{S} X̄_{1} where X̃_{2} = f̄^{−1}(X_{2}). Since the restriction of f̄ to Z_{1} is an isomorphism a splitting of T_{2} over V gives us a trivialization t of M on V ×_{S} Z_{1}.

Let U be an affine open neighborhood of V ×_{S} (Z_{1} ∐ X_{1,∞}) in V ×_{S} X̄. There exists a regular function h on U which is equal to 1 on V ×_{S} X_{1,∞} and to the regular function s/t on V ×_{S} Z_{1}. Let Y be the divisor of h. Its intersection with V ×_{S} X_{1,∞} is empty and its intersection with V ×_{S} Z_{1} is (V ∩ Z_{2}) ×_{S} Z_{1}. Hence the open subset V_{0} = ((V − V ∩ Z_{2}) ×_{S} X̄_{1}) ∩ (U − Y) is a neighborhood of (V − V ∩ Z_{2}) ×_{S} (Z_{1} ∐ X_{1,∞}) in (V − V ∩ Z_{2}) ×_{S} X̄_{1} and h gives an invertible function on U_{0}.

Let C be the corresponding element of H_{0}((V −V ∩Z_{2})×_{S} (X_{1}−Z_{1})/(V − V ∩ Z_{2})). We define our morphism ψ to be the composition

F (X_{1} − Z_{1}) →^{F (pr_{2})} F ((V − V ∩ Z_{2}) ×_{S} (X_{1} − Z_{1})) →^{ϕ(C)} F (V − V ∩ Z_{2}).

One can easily verify now that it satisfies the conditions of our proposition (we need the fact that F is of homological type to prove the commutativity of the upper triangle in our diagram).

Corollary 4.8 In the notations of Proposition 4.7 suppose that A is the category of abelian groups. Then the following statments hold.

1. Let a ∈ F (X_{2}−Z_{2}) be an element such that its restriction to F (X_{1}−Z_{1}) can be extended to F (X_{1}). Then for any closed point z of Z_{2} there exists an open neighborhood V of z in X_{2} such that the restriction of a to V − V ∩ Z_{2} can be extended to V.

2. Let a be an element it F (X_{1} − Z_{1}). Then for any closed point z of Z_{2} there exists an open neighborhood V of z in X_{2} and an element a_{0} ∈ F (V − V ∩ Z_{2}) such that the images of a and a_{0} in F (f^{−1}(V) − f^{−1}(V) ∩ Z_{1})/F (f^{−1}(V)) coincide.

Proposition 4.9 Let W be a smooth quasi-projective variety over a field k. Let Y ⊂ W be a closed reduced subscheme in W such that Y ≠ W and {y_{1}, …, y_{n}} be a finite set of closed points of Y. Then there exists an affine open neighborhood U of {y_{1}, …, y_{n}} in W and a standard triple (p̄ : X̄ → S, X_{∞}, Z) over a smooth affine variety S such that the pair (U, U ∩ Y) is isomorphic to the pair (X = X̄ − X_{∞}, Z).

Proof: See [11, Remark 4.13].

4.2 Pretheories in a neighborhood of a smooth subscheme.

Consider the inclusion of categories i : Sm/k → Sch/k where Sch/k is the category of Noetherian schemes over k. Then for any presheaf F on Sm/k with values in the category of abelian groups the restriction of the presheaf i^{∗}(F) on Sch/S to Sm/k is equal to F.

For any pretheory (F, ϕ, A) over k such that A is the category of abelian groups and any Noetherian scheme X over k we will use the notation F (X) instead of i^{∗}(F)(X).

We denote by F_{Zar} the sheaf in the Zariski topology on Sch/k associated with the presheaf i^{∗}(F).

Let k be a field, i : Z → X be a smooth pair over k and F be a presheaf of abelian groups on Sm/k. Denote the open subscheme X − Z by U and let j : U → X be the corresponding open embedding. Consider the exact sequence of presheaves on X_{Zar}:

F → j_{∗}j^{∗}(F) → coker → 0.

Let coker_{Zar} be the sheaf in the Zariski topology associated with the presheaf coker. One can easily see that the canonical morphism

coker_{Zar} → i_{∗}i^{∗}(coker_{Zar})

is an isomorphism. We denote by F_{(X,Z)} the sheaf i^{∗}(coker_{Zar}) on Z_{Zar}.

Remark 4.10 Since the associated sheaf functor is exact we have an exact sequence of the form

F_{Zar} → (j_{∗}j^{∗}F)_{Zar} → (coker)_{Zar} → 0

but in general the canonical morphism (j_{∗}j^{∗}F)_{Zar} → j_{∗}j^{∗}(F_{Zar}) is not an isomorphism since the associated sheaf functor does not commute with the direct image functor. For example if F is a sheaf on X_{Zar} we have (j_{∗}j^{∗}(H_{Zar}^{n}(−, F)))_{Zar} ≅ R^{n}j_{∗}(F) but (H_{Zar}^{n}(−, F))_{Zar} = 0.

Proposition 4.11 Let Z be a smooth scheme over a field k and (F, ϕ, A) be a homotopy invariant pretheory over k such that A is the category of abelian groups. Then there is a canonical isomorphism of sheaves on Z_{Zar} of the form

(F_{−1})_{Zar} → F_{(Z×A^{1},Z×{0})}.

Proof: Let us first define a morphism

(F_{−1})_{Zar} → F_{(Z×A^{1},Z×{0})}.

Let i : Z → Z × A^{1}, j : Z × (A^{1} − {0}) → Z × A^{1} be the obvious closed and open embeddings. Since the associated sheaf functor commutes with the inverse image functor it is sufficient to define a morphism of presheaves

F_{−1} → i^{∗}(coker(F → j_{∗}j^{∗}(F)))

on Z_{Zar}. Almost by definition a section of the right hand side presheaf over an open subset U of Z is an equivalence class of pairs of the form (V, a) where V is an open subset in Z × A^{1} such that U ⊂ Z ∩ V and a is an element of F (U − U ∩ (Z × {0})). Two pairs (V_{1}, a_{1}), (V_{2}, a_{2}) correspond to the same section of our presheaf if there exists an open subset V_{0} in V_{1} ∩ V_{2} such that U ⊂ V_{0}∩(Z×{0}) and the images of a_{1} and a_{2} in F (V_{0}−V_{0}∩(Z×{0}))/F (V_{0}) coincide.

Since a section of F_{−1} over an open subset U of Z is by definition an element of F (U × (A^{1} − {0}))/F (U × A^{1}) the definition of our morphism is obvious.

To prove that the corresponding morphism of associated sheaves in the Zariski topology on Z is an isomorphism it is sufficient to show that if Z is a local smooth scheme over k and V is an open neighborhood of Z × {0} in Z × A^{1} then the morphism

F (Z × (A^{1} − {0}))/F (Z × A^{1}) → F (U − U ∩ (Z × {0}))/F (U)

is an isomorphism.

Note that since U is a neighborhood of Z × {0} and we may assume Z to be local the triple T_{1} = (P_{Z}^{1} → Z, P_{Z}^{1} − U, Z × {0}) is a standard triple over Z by Lemma 4.3 and we may apply Proposition 4.7 to the standard triples T_{1}, T_{2} = (P_{Z}^{1} → Z, P_{Z}^{1} − A_{Z}^{1}, Z × {0}) and the identity morphism Id_{P_{Z}^{1}}. Since the standard triple T_{2} splits over A_{Z}^{1} it implies the result we need.

Up to the end of this section (F, ϕ, A) denote a homotopy invariant pretheory of homological type over a field k such that A is the category of abelian groups.

Proposition 4.12 Let f : X_{1} → X_{2} be an etale morphism of smooth schemes over k and Z be a closed subscheme in X_{2} such that the morphism f^{−1}(Z) → Z is an isomorphism. Then for any (F, ϕ, A) as above the following conditions hold:

1. Let a ∈ F (X_{2} − Z) be an element such that the preimage of a to X_{1} − f^{−1}(Z) can be extended to X_{1}. Then for any closed point x of X_{2} there exists an open neighborhood U in X_{2} such that the restriction of a to U − U ∩ Z can be extended to U.

2. For any element a ∈ F (X_{1}−f^{−1}(Z)) and any closed point x of X_{2} there exists an open neighborhood U of x on X_{2} and an element a′ ∈ F (U − U∩Z) such that the images of a and a′ in F (f^{−1}(U−U∩Z))/F (f^{−1}(U)) coincide.

Proof: Denote by f̄ : X′_{1} → X_{2} the finite morphism with normal X′_{1} associated with f and let Y be the closed subspace f̄(X′_{1} − X_{1}) in X_{2}. By Proposition 4.9 for any closed point x of X_{2} there is an open neighborhood U and a standard triple (p̄_{1} : W̄_{1} → S, W_{1,∞}, R_{1}) over a smooth scheme S such that the pair (U, U ∩(Z ∪Y)) is isomorphic to the pair (W_{1} = W̄_{1}−W_{1,∞}, R_{1}).

Let g be the morphism X_{1} ×_{X_{2}} U → X_{2}. Lemma 4.2 implies that there is a standard triple (p̄_{2} : W̄_{2} → S, W_{2,∞}, R_{2}) such that the pair (W_{2} = W̄_{2} − W_{2,∞}, R_{2}) is isomorphic to the pair (X_{1} ×_{X_{2}} U, (X_{1} ×_{X_{2}} U) ∩ f^{−1}(Z)). To finish the proof it is sufficient now to apply Corollary 4.8 to the standard triples (p̄_{1} : W̄_{1} → S, W_{1,∞}, U ∩ Z), (p̄_{2} : W̄_{2} → S, W_{2,∞}, R_{2}).

Corollary 4.13 Let f : X_{1} → X_{2} be an etale morphism of smooth schemes over k and Z be a closed subscheme in X_{2} such that the morphism f^{−1}(Z) → Z is an isomorphism. Then for any (F, ϕ, A) as above the canonical morphism of sheaves

F_{(X_{2},Z)} → F_{(X_{1},f^{−1}(Z))}

on Z_{Zar} = f^{−1}(Z)_{Zar} is an isomorphism.

Theorem 4.14 Let k and (F, ϕ, A) be as above, X be a smooth scheme over k and Z ⊂ X be a smooth closed subscheme of X. Let further x be a closed point of X and d be the codimension of Z in x. Then there exists an open neighborhood U and a family of isomorphisms

F_{(U×Y,(U∩Z)×Y)} ≅ F_{(A^{d}×(U∩Z)×Y,(U∩Z)×Y)}

of sheaves on ((U ∩ Z) × Y)_{Zar} given for all smooth schemes Y over k which are natural with respect to Y (here (U ∩ Z) × Y is considered as a closed subscheme of A^{d} ×(U ∩Z)×Y by means of the embedding which corresponds to the point (0, …, 0) of A^{d}).

In particular for d = 1 we have isomorphisms

F_{(U×Y,(U∩Z)×Y)} ≅ (F_{−1})_{Zar}.

Proof: The second statement follows from the first one by Proposition 4.11.

Since Z is a smooth subscheme on a smooth variety X there exists an open neighborhood U of x and an etale morphism p : U → A^{n} such that Z ∩ U = p^{−1}(A^{n−d} × {0}). We may suppose that U = X. Let p_{0} : Z → A^{n−d} be the restriction of p to Z. Consider the Cartesian square

W_{0} → Z × A^{d}
q ↓  ↓ p_{0} × Id_{A^{d}}
X →^{p}  A^{n}  = A^{n−d} × A^{d}.

Then q^{−1}(Z) ≅ Z ×_{A^{n−d}} Z and since p_{0} is etale the complement to the diagonal q^{−1}(Z(−Δ is a closed subset in q^{−1}(Z) and, hence in W_{0}. Let W be the complement W_{0} − (q^{−1}(Z) − Δ).

To finish the proof it is sufficient now to apply Corollary 4.13 to both arrows in the diagram

W × Y → Z × A^{d} × Y
↓
U × Y.

4.3 Pretheories on curves over a field.

Theorem 4.15 Let k be a field and (F, ϕ, A) be a homotopy invariant pretheory over k such that A is the category of abelian groups. Then for any open subscheme U in A_{k}^{1} one has:

H_{Zar}^{i}(U, F_{Zar}) = \{ \begin{matrix}F (U) & for & i = 0 \\ 0 & for & i ≠ 0 \end{matrix}

Proof: Consider an open covering of the form U = U_{1} ∪ U_{2}. Let Z_{i} be the reduced closed subscheme U − U_{i}. Then

T = (P_{k}^{1} → Spec(k), P_{k}^{1} − U, Z_{1} ∐ Z_{2})

is a standard triple over Spec(k) which obviously splits over U. Hence the sequence

0 → F (U) → F (U_{1}) ⊕ F (U_{2}) → F (U_{1} ∩ U_{2}) → 0

is exact by Lemma 4.6. Therefore, F considered as a functor on the category of Zariski open subsets of U is a sheaf which implies the statement of the theorem for i = 0. To prove the statement for i > 0 note that using induction on the number of open subsets in a covering we can show that Cech cohomology of any open U′ in U with coefficients in F are zero in positive dimensions. Then the standard spectral sequence which relates usual cohomology and Cech cohomology shows that H_{Zar}^{i}(U, F) = 0 for i > 0.

Corollary 4.16 Let k be a field and (F, ϕ, A) be a homotopy invariant pretheory over k such that A is the category of abelian groups. Then the morphism A_{k}^{1} → Spec(k) induces an isomorphism

F_{Zar}(Spec(k)) → F_{Zar}(A_{k}^{1}).

4.4 Pretheories on semi-local schemes.

Let W be an object of Sch/k. We say that W is a smooth semi-local scheme over k if there exists a smooth affine scheme X of finite type over k and a finite set {x_{1}, …, x_{n}} of points of X such that W is the inverse limit of open neighborhoods of this set.

Proposition 4.17 Let k be a field , (F, ϕ, A) be a homotopy invariant pretheory over k, W be a smooth quasi-projective variety over k and {x_{1}, …, x_{n}} be a finite set of points of W. Then for any nonempty open subset U of W there exists an open neighborhood V of {x_{1}, …, x_{n}} and a morphism F (U) → F (V) such that the following diagram commutes:

F (W) → F (U)
↓ ↙
F (V).

Proof: Let N be the reduced closed subscheme W −U of W. By Lemma 4.9 there exists an open neighborhood V_{0} of the set {x_{1}, …, x_{n}} and a standard triple T = (p̄ : X̄ → S, X_{∞}, Z) over a smooth affine variety S such that the pair (V_{0}, V_{0} ∩ N) is isomorphic to the pair (X = X̄ − X_{∞}, Z). Since the Picard group of an affine semi-local scheme is zero there exists an open neighborhood V of the set {x_{1}, …, x_{n}} in X such that T splits over V. The statement of our proposition follows now from Lemma 4.5.

Corollary 4.18 Let k be a field , (F, ϕ, A) be a homotopy invariant pretheory over k such that A is the category of abelian groups and W be a smooth semilocal scheme over k. Then for any nonempty open subscheme U of W the morphism F (W) → F (U) is a splitting monomorphism.

Corollary 4.19 Let k be a field , (F, ϕ, A) be a homotopy invariant pretheory over k such that A is the category of abelian groups and X be a smooth variety over k. Then for any nonempty open subscheme U of X the morphism F_{Zar}(X) → F_{Zar}(U) is a monomorphism.

Proposition 4.20 Let k be a field , (F_{i}, ϕ_{i}, A), i = 1, 2 be two homotopy invariant pretheories over k such that A is the category of abelian groups and f : F_{1} → F_{2} be a morphism of pretheories such that for any field extension k ⊂ E the morphism f_{E} : F_{1}(Spec(E)) → F_{2}(Spec(E)) is an isomorphism. Then the morphism of associated sheaves f_{Zar} : (F_{1})_{Zar} → (F_{2})_{Zar} is an isomorphism.

Proof: Consider the exact sequence of presheaves

0 → ker(f) → F_{1} → F_{2} → coker(f) → 0.

By Proposition 3.5 the presheaves ker(f) and coker(f) have canonical structures of homotopy invariant pretheories. Since the associated sheaf functor is exact it is sufficient to show that ker(f)_{Zar} = coker(f)_{Zar} = 0. It follows from our assumption and Corollary 4.18.

Proposition 4.21 Let k be a field and (F, ϕ, A) be a homotopy invariant pretheory over k such that A is the category of abelian groups. Then for any smooth scheme X over k one has:

F_{Zar}(X × A^{1}) = F_{Zar}(X).

Proof: It follows from Corollary 4.16 and Corollary 4.19.

For a presheaf of abelian groups F on Sch/k we denote by s(F) the separated in the Zariski topology presheaf associated with F. By definition it means that for any noetherian scheme X over k we have

s(F)(X) = F (X)/ ∪ (ker : F (X) → ⊕F (U_{i}))

where the union is taken over the partially ordered set of the Zariski open coverings {U_{i}} of X.

Proposition 4.22 Let k be a field and (F, ϕ, A) be a homotopy invariant pretheory over k such that A is the category of abelian groups. Then the presheaf s(F) has a unique structure of a homotopy invariant pretheory such that the canonical morphism of presheaves F → s(F) is a morphism of pretheories.

Proof: Note first that s(F) is obviously homotopy invariant. Let S be a smooth scheme over k, p : X → S be a smooth curve over S and Z be a closed reduced subscheme of X which belongs to c_{equi}(X/S, 0). To prove our proposition it is sufficient to show that if S is local and a ∈ F (X) is an element such that the image of a in s(F)(X) is equal to zero then ϕ_{X/S}(Z)(a) = 0.

By Corollary 4.18 our condition implies that there is an open neighborhood U of Z in X such that the restriction of a to U is equal to zero. Thus ϕ_{X/S}(Z)(a) = 0 by Proposition 3.12.

Proposition 4.23 Let W be a smooth semi-local scheme over a field k and W = U ∪ V be an open covering of W. Then for any pretheory (F, ϕ, A) over k such that A is the category of abelian groups the following sequence is exact:

0 → F (W) → F (U) ⊕ F (V) → F (U ∩ V) → 0.

Proof: It follows from Lemma 4.6 in exactly the same way as in the proof of Proposition 4.17.

Proposition 4.24 Let k be a field , (F, ϕ, A) be a homotopy invariant pretheory over k such that A is the category of abelian groups and W be a smooth semi-local scheme over k. Then one has F_{Zar}(W) = F (W).

Proof: By Proposition 4.22 and Corollary 4.19 we may suppose that F is a separated presheaf. Let W = ∪U_{i} be an open covering of W. It is sufficient to show that for any collection of elements a_{i} ∈ F (U_{i}) such that the restrictions of a_{i} and a_{j} to U_{i} ∩ U_{j} coincide there is an element a in F (W) such that its restrictions to U_{i} are equal to a_{i}.

We use induction on the number k of open subsets. For k = 2 our statement follows from Proposition 4.23.

For an arbitrary finite covering W = ∪U_{i} let J be the subset in the set of closed points of W which consists of points which belong to U_{1} ∪ U_{2}.

Proposition 4.23 implies that there is a closed subscheme Z in W such that Z does not contain points from J and an element b ∈ F (U_{1} ∪U_{2} −(U_{1} ∪U_{2})∩Z) such that the restriction of b to U_{l} − U_{l} ∩ Z_{l} coincides with the restrictions of a_{l} for l = 1, 2.

One can easily see that the collection of open subsets ((U_{1}∪U_{2}−(U_{1}∪U_{2})∩ Z), U_{3}, …, U_{k}) is a covering of W and we can use the inductive assumption to assign an element in F (W) to the collection of sections (b, a_{3}, …, a_{k}) on this open covering. Since F is separated Corollary 4.18 implies that this element satisfies the conditions we need. Proposition is proven.

Proposition 4.25 Let k be a field and (F, ϕ, A) be a homotopy invariant pretheory over k such that A is the category of abelian groups. Then the restriction of the sheaf F_{Zar} to the category Sm/k has a unique structure of pretheory such that the morphism of presheaves F → F_{Zar} is a morphism of pretheories.

Proof: By Proposition 4.22 we may assume that F is separated. Let S be a smooth scheme over k, p : X → S be a smooth curve over S and Z be a reduced irreducible subscheme of X which belongs to c_{equi}(X/S, 0). Consider a section a ∈ F_{Zar}(X). Since F is separated, Proposition 4.24 implies easily that there is an open covering {U_{i} → S} of S and open neighborhoods V_{i} of closed subschemes Z ∩ p^{−1}(U_{i}) in p^{−1}(U_{i}) such that the restrictions a_{i} of a to V_{i} belong to the subgroups F (V_{i}) of F_{Zar}(V_{i}). Let b_{i} be the elements ϕ_{V_{i}/U_{i}}(Z ∩V_{i})(a_{i}) in F (U_{i}). Since F is separated Corollary 4.19 implies easily that the restrictions of b_{i} and b_{j} to U_{i} ∩ U_{j} coincide and hence there is a global element b ∈ F (S). We set ϕ_{X/Z}(Z)(a) = b. One can verify easily (using again the fact that F is separated and Corollary 4.19) that it gives a structure of pretheory on F_{Zar}.

Proposition 4.26 Let k be a field and (F, ϕ, A) be a homotopy invariant pretheory over k such that A is the category of abelian groups. Then the pretheory F_{Zar} is a homotopy invariant pretheory of homological type.

Proof: It follows easily from Proposition 3.17, Proposition 4.21 and Corollary 4.19.

4.5 Zariski cohomology with coefficients in pretheories.

Theorem 4.27 Let k be a perfect field and (F, ϕ, A) be a homotopy invariant pretheory over k such that A is the category of abelian groups. Then for all i ≥ 0 the presheaves H_{Zar}^{i}(−, F_{Zar}) on Sm/k have canonical structures of homotopy invariant pretheories of homological type.

Proof: We use induction on i. For i = 0 our statement is proven in Proposition 4.26. Suppose that the theorem is proven for all i < n.

Lemma 4.28 Let k be a perfect field and (F, ϕ, A) be a homotopy invariant pretheory over k such that A is the category of abelian groups. Then under the inductive assumption for any smooth semi-local scheme W over k one has

H_{Zar}^{n}(W, F_{Zar}) = 0.

Proof: We use induction on the number j of closed points of W. For j = 1 our statement is trivial. Suppose that W is a smooth semi-local scheme, {x_{1}, …, x_{j}} the set of closed points of W and a an element of H_{Zar}^{n}(W, F_{Zar}).

By the inductive assumption on j there is a closed subscheme Z_{1} of W which does not contain x_{1} and a closed subscheme Z_{2} of W which does not contain any of the points x_{2}, …, x_{j} such that a becomes zero on U_{1} = W −Z_{1} and U_{2} = W − Z_{2}. One can easily see that Z_{1} ∩ Z_{2} = ∅ and hence we have a long exact sequence

… → H^{n−1}(U_{1}, F_{Zar}) ⊕ H^{n−1}(U_{2}, F_{Zar}) → H^{n−1}(U_{1} ∩ U_{2}, F_{Zar}) →
→ H^{n}(W, F_{Zar}) → H^{n}(U_{1}, F_{Zar}) ⊕ H^{n}(U_{2}, F_{Zar}) → ….

But since H^{n−1}(−, F_{Zar}) has a structure of homotopy invariant pretheory the homomorphism

H^{n−1}(U_{1}, F_{Zar}) ⊕ H^{n−1}(U_{2}, F_{Zar}) → H^{n−1}(U_{1} ∩ U_{2}, F_{Zar})

is surjective by Proposition 4.23 which implies that a = 0.

Lemma 4.29 Let k be a perfect field and (F, ϕ, A) be a homotopy invariant pretheory over k such that A is the category of abelian groups. Then under the inductive assumption the presheaf U → H_{Zar}^{n}(U, F_{Zar}) on Sm/k has a canonical structure of pretheory.

Proof: Let S be a smooth variety over k and p : X → S be a smooth curve over S. Consider an element A of c_{equi}(X/S, 0) and let Z = Supp(A) be the support of A which is a closed subset in X finite and surjective over S.

Let C_{A} be the full subcategory in the category X_{Zar} of open subsets of X which consists of open subsets U such, that

p^{−1}(p((X − U) ∩ Z)) ∩ Z = (X − U) ∩ Z.

One can easily see that C_{A} is closed under finite intersections. Consider the Grothendieck topology on C_{A} such that coverings are families of the form {U_{i} → U} satisfying U = ∪U_{i} and denote by α : X_{Zar} → C_{A} the obvious morphism of sites.

For any object U of C_{A} let β^{−1}(U) be the open subset S − p((X − U) ∩ Z) of S. One can easily see that β^{−1}(U ∩ V) = β^{−1}(U) ∩ β^{−1}(V) and that if U = ∪U_{i} then β^{−1}(U) = ∪β^{−1}(U_{i}). Therefore we have a morphism of sites β : S_{Zar} → C_{A}.

Lemma 4.28 implies easily that R^{i}α_{∗}(F_{Zar}) = 0 for i ≤ n. Therefore by the Leray spactral sequence for the morphism α we have

H_{Zar}^{n}(X, F_{Zar}) = H^{n}(C_{A}, α_{∗}(F_{Zar})).

On the other hand we have a canonical morphism

H^{n}(C_{A}, α_{∗}(F_{Zar})) → H_{Zar}^{n}(S, β^{∗}α_{∗}(F_{Zar})).

To define a homomorphism

ϕ_{X/S}^{n}(A) : H_{Zar}^{n}(X, F_{Zar}) → H_{Zar}^{n}(S, F_{Zar})

it is sufficient now to define a homomorphism β^{∗}α_{∗}F_{Zar} → F_{Zar} of sheaves on S_{Zar} or, equivalently, a homomorphism α_{∗}F_{Zar} → β_{∗}F_{Zar} of sheaves on C_{A}. It means that for any open subset U of X which belongs to C_{A} we have to define a homomorphism

F_{Zar}(U) → F_{Zar}(β^{−1}(U)).

We obviously have U ∩Z = p^{−1}(β^{−1}(U))∩Z and hence U ∩Z ∩p^{−1}(β^{−1}(U)) is finite over β^{−1}(U). Therefore A gives us an element A_{U} in c_{equi}(p^{−1}(β^{−1}(U))∩ U/β^{−1}(U), 0) and we can define our homomorphism to be the composition

F_{Zar}(U) → F_{Zar}(p^{−1}(β^{−1}(U)) ∩ U) → F_{Zar}(β^{−1}(U))

where the last arrow is the morphism which corresponds to A_{U} with respect to the canonical structure of pretheory on F_{Zar} defined in Proposition 4.25.

One can easily verify now that our construction indeed provides a structure of a pretheory on the presheaf H_{Zar}^{n}(−, F_{Zar}) on Sm/k.

Lemma 4.30 Under the assumptions of Lemma 4.29 the pretheory H_{Zar}^{n}(−, F_{Zar}) is of homological type.

Proof: Let S be a smooth scheme over k, f : X_{1} → X_{2} be a morphism of smooth curves over S and A be an element of c_{equi}(X_{1}/S, 0). Then f defines a morphism of sites ψ : C_{A} → C_{f_{∗}(A)} such that the following diagram is commutative

(X_{1})_{Zar} →^{f} (X_{2})_{Zar}
α_{1} ↓  ↓ α_{2}
C_{A}  →^{ψ} C_{f_{∗}(A)}
β_{1}  ↖ ↑ β_{2}
S_{Zar}.

We only have to verify that the homomorphisms

β_{1}^{∗}ψ^{∗}F_{Zar} → F_{Zar}
β_{2}^{∗}F_{Zar} → F_{Zar}

coincide with respect to the canonical isomorphism β_{1}^{∗}ψ^{∗} ≅ β_{2}^{∗}. One can easily see that it means exactly that F_{Zar} is a pretheory of homological type which was proven in Proposition 4.26.

We will need the following technical result.

Lemma 4.31 Let X be a smooth scheme of finite type over a perfect field k, Z ⊂ X be a closed subset of X such that codim(Z) ≥ 1 and x be a point of X. Then there is an open neighborhood U of x in X and a sequence of reduced closed subschemes ∅ ⊂ Y_{0} ⊂ Y_{1} ⊂ … ⊂ Y_{k} in U satisfying the following conditions:

1. The schemes Y_{i} − Y_{i−1} are smooth divisors on U − Y_{i−1}.

2. U ∩ Z is a subset of Y_{k}.

Proof: We may suppose that x is a closed point of X and that X is connected.

We proceed by induction on dim(X). Since X is smooth over k there is an open neighborhood U of x and a smooth morphism p : U → V of relative dimension one such that V is a smooth scheme of dimension dim(X)−1 over k. Consider the closure Z′ of p(Z ∩ U) in V.

If codim(Z′) ≥ 1 then by the inductive assumption there is a sequence of closed subschemes in V of the form ∅ ⊂ Y′_{0} ⊂ … ⊂ Y′_{n} such that the conditions of our lemma hold for this sequence with respect to Z′. One can easily see that these conditions also hold for the sequence ∅ ⊂ p^{−1}(Y′_{0}) ⊂ … ⊂ p^{−1}(Y′_{n}) of closed subschemes of U with respect to Z.

Suppose now that codim(Z′) = 0 i.e. Z′ = V. Let Z_{0} be the closed subset of Z ∩ U which is the union of irreducible components of Z of codimension > 1 and the closed subset of singular points of Z. Then the codimension of the closure of p(Z_{0}) is greater than zero hence there is a sequence ∅ ⊂ Y_{0} ⊂ … ⊂ Y_{n} of closed subschemes of U satisfying the conditions of the lemma with respect to Z_{0}. One can easily see now that the sequence

∅ ⊂ Y_{0} ⊂ … ⊂ Y_{n} ⊂ Y_{n} ∪ (Z ∩ U)

satisfies the conditions of the lemma with respect to Z. Lemma is proven.

To finish the proof of the theorem it is sufficient to show now that under the inductive assumption for any smooth scheme X over k the canonical projection X × A^{1} → X induces an isomorphism

H_{Zar}^{n}(X, F_{Zar}) → H_{Zar}^{n}(X × A^{1}, F_{Zar}).

Since the projection p : X×A^{1} → X splits our morphism is a monomorphism. The Leray spectral sequence for p together with the inductive assumption shows that it is sufficient to verify that any element a ∈ H_{Zar}^{n}(X × A^{1}, F_{Zar}) becomes zero on an open covering X = ∪U_{i} of X. By Theorem 4.15 there is a nonempty open subscheme U of X such that a equals zero on U × A^{1}. Together witth Lemma 4.31 it implies that one has to show that for a local smooth X an a smooth divisor Z on X the morphism H_{Zar}^{n}(X ×A^{1}, F_{Zar}) → H_{Zar}^{n}((X − Z) × A^{1}, F_{Zar}) is a monomorphism.

Lemma 4.32 Consider the open embedding j : (X − Z) × A^{1} → X × A^{1} and suppose that X is a local scheme. Then under our assumptions we have

R^{m}j_{∗}(F_{Zar}) = 0

for m < n.

Proof: Let i : Z × A^{1} → X × A^{1} be the complimentary closed embedding. By definition R^{m}j_{∗}(F_{Zar}) is the sheaf associated with the presheaf j_{∗}j^{∗}(H_{Zar}^{m}(−, F_{Zar})). Since the sheaf associated with the presheaf H_{Zar}^{m}(−, F_{Zar}) is zero we have R^{m}j_{∗}(F_{Zar}) = i_{∗}i^{∗}(R^{m}j_{∗}(F_{Zar})) and

i^{∗}(R^{m}j_{∗}(F_{Zar})) = (H_{Zar}^{m}(−, F_{Zar}))_{(X×A^{1},Z×A^{1})}.

By our assumption and Lemma 4.30 the presheaf H_{Zar}^{m}(−, F_{Zar}) has a structure of a homotopy invariant pretheory of homological type. Therefore since X is local we have (by Theorem 4.14) an isomorphism

i^{∗}(R^{m}j_{∗}(F_{Zar})) ≅ ((H_{Zar}^{m}(−, F_{Zar}))_{−1})_{Zar}.

Denote by p_{0} : Z × (A^{1} − {0}) → Z the canonical projection. The right hand side sheaf is isomorphic by definition to R^{m}(p_{0})_{∗}(F_{Zar}). Again by the inductive assumption and Corollary 4.19 this sheaf on Z_{Zar} is separated. On the other hand it is zero in the generic point by Theorem 4.15. Lemma is proven.

Consider the exact sequence of sheaves on X × A^{1}

0 → F_{Zar} → j_{∗}j^{∗}(F_{Zar}) → coker → 0

(the first morphism is a monomorphism by Corollary 4.19). Since the restriction of a to (X − Z) × A^{1} is equal to zero Lemma 4.32 and the Leray spectral sequence for the open embedding j imply that the image of a in H_{Zar}^{n}(X × A^{1}, j_{∗}j^{∗}(F_{Zar})) is equal to zero. Therefore a belongs to the image of the homomorphism

H_{Zar}^{n−1}(X × A^{1}, coker) → H_{Zar}^{n}(X × A^{1}, F_{Zar}).

Since coker ≅ i_{∗}i^{∗}coker and the higher direct images for closed embeddings are zero the left hand side group is isomorphic to H_{Zar}^{n−1}(Z ×A^{1}, i^{∗}coker). We may suppose that X is local. Then by Proposition 4.16 and Theorem 4.14 the sheaf i^{∗}(coker) is isomorphic to the sheaf (F_{−1})_{Zar}.

If n > 1 it implies immediately that a = 0 by the induction. For n = 1 we have a homomorphism

H^{0}(X × A^{1}, j_{∗}j^{∗}(F_{Zar})) → H^{0}(X × A^{1}, coker)

The left hand side group is equal to F_{Zar}(X − Z) since F_{Zar} is homotopy invariant and the right hand side group is isomorphic to F_{Zar}(Z × (A^{1} − {0}))/F_{Zar}(F). Therefore our homomorphism is surjective by Theorem 4.14 since X and Z are local. Theorem is proven.

Corollary 4.33 Let k be a perfect field and (F, ϕ, A) be a homotopy invariant pretheory over k such that A is the category of abelian groups. Let further X be a smooth scheme over k and j : U → X be an open subscheme of X such that X − U is a smooth divisor on X. Then R^{n}j_{∗}(F_{Zar}) = 0 for n > 0.

Proof: See Lemma 4.32.

4.6 The Gersten resolution.

Proposition 4.34 Let k be a perfect field and (F, ϕ, A) be a homotopy invariant pretheory over k such that A is the category of abelian groups. Then for any i ≥ 0 one has a canonical isomorphism of pretheories

(H_{Zar}^{i}(−, F_{Zar}))_{−1} ≅ H_{Zar}^{i}(−, (F_{−1})_{Zar}).

Proof: Let us first consider the case i = 0. We have to show that the canonical morphism of presheaves (F_{−1})_{Zar} → (F_{Zar})_{−1} is an isomorphism. By Proposition 4.26 both presheaves have canonical structures of homotopy invariant pretheories and obviously this morphism is a morphism of pretheories. Our statement follows now from Theorem 4.15 and Proposition 4.20.

Suppose now that i > 0. One can easily see that it is sufficient to show that for any smooth scheme X over k one has R^{i}p_{∗}(F_{Zar}) = 0 for i > 0 where p is the projection X × (A^{1} − {0}) → X. It follows from Theorem 4.27, Proposition 4.18 and Theorem 4.15.

Lemma 4.35 Let k, (F, ϕ, A) be as in Proposition 4.34 and S be a smooth local scheme over k then one has:

H_{Zar}^{i}(S × (A^{n} − \{0\}), F_{Zar})/H_{Zar}^{i}(S, F_{Zar}) = \{ \begin{matrix}F_{−n}(S) & for & i = n − 1 \\ 0 & for & i ≠ n − 1 \end{matrix}

Proof: It follows easily by induction from Proposition 4.34, Theorem 4.27 and the Mayer-Vietoris long exact sequence for the Zariski cohomology.

For any point x of a smooth scheme X over k we denote by H_{x}^{i}(X, F) the group lim_{→} H_{¯ {x}∩U}^{i}(U, F) where limit is taken over the partially ordered set of open neighborhoods of x in X.

Lemma 4.36 Let k be a perfect field, X be a smooth scheme over k and x be a point of codimension p on X. Then for any F as above one has:

H_{x}^{i}(X, F) = \{ \begin{matrix}F_{−p}(Spec(k_{x})) & for & i = p \\ 0 & for & i ≠ p \end{matrix}

Proof: Let us recall that for any open embedding j : U → X and any presheaf F on X_{Zar} we defined in §4.2 a sheaf F_{(X,X−U)} on (X − U)_{Zar} as i^{∗}(coker : F → j_{∗}j^{∗}(F))_{Zar} where i : X −U → X is the corresponding closed embedding.

Let X_{x} be the local scheme of x in X. For p = 0 or i = 0 our statement follows from Corollary 4.17. Assume that i, p > 0. Consider the sheaf H^{i−1}(−, F)_{(X_{x},x)} on Spec(k_{x})_{Zar}. Clearly it is the constant sheaf which corresponds to the abelian group H_{x}^{i}(X, F). Applying Theorem 4.14 to H^{i−1}(−, F) we conclude that

H_{x}^{i}(X, F) = (H^{i−1}(−, F))_{(A_{kx}^{p},(0,...,0))}(Spec(k_{x})) = H_{(0,...,0)}^{i}(A_{k_{x}}^{p}, F).

We have the long exact sequence for the cohomology with supports

.. → H_{(0)}^{i}(A_{k_{x}}^{p}, F) → H^{i}(A_{k_{x}}^{p}, F) → H^{i}(A_{k_{x}}^{p} − {0}, F) → H_{(0)}^{i+1}(A_{k_{x}}^{p}, F) → ..

By Theorem 4.27 we have isomorphisms H^{i}(A_{k_{x}}^{p}, F) → H^{i}(Spec(k_{x}), F) and therefore homomorphisms H^{i}(A_{k_{x}}^{p}, F) → H^{i}(A_{k_{x}}^{p} −{0}, F) are injective. Our statement follows now from Lemma 4.35.

Theorem 4.37 Let k be a perfect field , (F, ϕ, A) be a homotopy invariant pretheory over k such that A is the category of abelian groups and F is a sheaf in the Zariski topology. Let further X be a smooth scheme of dimension n over k. Denote by X^{(k)} the set of points of X of codimension k. Then there is a canonical exact sequence of sheaves on X_{Zar} of the form:

0 → F → ⊕_{x∈X^{(0)}}(i_{x})_{∗}(F) → … → ⊕_{x∈X^{(n)}}(i_{x})_{∗}(F_{−n}) → 0.

Proof: For any sheaf F we have a spectral sequence

(E_{1}^{pq} = ⊕_{x∈X^{(p)}}H_{x}^{p+q}(X, F)) ⇒ H^{p+q}(X, F).

Consider the corresponding spectral sequence of sheaves on X_{Zar}. It converges to the sheaves in the Zariski topology associated with the presheaves H^{p+q}(−, F) which equal zero for p + q ≠ 0 and F for p + q = 0. On the other hand Lemma 4.36 implies that it degenerates to a complex of the form

⊕_{x∈X^{(0)}}(i_{x})_{∗}(F) → … → ⊕_{x∈X^{(n)}}(i_{x})_{∗}(F_{−n}) → 0

which immediately implies the statement of our theorem.

5 Sheaves in Nisnevich and etale topologies associated with pretheories.

5.1 Pretheoretical structures on Nisnevich cohomology.

Definition 5.1 Let k be a field and (F, ϕ, A) be a pretheory over k such that A is the category of abelian groups. It is called a reasonable pretheory if the following condition holds. Let S be a smooth local scheme over k, f : X_{1} → X_{2} be an etale morphism of smooth curves over S and Z be a reduced irreducible subscheme of X_{1} which belongs to c_{equi}(X/S, 0) such that Z → f(Z) is an isomorphism. Then one has

ϕ_{X_{2}/S}(f(Z)) = ϕ_{X_{1}/S}(Z) ∘ F (f).

Proposition 5.2 Let k be a field and (F, ϕ, A) be a homotopy invariant pretheory over k such that A is the category of abelian groups. Then F is reasonable.

Proof: It follows from Corollary 4.19 and Proposition 3.17.

Theorem 5.3 Let k be a field and (F, ϕ, A) be a reasonable pretheory over k. Then the presheaves U → H_{Nis}^{i}(U, F_{Nis}) have canonical structures of (reasonable) pretheories.

Proof: Let us consider first the case i = 0. Denote by s_{Nis}(F) the separated in the Nisnevich topology presheaf associated with F. Let us show that it has a structure of a reasonable pretheory.

It is sufficient to show that for any smooth henselian local scheme S over k, smooth curve p : X → S over S and a reduced irreducible subscheme Z in c_{equi}(X/S, 0) we have ϕ_{X/S}(Z)(a) = 0 for any a ∈ ker(F (X) → s_{Nis}(F)(X)). It follows immideately from the fact any scheme finite over S is a local henselian scheme and the fact that F is reasonable.

We may assume now that F is a separated presheaf in the Nisnevich topology. Let p : X → S be a smooth curve over a smooth scheme S over k, Z be a closed subscheme in X which belongs to c_{equi}(X/S, 0) and a be an element in F_{Nis}(X). We want to construct an element b = ϕ_{X/S}(Z)(a) in F_{Nis}(S).

Let {U_{i} → X} be a finite Nisnevich covering of X such that the restrictions of a to U_{i} belong to F (U_{i}). Since a scheme finite over a local henselian scheme is a local henselian scheme there is a Nisnevich covering {V_{j} → S}_{1≤j≤m} of S such that for any j = 1, …, m the covering

{U_{i} ×_{S} V_{j} → X ×_{S} V_{j}}

splits over Z_{j} = Z ×_{S} V_{j}.

Let Z̃_{j} be the image of such a splitting, i.e. a closed subscheme in ∐_{i} U_{i}×_{S} V_{j} which maps isomorphically on Z_{j}. Then ∐_{i} U_{i} ×_{S} V_{j} → V_{j} is a smooth curve over V_{j} and Z̃_{j} belongs to c_{equi}((∐_{i} U_{i} ×_{S} V_{j})/V_{j}, 0). Consider the elements b_{j} = ϕ(Z̃_{j})(F (pr_{1})(∐ a_{i})).

One can easily see that b_{j} define a global element b in F_{Nis}(S) and that this construction indeed provides us with a structure of a reasonable pretheory on F_{Nis}.

Consider now the case i > 0. Let p : X → S be a smooth curve over a smooth scheme S over k and Z be a reduced closed subscheme of X which belongs to c_{equi}(X/S, 0). Consider the following diagram of morphisms of sites:

Z_{Nis}  →^{i} X_{Nis}
p_{0} ↘ ↓ p
S_{Nis}

Since p_{0} is finite the higher direct images in the Nisnevich topology with respect to p_{0} are zero. It implies that one has only to construct a morphism of sheaves (p_{0})_{∗}i^{∗}(F_{Nis}) → F_{Nis} on S_{Nis}. It can be easily done using our theorem for i = 0 and the fact that a scheme finite over a henselian local scheme is a disjoint union of henselian local schemes.

5.2 Cohomology of affine space and the comparison theorem.

Proposition 5.4 Let k be a field, (F, ϕ, A) be a homotopy inavriant pretheory over k such that A is the category of abelian groups and U be an open subscheme in A_{k}^{1}. Then one has:

H_{Nis}^{i}(U, F_{Nis}) = \{ \begin{matrix}F (U) & for & i = 0 \\ 0 & for & i ≠ 0 \end{matrix}

Proof: Let us consider first the case i = 0. First of all by Theorem 4.15 we have F ≅ F_{Zar} on U_{Zar}. Hence the morphism F (U) → F_{Nis}(U) is a monomorphism by Corollary 4.18 since any Nisnevich covering contains at least one non-empty open subset. It implies that we may replace F by s_{Nis}(F) and assume that F is a separated presheaf in the Nisnevich topology.

One can easily see now that it is sufficient to show that for any element a in F_{Nis}(U) and any closed point x of U there is an open neighborhood V such that the restriction of a to V belongs to F (V) ⊂ F_{Nis}(V). It follows immediately from Proposition 4.12.

To prove our proposition for i > 0 it is obviously sufficient to consider the case i = 1. The corresponding result again follows easily from Proposition 4.12.

Proposition 5.5 Let k be a field and (F, ϕ, A) be a homotopy invariant pretheory over k such that A is the category of abelian groups. Then the canonical morphism of presheaves F_{Zar} → F_{Nis} is an isomorphism.

Proof: Note first that Theorem 5.3 together with Proposition 5.4 imply that F_{Nis} is a homotopy invariant pretheory and our morphism is obviously a morphism of pretheories. To finish the proof it is sufficient to apply Proposition 4.20 to this morphism.

Theorem 5.6 Let k be a perfect field, (F, ϕ, A) be a homotopy invariant pretheory over k such that A is the category of abelian groups and X be a smooth scheme over k. Then the canonical projection X × A^{1} → X induces isomorphisms

H_{Nis}^{i}(X, F_{Nis}) → H_{Nis}^{i}(X × A^{1}, F_{Nis}).

Proof: The proof is strictly parallel to the proof of Theorem 4.27 with considerable simplification due to Theorem 5.3.

Theorem 5.7 Let k be a perfect field, (F, ϕ, A) be a homotopy invariant pretheory over k such that A is the category of abelian groups and X be a smooth scheme over k. Then the canonical morphisms H_{Zar}^{i}(X, F_{Zar}) → H_{Nis}^{i}(X, F_{Nis}) are isomorphisms.

Proof: The case i = 0 is proven in Proposition 5.5. Suppose that i > 0. By Theorems 5.6 and 5.3 the presheaves H_{Nis}^{i}(−, F_{Nis}) are homotopy invariant pretheories. It is obviously sufficient to show that the sheaves in the Zariski topology associated with these presheaves are zero. It follows immediately from Corollary 4.18 since the Nisnevich and the Zariski cohomology coincide for fields.

Remark 5.8 Unfortunately the analogs of the previous results do not hold for the flat version of the Nisnevich topology. Consider the following example. Let F be the cokernel of the morphism of sheaves G_{m} →^{z^{2}} G_{m} in the Zariski topology. Our results imply that F has a canonical structure of a homotopy invariant pretheory of homological type.

Consider the Nis-flat covering of A^{1} by the open subset U_{0} = A^{1} − {0} and the flat morphism U_{1} = A^{1} →^{z^{2}} A^{1}. Let a be the canonical section of F on U_{0}. Together with the zero section on U_{1} they give us a nontrivial global section of F_{Nis−fl} on A^{1}. It implies immediately that our theory does not work in this situation.

5.3 Applications to Suslin homology.

We will show in this section that our results on cohomology of sheaves associated with pretheories imply some important vanishing properties for the sheaves \underline{h}_{i}(−)_{Zar} (see the end of §3.2). As an application we will prove the Mayer-Vietoris theorem for the Suslin homology.

Theorem 5.9 Let k be a perfect field and (F, ϕ, A) be a pretheory over k with values in the category of abelian groups. Then the following conditions are equivalent.

1. The sheaves (\underline{h}_{i}(F))_{Zar} are zero for all i ≤ n.

2. For any homotopy invariant pretheory G which is a sheaf in the Nisnevich topology one has Ext^{i}(F_{Nis}, G) = 0 for all i ≤ n.

Proof: Note that by Proposition 5.5 one has (\underline{h}_{i}(F))_{Zar} = (\underline{h}_{i}(F))_{Nis} and that (by 4.26) these presheaves are homotopy invariant pretheories. (1− > 2). Consider the reduced singular simplicial complex ˜ \underline{C}_{∗}(F) of F where

˜\underline{C}_{0}(F)(X) = F (X) = \underline{C}_{0}(F)(X)

˜\underline{C}_{k}(F)(X) = (coker : F (X) → F (X × A^{k})) = (\underline{C}_{k}(F)/F)(X) for k > 0.

Clearly the obvious morphism \underline{C}_{∗}(F) → ˜ \underline{C}_{∗}(F) is a quasi-isomorphism. The reason why ˜ \underline{C}_{∗} is sometimes more convinient to work with is the following lemma.

Lemma 5.10 Let G be a Nisnevich sheaf which is a homotopy invariant pretheory. Then for any j > 0 and any i ≥ 0 one has

Ext^{i}(( ˜\underline{C}_{j}(F))_{Nis}, G) = 0.

Proof: It follows easily from the fact that the presheaf ˜ \underline{C}_{j}(F) is contractible (and hence so is the associated sheaf) and Theorem 5.6.

Lemma 5.10 implies that for any G as above and any i the obvious morphism

Hom_{D}((\underline{C}_{∗}(F))_{Nis}, G[i]) = Hom_{D}((\underline{C̃}_{∗}(F))_{Nis}, G[i]) → Ext^{i}(F_{Nis}, G) is an isomorphism (here Hom_{D} are morphisms in the derived category of Nisnevich sheaves on Sm/k). If (\underline{h}_{j}(F))_{Nis} = 0 for j ≤ n then the left hand side groups are obviously zero for all i ≤ n.

(2− > 1) Let (\underline{h}_{k}(F))_{Nis} be the first nontrivial cohomology sheaf of \underline{C}_{∗}(F). Then there is a nonzero morphism in the derived category

(\underline{C}_{∗}(F))_{Nis} → (\underline{h}_{k}(F))_{Nis}[k].

Since (\underline{h}_{k}(F))_{Nis} is a homotopy invariant pretheory it gives us as above a nontrivial element in Ext^{k}(F_{Nis}, (\underline{h}_{k}(F))_{Nis}) and hence by our condition k > n.

Corollary 5.11 Let k be a perfect field and F be a pretheory over k with values in the category of abelian groups such that F_{Nis} = 0. Then for any i ≥ 0 one has \underline{h}_{i}(F)_{Zar} = 0.

Corollary 5.12 Let k be a perfect field and

0 → F → G → H → 0

be a sequence of reasonable pretheories such that the corresponding sequence of associated sheaves in the Nisnevich topology is exact. Then one has a canonical long exact sequence of sheaves of the form

… → (\underline{h}_{i}(F))_{Zar} → (\underline{h}_{i}(G))_{Zar} → (\underline{h}_{i}(H))_{Zar} → (h_{i−1}(F))_{Zar} → ….

Proof: It follows immediately from Corollary 5.11 and Theorem 5.3.

Let k be a field, X be a scheme of finite type over k and U be a smooth scheme over k. Denote by c_{equi}(X/Spec(k), 0)(U) the free abelian group generated by integral closed subschemes of X ×U which are finite over U and surjective over an irreducible component of U. As was shown in [8] for any morphism of smooth schemes f : U′ → U one can define a homomorphism

cycl(f) : c_{equi}(X/Spec(k), 0)(U) → c_{equi}(X/Spec(k), 0)(U′)

such that c_{equi}(X/Spec(k), 0)(−) becomes a presheaf on the category Sm/k and the obvious analogs of properties (1)-(3) given at the beginning of §2 hold.

Proposition 5.13 Let k be a field and X be a scheme of finite type over k. Then the presheaf c_{equi}(X/Spec(k), 0) on Sm/k is a sheaf in the etale (and a fortiori in the Nisnevich) topology.

Proof: It follows immediately from our definition of c_{equi}(X/Spec(k), 0) and property (2) of the base change homomorphisms.

Proposition 5.14 For any scheme of finite type X over a field k the presheaf of abelian groups c_{equi}(X/Spec(k), 0) on Sm/k has a canonical structure of a pretheory of homological type.

Proof: We have to define for any smooth curve C/S over a smooth scheme S over k a pairing of the form

c_{equi}(C/S, 0) ⊗ c_{equi}(X/Spec(k), 0)(C) → c_{equi}(X/Spec(k), 0)(S).

This is a special case of the correspondence pairings considered in [8, Section 3.7]. The fact that these pairing give us a structure of a pretheory follows from [8, Th. 3.7.3].

Proposition 5.15 Let k be a field, X a separated scheme over k and X = U ∪ V an open covering of X. Then the sequence

0 → c_{equi}(U ∩ V) → c_{equi}(U) ⊕ c_{equi}(V) → c_{equi}(X) → 0

is exact in the Nisnevich topology (we abbreviated c_{equi}(−/Spec(k), 0) to c_{equi}(−)).

Proof: It is a special case of [8, Prop. 4.3.7].

Definition 5.16 Let k be a field and X a scheme of finite type over k. The groups \underline{h}_{i}(c_{equi}(X/Spec(k), 0))(Spec(k)) are called the Suslin homology of X. We denote them by H_{i}^{S}(X/k).

Theorem 5.17 Let k be a perfect field and X a scheme of finite type over k. Then for any open covering X = U ∪ V of X there is a canonical long exact sequence of the form

… → H_{i}^{S}(U ∩ V) → H_{i}^{S}(U) ⊕ H_{i}^{S}(V) → H_{i}^{S}(X) → H_{i−1}^{S}(U ∩ V) → ….

Proof: It follows from Corollary 5.12 and Proposition 5.15.

5.4 Cohomology of blowups.

For any smooth scheme X over a field k denote by Z_{Nis}(X) the sheaf of abelian groups on (Sm/k)_{Nis} freely generated by the sheaf of sets representable by X. Let Ab((Sm/k)_{Nis}) be the category of sheaves of abelian groups on (Sm/k)_{Nis}. Then for any object F of this category we have canonical isomorphisms:

Ext_{Ab((Sm/k)_{Nis})}^{i}(Z_{Nis}(X), F) ≅ H_{Nis}^{i}(X, F).

Proposition 5.18 Let k be a field, f : X → Y an etale morphism of smooth schemes over k and Z a closed subscheme in Y such that f^{−1}(Z) → Z is an isomorphism. Then the canonical morphism of sheaves

Z_{Nis}(X)/Z_{Nis}(X − f^{−1}(Z)) → Z_{Nis}(Y)/Z_{Nis}(Y − Z)

on (Sm/k)_{Nis} is an isomorphism.

Proof: Note first that (Y −Z) ∐ X → Y is a Nisnevich covering of Y. Therefore our morphism is an epimorphism. To prove that it is a monomorphism it is sufficient to prove that it is a monomorphism on the corresponding freely generated presheaves (since the associated sheaf functor is exact). Consider the diagram:

0 → Z(X − f^{−1}(Z)) →^{i} Z(X) → Z(X)/Z(X − f^{−1}(Z)) → 0
↓ ↓ ↓
0 → Z(Y − Z) → Z(Y) → Z(Y)/Z(Y − Z) → 0

where Z(−) is the presheaf which sends a smooth scheme U to the direct sum of free abelian groups generated by sets of morphisms from the connected components of U to −. We have to show that ker(Z(f)) lies in Im(i). Let W be a connected scheme. Then ker(Z(f)) is the group of formal sums of the form ∑_{i∈I} n_{i}g_{i}, where g_{i} are morphisms from W to X such that there exists a decomposition I = ∐ I_{k} satisfying f ∘ g_{i} = f ∘ g_{j} for i, j ∈ I_{k} and ∑_{i∈I_{k}} n_{i} = 0 for any k. Therefore, we have to prove only that if f ∘ g = f ∘ h for some g, h : W → X then either g = h or g and h can be factored through X − f^{−1}(Z). Let g, h be such morphisms. Then there exists a morphism g × h : W → X ×_{Y} X whose compositions with the projections are the morphisms g and h respectively. To finish the proof it is sufficient to notice that since f is an etale morphism the diagonal in X ×_{Y} X is a connected component i.e. there is a decomposition of the form X ×_{Y} X = Δ(X) ∐ X_{0}. But since f^{−1}(Z) → Z is an isomorphism the projections pr_{1}, pr_{2} : X_{0} → X can be factored through X − f^{−1}(Z). Proposition is proven.

Proposition 5.19 Let X be a smooth scheme over a field k and Z → X be a smooth subscheme of X. Denote by p_{Z} : X_{Z} → X the blowup of X in Z and consider the following Cartesian square:

p_{Z}^{−1}(Z) →^{i} X_{Z}
q ↓  ↓ p_{Z}
Z  → X

Then the canonical morphism of sheaves ker(Z_{Nis}(q)) → ker(Z_{Nis}(p_{Z})) on Sm/k is an isomorphism (and a fortiori the same holds for any stronger topology).

Proof: Obviously our morphism is a monomorphism. It is sufficient to show that it is an epimorphism. For any X ∈ ob(Sm/k) let L(X) be the sheaf of sets representable by X. Note that there is an obvious long exact sequence of the form

… → Z_{Nis}(L(X_{Z}) ×_{L(X)} L(X_{Z})) → Z_{Nis}(L(X_{Z})) → Z_{Nis}(L(X))

It is sufficient to show that the morphism

L(i) × L(i) ∐ L(Δ) : L(p_{Z}^{−1}(Z) ×_{Z} p_{Z}^{−1}(Z)) ∐ L(X_{Z}) → L(X_{Z}) ×_{L(X)} L(X_{Z})

is an epimorphism. It is obvious since as a scheme X_{Z} ×_{X} X_{Z} is a union of the diagonal and the image of p_{Z}^{−1}(Z) ×_{Z} p_{Z}^{−1}(Z).

Remark 5.20 The statement of the proposition above is false for sheaves in the Nisnevich (or Zariski) topology on Sch/k which gives in particular an example of the fact that functor of the inverse image on the categories of sheaves associated with the canonical embedding of categories Sm/k → Sch/k is not left exact. Nevertheless the proposition is still valid for all schemes if we consider qfh- or cdh-topology (see [10], [8]).

Proposition 5.21 In the notations of Proposition 5.19 suppose that k is a perfect field and let (F, ϕ, A) be a homotopy invariant pretheory over k such that A is the category of abelian groups. Then for any i ≥ 0 one has:

Ext^{i}(coker(Z_{Nis}(p_{Z})), F_{Nis}) = 0

Proof: One can easily see that our problem is local with respect to X. Denote by U the open subscheme X − Z of X and consider the diagram of sheaves on (Sm/k)_{Nis}:

0 → Z_{Nis}(U) → Z_{Nis}(X_{Z}) → coker_{1} → 0
↓ ↓ ↓
0 → Z_{Nis}(U) → Z_{Nis}(X) → coker_{2} → 0

One can easily see that coker(Z(p_{Z})) = coker(coker_{1} → coker_{2}). Together with Proposition 5.18 and the fact that Z is smooth it implies that locally with respect to X we have:

coker(Z(p_{Z})) ≅ coker(Z(Z × A_{(Z×{0})}^{d}) → Z(Z × A^{d}))

where d is the codimension of Z. We have a projection Z × A_{(Z×{0})}^{d} → Z × P^{d−1} which is a locally trivial in the Zariski topology fibration with the fiber A^{1}. By Theorem 5.6 it implies that we have:

Ext^{i}(Z(Z × A_{(Z×{0})}^{d}), F) = H^{i}(Z × P^{d−1}, F)
Ext^{i}(Z(Z × A^{d}), F) = H^{i}(Z, F).

Our statement follows now trivially from Proposition 5.19.

Corollary 5.22 Let k be a perfect field, X be a smooth scheme over k and Z be a smooth closed subscheme in X. Denote by p : X_{Z} → X the blow-up of Z in X. Then for any homotopy invariant pretheory F with values in the category of abelian groups there is a long exact sequence of the form:

… → H^{i}(X, F_{Zar}) → H^{i}(X_{Z}, F_{Zar}) ⊕ H^{i}(Z, F_{Zar}) →
→ H^{i}(p^{−1}(Z), F_{Zar}) → H^{i+1}(X, F_{Zar}).

(Note that it is true for both the Zariski and the Nisnevich cohomology groups by Theorem 5.7)

Proof: Consider the following diagram of morphisms of sheaves in the Nisnevich topology:

Z_{Nis}(p_{Z}^{−1}(Z)) →^{i} Z_{Nis}(X_{Z})
q ↓ ↓ Z_{Nis}(p_{Z})
Z_{Nis}(Z) → Z_{Nis}(X)

The left vertical arrow is an epimorphism since p^{−1}(Z) → Z is a locally trivial in Zarski topology bundle over Z. This fact and Proposition 5.19 imply immediately that we have an exact sequence of the form:

0 → Z_{Nis}(p_{Z}^{−1}(Z)) → Z_{Nis}(X_{Z}) ⊕ Z_{Nis}(Z) → Z_{Nis}(X) → coker(Z_{Nis}(p_{Z})) → 0

and the corollary follows from Proposition 5.21.

Remark 5.23 It can be shown (see [9]) that the exact sequence from Corollary 5.22 splits canonically and one has:

H^{i}(X_{Z}, F_{Zar}) = H^{i}(X, F_{Zar}) ⊕ ⨁_{j=1}^{c−1} H^{i−2j}(Z, (F_{−j})_{Zar}).

5.5 Sheaves in the etale topology associated with pretheories.

Proposition 5.24 Let k be a field and (F, ϕ, A) be a homotopy invariant pretheory over k such that A is the category of Q-vector spaces. Then the canonical morphism of presheaves on Sm/k

F_{Nis} → F_{et}

is an isomorphism.

Proof: It follows trivially from Corollary 4.18 and the results of §3.3.

The corresponding result with intergral coefficients is wrong. Instead we have the following “rigidity” theorem.

Theorem 5.25 Let k be a field of exponential characteristic p and (F, ϕ, A) be a homotopy invariant pretheory over k such that A is the category of torsion abelian groups of torsion prime to p. Then the sheaf F_{et} on Sm/k is locally constant.

Proof: The proof is strictly parallel to the proof of the rigidity theorem in [7].

Corollary 5.26 Let k be a field of exponential characteristic p and (F, ϕ, A) be a homotopy inavriant pretheory over k such that A is the category of Z[1/p]-modules. Consider the exact sequence of sheaves

0 → (F_{tors})_{et} → F_{et} → F_{et} ⊗ Q̄ → (F_{cotors})_{et} → 0.

Then the sheaves (F_{tors})_{et}, (F_{cotors})_{et} are locally constant.

Proof: It follows from Theorem 5.25 and Proposition 3.5.

Proposition 5.27 Let X be a scheme and F be a sheaf of Q-vector spaces on X_{et} then the canonical morphisms

H_{Nis}^{i}(X, F) → H_{et}^{i}(X, F)

are isomorphisms.

Proof: It follows immediately from the Leray spectral sequence and the fact that the etale cohomology of any henselian local scheme are torsion groups.

Proposition 5.28 Let k be a perfect field and (F, ϕ, A) be a homotopy invariant pretheory over k such that A is the category of abelian groups. Then the canonical morphisms of pretheories

H_{Zar}^{i}(−, F_{Zar} ⊗ Q) → H_{et}^{i}(−, F_{et} ⊗ Q)

are isomorphisms.

Proof: It follows from Theorem 5.7, Proposition 5.28 and Proposition 5.24.

Corollary 5.29 Let k be a perfect field of exponential characteristic p, (F, ϕ, A) a homotopy invariant pretheory over k such that A is the category of Z[1/p]-modules and X a smooth scheme over k. Then the canonical morphisms

H_{et}^{i}(X, F_{et}) → H_{et}^{i}(X × A^{1}, F_{et})

are isomorphisms.

Proof: It follows from our comparison theorem for the rational coefficients, Theorem 5.25 and homotopy invariance of the etale cohomology with locally constant torsion coefficients.

Proposition 5.30 Let k be a perfect field and F : (Sm/k)^{op} → A a contravariant functor from the category Sm/k to a Q-linear additive category A. Suppose that there are two pretheory structures ϕ^{1}, ϕ^{2} of on F. Then ϕ^{1} and ϕ^{2} coincide semi-locally. In particular if F is a separated presheaf of Q-vector spaces in the Zariski topology, then ϕ^{1} and ϕ^{2} coincide.

Proof: Let S be a smooth affine variety over k and p : X → S a smooth curve over S. We have to show that for any finite set {s_{1}, …, s_{n}} of points of S and any element A in c_{equi}(X/S, 0) there is an open neighborhood U of {s_{1}, …, s_{n}} such that the following two compositions coincide

F (X) →^{ϕ_{X/S}^{1}(A)} F (S) → F (U)

F (X) →^{ϕ_{X/S}^{2}(A)} F (S) → F (U).

By Proposition 4.17 it is sufficient to show that there exists a nonempty open subset U satisfying this property. It means that we may suppose that S = Spec(K) where K is a field, X is a curve over K and A is a closed point of X. In this case our statement follows easily from the results of §3.3.

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