Generic Injectivity for Étale Cohomology and Pretheories

by Alexander Schmidt and Kirill Zainoulline

Let k be a field. We call W a smooth semi-local k-scheme if there exists a smooth affine k-scheme Y and finitely many closed points y_{1}, …, y_{n} on Y such that W is the inverse limit of all Zariski open neighbourhoods of {y_{1}, …, y_{n}} in Y. The objective of this paper is to show the following

Theorem 1 Let W be a connected smooth semi-local scheme over a field k and let η be its generic point. Let X → W be a proper smooth morphism, n an integer prime to char(k) and K^{•} a complex of sheaves of ℤ/nℤ-modules on X_{et} whose cohomology sheaves are locally constant constructible and bounded below. Then for every q ∈ ℤ the canonical map

H_{et}^{q}(X, K^{•}) ⟶ H_{et}^{q}(X_{η}, K^{•})

is a universal monomorphism.

This injectivity result for étale cohomology was known before by the work of J.-L. Colliot-Thélène, R. Hoobler and B. Kahn [CHK] in the following cases:

- dim W = 1 ([CHK], Corollary B.3.3)

- K^{•} = μ_{n}^{⊗i} (concentrated in degree zero) and X = W ×T with T a smooth

(not necessarily proper) variety over k ([CHK], Theorem 8.1.1). The reader should also compare Theorem 1 with O. Gabber’s injectivity result [Ga] for henselizations.

The word “universal” in the statement of the theorem has the following meaning: Let ϕ : M ↪ N be a monomorphism in an abelian category A which has the property that filtered direct limits exist and are exact. ϕ is called universal monomorphism if for every abelian category B in which filtered direct limits exist and are exact and for every additive functor T : A → B commuting with filtered direct limits, the homomorphism T (M) → T (N) is again monomorphic. A typical example of a universal monomorphism is a filtered direct limit of split injections.

The techniques used in the proof of theorem 1 are those of Voevodsky [Vo], §§4.3,4.4. Let X be a smooth scheme over k and let X-Sm(k) be the category of X-schemes which are smooth over the field k. Let F : X-Sm(k)^{op} → Ab be a presheaf of abelian groups which can be endowed with the structure of a homotopy invariant pretheory over X (see section 2). As usual, we extend F to pro-objects by setting

F (lim_{⟵} Y_{i}) = lim_{⟶} F (Y_{i}).

If F satisfies the additional property of being extensible (see Definition 2.5), we show the

Theorem 2 Let X be a smooth scheme over a field k and let F be an extensible homotopy invariant pretheory over the k-scheme X. Then for every smooth semi-local scheme W over X and each dense open subscheme U ⊂ W, the restriction homomorphism

F (W) ⟶ F (U)

is a universal monomorphism. In particular, the natural homomorphism

F_{Zar} ⟶ ⨁_{x∈X^{0}}(i_{x})_{∗}F (k(x))

of sheaves on X_{Zar} is injective.

If F comes by base change from a homotopy invariant pretheory over k, Theorem 2 is a result of V. Voevodsky ([Vo], Cor.4.18) and, if k is perfect, the given injection of Zariski sheaves is the first arrow of the Gersten resolution ([Vo], Theorem 4.37) for F_{Zar}. The essential difficulty in generalizing Voevodsky’s result to the relative case is ([Vo] Proposition 4.9) which says that any finite set of closed points on a smooth quasiprojective variety over a field has an open neighbourhood being part of a “standard triple”. This does not remain true in the relative case. The key idea of the present paper is the observation that one can overcome this difficulty if the pretheory F is “extensible”. Étale cohomology is naturally equipped with the structure of a pretheory and we show that this pretheory structure is extensible. Therefore Theorem 1 follows from Theorem 2 and from the smooth-proper base change theorem.

This article is based on ideas of an earlier unpublished preprint of the second author. The second author wants to thank I. Panin for helpful discussions on the subject, in particular, for his suggestion to consider the natural transformation Ψ of section 4. The second author also wants to thank J.-L. Colliot-Thélène for his comments on the subject and the TMR project ERB FMRX CT-97-0107 and EPDI for financial support.

1 Relative Curves and Standard Triples

In the present section we recall some definitions and facts of [Vo], §2. We tacitly assume that all occurring schemes are noetherian.

To begin with, let S be a regular connected scheme. Let p : X → S be a curve, i.e. a dominant morphism of finite type such that all nonempty fibers are of dimension 1. By c_{equi}(X/S, 0) we denote the free abelian group generated by the set of closed integral subschemes Z ⊂ X such that the projection Z → S is finite and surjective. As was shown in [SV1], for any morphism of connected regular schemes f : S′ → S one can define a homomorphism

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

If f is dominant and Z is a closed integral subscheme in the group c_{equi}(X/S, 0), then

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

where Cycl_{X×_{S}S′}(Z ×_{S} S′) is the cycle of the closed subscheme Z ×_{S} S′ in X ×_{S} S′. Let g : X_{1} → X_{2} be a morphism of curves over S. Then the direct image homomorphism

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

is defined by setting g_{∗}(Z) = n(Z, g)g(Z), where Z is an integral closed subscheme in X_{1} which belongs to c_{equi}(X_{1}/S, 0) and n(Z, g) is the degree of the finite extension of function fields k(Z)|k(g(Z)). By ([SV1], 3.6.2), the homomorphisms g_{∗} and cycl(f) respect each other, i.e. for a morphism g : X_{1} → X_{2} of curves over S and a morphism f : S′ → S the diagram

\xymatrix{c_{equi}(X_{1}/S, 0) \ar[d]_{g_{∗}} \ar[r]^{cycl(f)} & c_{equi}(X_{1} ×_{S} S′/S′, 0) \ar[d]^{g_{∗}} \\ c_{equi}(X_{2}/S, 0) \ar[r]^{cycl(f)} & c_{equi}(X_{2} ×_{S} S′/S′, 0)}

commutes. Two elements Z_{0}, Z_{1} of c_{equi}(X/S, 0) are equivalent (homologous) if there is an element Z in the group c_{equi}(𝔸_{X}^{1}/𝔸_{S}^{1}, 0) such that cycl(i_{0})(Z) = Z_{0} and cycl(i_{1})(Z) = Z_{1}, where i_{0}, i_{1} : S → 𝔸_{S}^{1} are the closed embeddings corresponding to the points 0 and 1, respectively. The group of equivalence classes of elements of c_{equi}(X/S, 0) with respect to this equivalence relation will be denoted by h_{0}(X/S). It follows from the definition that the homomorphisms cycl(f) induce homomorphisms h_{0}(X/S) → h_{0}(X ×_{S} S′/S′) and that the groups h_{0}(X/S) are covariantly functorial with respect to morphisms X_{1} → X_{2} of curves over S.

According to ([SV2], §2), a good compactification of a smooth curve X/S is a pair (p̄ : X̄ → S, j : X → X̄) where X̄ is a normal 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 neighbourhood which is affine over S.

Suppose that X → S is quasi-affine and that we are given a line bundle L on X̄ which is trivial over an open neighbourhood U of X_{∞} in X̄. Any trivialization of L over U, considered as a rational section of L over X̄ defines a divisor on X whose class is in h_{0}(X/S). We will use the following result ([Vo], 2.6).

Proposition 1.1 Let p : X → S be a smooth quasi-affine curve over a regular scheme S and let (p̄: X̄ → S, j : X → X̄) be a good compactification of X. Let X_{∞} be the reduced subscheme X̄−X, L a line bundle on X̄ and s : O_{X_{∞}} → L|_{X_{∞}} a trivialization of L over X_{∞}. Then the following statements hold.

(i) For any two extension s̃_{1}, s̃_{2} of the trivialization s to an open neighbourhood U of X_{∞} the cycles Cycl_{X}(D(L, U, s̃_{1})) and Cycl_{X}(D(L, U, s̃_{2})) give the same element in h_{0}(X/S) (here D(L, U, s̃_{i}) is the associated divisor).

(ii) If S is affine, there exists an affine open neighbourhood U of X_{∞} in X̄ and an extension s̃: O_{U} → L|_{U} of s to a trivialization of L on U.

Following ([Vo], 4.1), we recall the notion of a standard triple.

Definition 1.2 A standard triple (p̄ : X̄ → S, X_{∞}, Z) over a regular scheme S is a proper normal curve p̄ : X̄ → S together with a pair of reduced closed subschemes Z, X_{∞} in X̄ such that the closed subset Z ∪ X_{∞} has an open neighbourhood in X̄ which is affine over S, Z ∩X_{∞} = ∅ and the scheme X = X̄ −X_{∞} is quasi-affine and smooth over S.

If (p̄: X̄ → S, X_{∞}, Z) is a standard triple, then p̄: X̄ → S represents a good compactification of X = X̄ − X_{∞} and of X − Z. In the case of smooth schemes over a field, the existence of standard triples is provided by the following fact ([Wa], 4.13 or [Vo], 4.9).

Proposition 1.3 Let W be a smooth quasi-projective variety over a field k. Let N ⊂ W be a closed reduced subscheme in W such that N ≠ W and let {x_{1}, …, x_{n}} be a finite set of closed points of N. Then there exists an affine open neighbourhood V of {x_{1}, …, x_{n}} in W and a standard triple (p̄: X̄ → S, X_{∞}, Z) over a smooth affine variety S such that the pair (V, V ∩ N) is isomorphic to the pair (X = X̄ − X_{∞}, Z).

For an open subscheme U ⊂ X we denote by Δ_{X,U} the image of the canonical morphism U → X ×_{S} U. Let π : Y → X ×_{S} U be a finite étale morphism having a section δ : Δ_{X,U} → Y over the closed subscheme Δ_{X,U}. The image δ(Δ_{X,U}) is a divisor on Y (isomorphic to U) and we denote the associated line bundle by L_{δ}. The following definition generalizes ([Vo],4.4).

Definition 1.4 A standard triple T = (p̄ : X̄ → S, X_{∞}, Z) splits over (U, π : Y → X ×_{S} U) if the restriction of L_{δ} to π^{−1}(Z ×_{S} U) is trivial. A trivialization of L_{δ}|_{π^{−1}(Z×_{S}U)} is called a splitting of T over (U, π : Y → X ×_{S} U).

Lemma 1.5 Let T = (p̄: X̄ → S, X_{∞}, Z) be a standard triple which splits over (U, π : Y → X ×_{S} U, δ : Δ_{X,U} → Y), where U is affine. Then there exists an element in h_{0}(π^{−1}((X − Z) ×_{S} U)/_{pr_{2}∘π}U) whose direct image in h_{0}(Y/_{pr_{2}∘π}U) coincides with the class of the U-point δ_{U} : U → Y which is given as the composite of the canonical U-point U → X ×_{S} U and δ : Δ_{X,U} → Y.

Proof: Consider the standard triple T_{U} = (pr_{2} : X̄ ×_{S} U → U, X_{∞} ×_{S} U, Z ×_{S} U) which is given by base change of the standard triple T along U → S. Let Ȳ be the normalization of X̄ ×_{S} U in the function field of Y. The projection π : Y → X ×_{S} U extends to a finite morphism π̄ : Ȳ → X̄ ×_{S} U and pr_{2} ∘ π̄ : Ȳ → U is a good compactification of pr_{2} ∘ π : Y → U and also of pr_{2} ∘ π :

Y −π^{−1}(Z×_{S}U) → U. The divisor δ(Δ_{X,U}) ⊂ Y is closed in Ȳ and we denote by L_{δ} the associated line bundle. Put Y_{∞} = (Ȳ −Y)∪π^{−1}(Z×_{S}U). By assumption, the restriction of L_{δ} to π^{−1}(Z ×_{S} U) is trivial. Since (Ȳ − Y) ∩ π^{−1}(Z ×_{S} U) = ∅, we can choose a trivialization of L_{δ} over Y_{∞} whose restriction to Ȳ − Y coincides with the canonical trivialization. By proposition 1.1 (ii) applied to the compactification Ȳ of Y − π^{−1}(Z ×_{S} U), we obtain an extension of the chosen trivialization to an open neighbourhood of Y_{∞} and therefore we get an element in h_{0}(π^{−1}((X − Z) ×_{S} U)/_{pr_{2}∘π}U). By proposition 1.1 (i), the image of this element in h_{0}(Y/_{pr_{2}∘π}U) has the required property. □

Lemma 1.6 Let T = (p̄ : X̄ → S, X_{∞}, Z) be a standard triple, U ⊂ X a nonempty open affine subscheme and π : Y → X ×_{S} U a finite étale morphism having a splitting δ : Δ_{X,U} → Y over Δ_{X,U}. Let {x_{1}, …, x_{n}} be a finite set of closed points of U. Then there exists an affine open neighbourhood U′ of {x_{1}, …, x_{n}} in U such that the triple T splits over (U′, π : π^{−1}(X ×_{S} U′) → X ×_{S} U′).

Proof: Since Z is proper over S and has an open neighbourhood in X̄ which is affine over S, the projection Z → S is finite. Consider the semi-local scheme U = SpecO_{U,{x_{1},...,x_{n}}}. The projection Z ×_{S} U ⟶^{pr_{2}} U is finite and therefore also the composite π^{−1}(Z ×_{S} U) ⟶^{π} Z ×_{S} U ⟶^{pr_{2}} U is finite. Thus π^{−1}(Z ×_{S} U) is a semi-local affine scheme. Since every line bundle on a semi-local affine scheme is trivial, the restriction of L_{δ} to π^{−1}(Z ×_{S} U) is trivial. Consequently we find an affine open neighbourhood U′ of {x_{1}, …, x_{n}} in U such that the restriction of L_{δ} to π^{−1}(Z ×_{S} U′) is trivial. □

2 Extensible Pretheories

In this section we translate the definition of a homotopy invariant pretheory and some of its properties given in ([Vo], §3) to the relative case. We introduce the notion of an extensible pretheory.

From this point on we fix a connected smooth scheme B over a field k and we consider the category B-Sm(k) of B-schemes which are smooth over k. Similar to ([Vo], 3.1.) we define

Definition 2.1 A pretheory (F, ϕ) over B is the following collection of data:

1. A presheaf of abelian groups F : B-Sm(k)^{op} ⟶ Ab.

2. For any object S ∈ B-Sm(k) and any smooth curve p : X → 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 ∈ B-Sm(k), any smooth curve p : X → 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 in the category B-Sm(k) and let p : X_{2} → S_{2} be a smooth curve over S_{2}. Consider the Cartesian square

\xymatrix{X_{1} \ar[r]^{g} \ar[d] & X_{2} \ar[d]^{p} \\ S_{1} \ar[r]^{f} & S_{2}.}

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}}(cycl(f)(Z)) ∘ F (g).

3. For any pair X, Y of objects in B-Sm(k) the canonical morphism

F (X ⨿ Y) → F (X) ⊕ F (Y)

is an isomorphism.

Definition 2.2 A pretheory (F, ϕ) be over B is called homotopy invariant if for any object X of B-Sm(k) the projection 𝔸_{X}^{1} → X induces an isomorphism F (X) → F (𝔸_{X}^{1}).

The following properties of homotopy invariant pretheories ( [Vo], 3.11, 3.12) remain true with the same proofs.

Proposition 2.3 A pretheory (F, ϕ) is homotopy invariant if and only if for any object S ∈ B-Sm(k) and any smooth curve X → S the morphism

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

factors through the natural projection c_{equi}(X/S, 0) → h_{0}(X/S).

Proposition 2.4 Let (F, ϕ) be a homotopy invariant pretheory over B, S an object of B-Sm(k) and j : U → X an open embedding of smooth curves over S. Then for any element a ∈ c_{equi}(U/S, 0) one has

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

Now we introduce the notion of an extensible pretheory. Let X → B be smooth and let (F, ϕ) be a pretheory over X. Consider the two projections pr_{1}, pr_{2} : X ×_{B} X → X. The pull-back by means of pr_{1} and pr_{2} gives two different presheaves pr_{1}^{∗}F, pr_{2}^{∗}F on the category (X ×_{B} X)-Sm(k).

Definition 2.5 We say that a pretheory (F, ϕ) over X is an extensible pretheory over the B-scheme X if for any finite set {x_{1}, …, x_{n}} of closed points of X there

exist an open neighbourhood U ↪^{i} X of {x_{1}, …, x_{n}} in X and 1. A connected scheme Y together with a finite étale morphism π : Y →

X ×_{B} U having a section δ : Δ_{X,U} → Y over Δ_{X,U}.

2. A homomorphism of presheaves on Y -Sm(k)

Ψ : p_{1}^{∗}F → p_{2}^{∗}F,

where p_{1} = pr_{1} ∘ π and p_{2} = i ∘ pr_{2} ∘ π, such that the restriction of Ψ to

the U-point δ_{U} : U ⟶^{can} Δ_{X,U} →^{δ} Y

Ψ_{δ_{U}} : δ_{U}^{∗}(p_{1}^{∗}F) ⟶ δ_{U}^{∗}(p_{2}^{∗}F)

is the canonical isomorphism id : i^{∗}F ⟶^{∼} i^{∗}F over U.

If (F, ϕ) comes by base change from a pretheory over B, then the presheaves p_{1}^{∗}F and p_{2}^{∗}F on the category (X ×_{B} X)-Sm(k) are canonically isomorphic, and hence (F, ϕ) is extensible over B. More general, we have the following lemma whose proof is straightforward.

Lemma 2.6 Assume we are given a commutative diagram of smooth morphisms

\xymatrix{X′ \ar[r]^{f} \ar[d] & X \ar[d] \\ B′ \ar[r] & B.}

If (F, ϕ) is an extensible pretheory over the B-scheme X, then the presheaf f^{∗}F has a canonical structure of an extensible pretheory over the B′-scheme X′.

3 Proof of Theorem 2

Theorem 2 of the introduction is a straightforward consequence of the following

Theorem 3.1 Let W be a smooth quasi-projective variety over a field k. Let (F, ϕ) be an extensible homotopy invariant pretheory over the k-scheme W. Let {x_{1}, …, x_{n}} be a finite set of points of W. Then for any nonempty open subset V of W there exists an open neighbourhood U of {x_{1}, …, x_{n}} and a homomorphism F (V) → F (U) such that the following diagram commutes:

\xymatrix{F (W) \ar[r] \ar[d] & F (V) \ar[dl] \\ F (U).}

Proof: Let N be the reduced closed subscheme W − V of W. By Proposition 1.3 there exists an open affine neighbourhood W′ of {x_{1}, …, x_{n}} and a standard triple T = (p̄ : X̄ → S, X_{∞}, Z) over a smooth affine variety S such that the pair (W′, W′ ∩ N) is isomorphic to the pair (X = X̄ − X_{∞}, Z). Note that X − Z = W′ ∩ V. Applying Lemma 2.6 to the diagram

\xymatrix{X \ar[r]^{incl} \ar[d] & W \ar[d] \\ S \ar[r] & Spec(k),}

the pull-back (incl)^{∗}F has a canonical structure of an extensible pretheory over the S-scheme X. We denote it again by (F, ϕ). By Definition 2.5, there exist an affine open neighbourhood U of {x_{1}, …, x_{n}} in X and a finite étale morphism π : Y → X ×_{S} U together with a section δ : Δ_{X,U} → Y over Δ_{X,U} and a homomorphism of presheaves Ψ : p_{1}^{∗}F → p_{2}^{∗}F over Y such that the restriction of Ψ to the U-point δ_{U} : U ⟶^{can} Δ_{X,U} ⟶^{δ} Y

Ψ_{δ_{U}} : δ_{U}^{∗}(p_{1}^{∗}F) ⟶ δ_{U}^{∗}(p_{2}^{∗}F)

is the identity id : i^{∗}F ⟶^{∼} i^{∗}F over U. By Lemma 1.6 we may assume (making U smaller, if necessary) that the standard triple T = (p̄: X̄ → S, X_{∞}, S) splits over (U, π : Y → X ×_{S} U, δ : Δ_{X,U} → Y).

Setting Z′ = π^{−1}(Z ×_{S} U), Lemma 1.5 yields an element τ in the group h_{0}(Y − Z′/_{pr_{2}∘π}U) whose image in h_{0}(Y/_{pr_{2}∘π}U) coincides with the class of the U-point δ_{U}. Since (F, ϕ) is homotopy invariant, τ defines (cf. Proposition 2.3) a homomorphism

ϕ(τ) : F (Y − Z′/_{p_{2}∘j′}X) → F (U),

where the structure of an X-scheme on Y − Z′ is given by the composite

Y − Z′ ↪^{j′} Y ⟶^{p_{2}} X.

Proposition 2.4 applied to the open embedding j′ : (Y − Z′) → Y, yields the commutative diagram

\xymatrix{F (Y/_{p_{2}}X) \ar[r]^{F (j′)} \ar[dr]_{F (δ_{U})=ϕ(j′_{∗}(τ))} & F (Y − Z′/_{p_{2}∘j′}X) \ar[d]^{ϕ(τ)} \\  & F (U).}

We therefore obtain the commutative diagram

F (W)  ⟶^{res}  F (V)
↓^{res}  ↓^{res}
F (X)  ⟶^{res}  F (X − Z)
↓^{pr_{1}^{∗}}  ↓^{pr_{1}∗}
F (X ×_{S} U/_{pr_{1}}X) ⟶^{res} F ((X − Z) ×_{S} U/_{pr_{1}}X)
↓_{π^{∗}}  ↓_{π^{∗}}
p_{1}^{∗}F (Y)  ⟶^{res}  p_{1}^{∗}F (Y − Z′)
↓^{Ψ}  ↓^{Ψ}
p_{2}^{∗}F (Y)  ⟶^{res}  p_{2}^{∗}F (Y − Z′)
∥  ∥
F (Y/_{p_{2}}X)  ⟶^{res}  F (Y − Z′/_{p_{2}∘j′}X)
↓^{δ_{U}^{∗}}  ↓^{ϕ(τ)}
F (U)  ⟶^{id}  F (U).

By the defining property of Ψ, the composite of all vertical arrows in the left hand column coincides with res : F (W) → F (U). We obtain the required homomorphism F (V) → F (U) as the composite of all vertical arrows in the right hand column □

4 Application to Étale Cohomology

Let X be a smooth scheme over a field k, n an integer prime to char(k) and K^{•} a bounded complex of locally constant constructible sheaves of ℤ/nℤ-modules on X_{et}. For an integer q we consider the presheaf F on X-Sm(k) which is given by

F (Y ⟶^{f} X) = H_{et}^{q}(Y, f^{∗}K^{•}),

where the object on the right hand side is étale hypercohomology. We will show that F carries in a natural way the structure of an extensible homotopy invariant pretheory over X.

First of all, F is homotopy invariant (see, e.g., [Mi] VI, 4.20). Next we will construct natural trace maps which will endow F with the structure of a pretheory over X. This is well known, but knowing no good reference, we give a construction of the trace maps below:

Let g : Y → S be a smooth curve in X-Sm(k). Since the base scheme X does not play any further role, we denote the pull-back of K^{•} to S by the same letter. Using the isomorphism Rg^{!}K^{•} ≅ g^{∗}K^{•}(1)[2] ([SGA4], XVIII, thm.3.2.5), we obtain

Hom(g^{∗}K^{•}, g^{∗}K^{•}) ≅ Hom(Rg_{!}g^{∗}K^{•}(1)[2], K^{•}),

and thus the identity of g^{∗}K^{•} induces a natural homomorphism

Tr_{g} : H_{c,et}^{q+2}(Y, g^{∗}K^{•}(1)) ⟶ H_{et}^{q}(S, K^{•}).

Here cohomology with compact support of Y is meant with respect to its Sscheme structure (and not as a variety over k). Let i : Z ↪ Y be an integral subscheme of codimension one in Y such that f = g ∘ i : Z → S is quasifinite and let cl(Z) ∈ H_{Z}^{2}(Y, ℤ/nℤ(1)) be its fundamental class (see [SGA4.5], (cycle 2.3)). The composition of the cup product ([SGA4.5], (cycle 1.2.2.2))

cl(Z)∪? : H_{c,et}^{q}(Z, f^{∗}K^{•}) → H_{c,et}^{q+2}(Y, g^{∗}K^{•}(1))

with Tr_{g} induces trace maps

Tr_{f} : H_{c,et}^{q}(Z, f^{∗}K^{•}) ⟶ H_{et}^{q}(S, K^{•}).

Finally, if f : Z → S is finite and surjective, then we get the required trace map ϕ_{Y/S}(Z) : F (Y) → F (S) as the composite map

H_{et}^{q}(Y, g^{∗}K^{•}) ⟶^{i^{∗}} H_{et}^{q}(Z, f^{∗}K^{•}) ⟶^{Tr_{f}} H_{et}^{q}(S, K^{•}).

All necessary compatibilities follow from the properties of the fundamental class cl(Z) proven in [SGA4.5], (cycle, 2.3).

Remark: A more “advanced” way to construct the trace maps would consist of the following steps

1. Since K^{i} is represented by an étale X-scheme for all i, we can extend K^{•} to a complex of qfh-sheaves ([SV2], appendix) on the category X-Nor(k) of X-schemes which are normal k-schemes of finite type.

2. Since qfh-sheaves admit transfer maps ([SV2],§5), the presheaf of abelian groups G(Y) = H_{qfh}^{q}(Y, K^{•}) on X-Nor(k) admits natural transfer maps.

3. The isomorphism H_{qfh}^{q}(Y, K^{•}) ≅ H_{et}^{q}(Y, K^{•}) ([SV2], thm.10.3) shows that the restriction of G to X-Sm(k) coincides with F.

Finally we have to show that F is extensible over k. Let X̃ be a finite Galois covering with Galois group G such that the pull-back of K^{•} to X̃ is a complex of constant sheaves. Consider the étale covering X̃ ×_{k} X̃ → X ×_{k} X with Galois group G × G. Let { \atop X ×_{k} X} = (X̃ ×_{k} X̃)_{G} be the unique intermediate covering associated with the diagonal subgroup G = (g, g) ∈ G × G. The diagonal map X̃ → X̃ ×_{k} X̃ induces a map δ : X → { \atop X ×_{k} X} which is a section to the projection { \atop X ×_{k} X} → X ×_{k} X over the diagonal X ≅ Δ_{X} ⊂ X ×_{k} X. Let Y be the connected component of X = im(δ) in { \atop X ×_{k} X}. Then Y is a connected Galois covering of the connected component of the diagonal Δ_{X} ⊂ X ×_{k} X having a section over Δ_{X}. The projections pr_{1} and pr_{2} on the first and second factor induce two structures of an X-scheme on Y and we use the notation Y_{1} and Y_{2} for Y in order to indicate the X-scheme structure on Y.

We will now extend the identity id : δ^{∗}pr_{1}^{∗}F ⟶^{∼} δ^{∗}pr_{2}^{∗}F over X = im(δ) ⊂ Y to the full scheme Y, thus verifying the condition of definition 2.5 with U = X.

By definition, for i = 1, 2, pr_{i}^{∗}F is the presheaf on Y -Sm(k) given by (f : U → Y) ↦ H_{et}^{q}(U, (pr_{i} ∘ f)^{∗}K^{•}). It therefore suffices to construct compatible isomorphisms of sheaves on Y -Sm(k)_{et}

(pr_{1} ∘ f)^{∗}K^{i} ⟶^{∼} (pr_{2} ∘ f)^{∗}K^{i}.

for all i. Let K^{i} be the finite étale X-scheme representing K^{i} on X_{et}. Then we have to construct a natural isomorphism

Y_{1} ×_{X} K^{i} ⟶^{∼} Y_{2} ×_{X} K^{i}.

Since K^{i} becomes constant over X̃, we are reduced to show that there exists a natural G-invariant isomorphism of Y -schemes

Y_{1} ×_{X} X̃ ⟶^{∼} Y_{2} ×_{X} X̃,

where G acts from the right on the second factors. We obtain this by restricting a natural G-invariant isomorphism of { \atop X ×_{k} X}

Ψ : ({ \atop X ×_{k} X})_{1} ×_{X} X̃ ⟶^{∼} ({ \atop X ×_{k} X})_{2} ×_{X} X̃.

to be constructed below to Y. Let us give the isomorphism Ψ on (geometric) points. A point on ({ \atop X ×_{k} X})_{1} ×_{X} X̃ is a pair ((a, b)G, c) where (a, b)G is a G-orbit (diagonal action) of points in X̃ ×_{k} X̃ and c is a point on X̃ such that a and c project to the same point in X. Let g ∈ G be the unique element with c = ag. We define Ψ(((a, b)G, c)) as the pair ((a, b)G, bg), which is a point in ({ \atop X ×_{k} X})_{2} ×_{X} X̃. If (a′, b′)G = (a, b)G, then there exists an element h ∈ G with ah = a′, bh = b′ and c = a′h^{−1}g, bg = bhh^{−1}g = b′h^{−1}g. Therefore Ψ is correctly defined. If one wants to obtain Ψ in a more formal way, one can give it as a G-invariant map of the (G × G) × G-sets associated with the schemes in question. Finally note that the diagram

im(δ)_{1} ×_{X} X̃ ⟶^{Ψ} im(δ)_{2} × X̃
↑^{≀}  ↑^{≀}
X ×_{X} X̃  ⟶^{id} X ×_{X} X̃

commutes. Therefore Ψ respects the connected component of im(δ) and induces an isomorphism

Y_{1} ×_{X} X̃ ⟶^{∼} Y_{2} ×_{X} X̃.

This shows that F carries in a natural way the structure of an extensible homotopy invariant pretheory over the k-scheme X.

Now it is easy to prove Theorem 1 of the introduction. Since étale cohomology commutes with inductive limits, we may suppose that the cohomology of K^{•} is also bounded from above. Denoting the projection by π : X → W, the complex Rπ_{∗}K^{•} has locally constant constructible cohomology sheaves bounded in both directions (see, e.g., [Mi], VI.4.2). By the main result of [PS], a bounded complex of sheaves on W_{et} with locally constant constructible cohomology sheaves is in the derived category isomorphic to a bounded complex of locally constant constructible sheaves. Therefore the assignment

(U ⟶^{f} W) ⟼ H_{et}^{q}(U, f^{∗}Rπ_{∗}K^{•}),

defines an extensible homotopy invariant pretheory over the smooth semi-local k-scheme W. Applying Theorem 2, we obtain Theorem 1. □

Finally, we mention the following variant of Theorem 1, which can be deduced by a straightforward limit argument.

Theorem 4.1 Let W be the spectrum of the henselization of the local ring of a closed point on a smooth k-scheme and let η ∈ W be the generic point. Let X → W be a proper smooth morphism, n an integer prime to char(k) and K^{•} a complex of sheaves of ℤ/nℤ-modules on X_{et} whose cohomology sheaves are locally constant constructible and bounded below. Then the canonical map

H_{et}^{q}(X, K^{•}) ⟶ H_{et}^{q}(X_{η}, K^{•})

is a universal monomorphism for all q ∈ ℤ .

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Alexander Schmidt Kirill Zainoulline Mathematisches Institut Steklov Mathematical Institute der Universität Heidelberg at St.Petersburg (POMI) Im Neuenheimer Feld 288 Fontanka 27 69120 Heidelberg 191011 St.Petersburg Germany Russia schmidt@mathi.uni-heidelberg.de kirill@pdmi.ras.ru