Milne’s Correcting Factor

and Derived De Rham Cohomology II

Baptiste Morin

Received: December 1, 2016

Communicated by Stephen Lichtenbaum

Abstract. Milne’s correcting factor, which appears in the Zetavalue at s = n of a smooth projective variety X over a finite field 𝔽_{q}, is the Euler characteristic of the derived de Rham cohomology of X/ℤ modulo the Hodge filtration F^{n}. In this note, we extend this result to arbitrary separated schemes of finite type over 𝔽_{q} of dimension at most d, provided resolution of singularities for schemes of dimension at most d holds. More precisely, we show that Geisser’s generalization of Milne’s factor, whenever it is well defined, is the Euler characteristic of the eh-cohomology with compact support of the derived de Rham complex relative to ℤ modulo F^{n}.

2010 Mathematics Subject Classification: 14G10, 14F40, 11S40, 11G25 Keywords and Phrases: Zeta functions, Special values, Derived de Rham cohomology, eh-cohomology

1 Introduction

For any separated scheme X of finite type over the finite field 𝔽_{q}, the special values of the zeta function Z(X, t) := ∏_{x∈X_{0}}(1 − t^{deg(x)})^{−1} are conjecturally given by

lim_{t→q^{−n}}Z(X, t) · (1 − q^{n}t)^{ρ_{n}} = ±χ(H_{W,c}^{∗}(X, ℤ(n)), ∪e) · q^{χ_{c}^{eh}(X/𝔽_{q},O,n)}. (1)

Here H_{W,c}^{∗}(X, ℤ(n)) denotes Geisser’s ”arithmetic cohomology with compact support”, ∪e is cup-product with the fundamental class e ∈ H^{1}(W_{𝔽_{q}}, ℤ) and q^{χ_{c}^{eh}(X/𝔽_{q},O,n)} is Geisser’s generalization of Milne’s correcting factor. The factor q^{χ_{c}^{eh}(X/𝔽_{q},O,n)} is well defined under the assumption that resolution of singularities for schemes of dimension ≤ dim(X) holds. The same assumption guaranties that, for X smooth projective, H_{W,c}^{∗}(X, ℤ(n)) coincides with Weilétale motivic cohomology and q^{χ_{c}^{eh}(X/𝔽_{q},O,n)} coincides with Milne’s correcting factor. For arbitrary X, the definitions of H_{W,c}^{∗}(X, ℤ(n)) and q^{χ_{c}^{eh}(X/𝔽_{q},O,n)} involve eh-cohomology with compact support. For instance

χ_{c}^{eh}(X/𝔽_{q}, O, n) := ∑_{i≤n,j∈ℤ}(−1)^{i+j} · (n − i) · dim_{𝔽_{q}}H_{c}^{j}(X_{eh}, Ω^{i})

where H_{c}^{i}(X_{eh}, Ω^{i}) denotes eh-cohomology with compact support of the sheaf of differentials Ω^{i}. Let Sch^{d}/𝔽_{q} be the category of separated schemes of finite type over 𝔽_{q} of dimension at most d. We say that R(d) holds if any X ∈ Sch^{d}/𝔽_{q} admits resolution of singularities (see [2] Definition 2.4 for a precise statement). T. Geisser has shown in [2] that, if R(d) holds and if the groups H_{W}^{i} (Y, ℤ(n)) are finitely generated for any smooth projective variety Y of dimension at most d, then χ_{c}^{eh}(X/𝔽_{q}, O, n) is well defined and (1) holds for any X ∈ Sch^{d}/𝔽_{q}. It was pointed out in [6] that, for X smooth projective, Milne’s correcting factor is the (multiplicative) Euler-Poincaré characteristic of the derived de Rham cohomology complex RΓ(X_{Zar}, LΩ_{X/ℤ}^{∗}/F^{n}) and that (1) can be restated in terms of a certain fundamental line. The aim of this note is to show that this remark applies for arbitrary separated schemes of finite type over 𝔽_{q}. More precisely, we denote by Sh_{eh}(Sch^{d}/𝔽_{q}) the category of sheaves of sets on the category Sch^{d}/𝔽_{q} endowed with the eh-topology. The resulting eh-topos Sh_{eh}(Sch^{d}/𝔽_{q}) is endowed with a structure ring O^{eh}, which is defined as the eh-sheafification of the presheaf X ↦ O_{X}(X) on Sch^{d}/𝔽_{q}. We denote by LΩ_{O^{eh}/ℤ}^{∗}/F^{n} the derived de Rham complex modulo the Hodge filtratio n F^{n} associated with the morphism of ringed topoi

(Sh_{eh}(Sch^{d}/𝔽_{q}), O^{eh}) ⟶ (Spec(ℤ), O_{Spec(ℤ)})

where O_{Spec(ℤ)} is the usual structure sheaf on Spec(ℤ). Then we consider its cohomology with compact support RΓ_{c}(X_{eh}, LΩ_{O^{eh}/ℤ}^{∗}/F^{n}). Under the assumption of Theorem 1.1(4) below, one may define the fundamental line

Δ(X/ℤ, n) := det_{ℤ}RΓ_{W,c}(X, ℤ(n)) ⊗_{ℤ} det_{ℤ}RΓ_{c}(X_{eh}, LΩ_{O^{eh}/ℤ}^{∗}/F^{n})

and its trivialization

λ_{X} : ℝ ⟶^{∼} Δ(X/ℤ, n) ⊗_{ℤ} ℝ

which is induced by the acyclic complex

⋯ ⟶^{∪θ} H_{W,c}^{i}(X, ℤ(n))_{ℝ} ⟶^{∪θ} H_{W,c}^{i+1}(X, ℤ(n))_{ℝ} ⟶^{∪θ} ⋯

Here the fundamental class θ = Id_{ℝ} ∈ H^{1}(ℝ, ℝ) = ”H^{1}(W_{𝔽_{1}}, ℝ)” is in some sense analogous to e ∈ H^{1}(W_{𝔽_{q}}, ℤ). We denote by ζ^{∗}(X, n) the leading coefficient in the Taylor development of ζ(X, s) = Z(X, q^{−s}) near s = n.

Theorem 1.1. Let X be a separated scheme of finite type over 𝔽_{q} and let n ∈ ℤ be an integer. Assume that X has dimension d and that R(d) holds.

1. If X is smooth projective, the canonical map

RΓ(X_{Zar}, LΩ_{X/ℤ}^{∗}/F^{n}) → RΓ_{c}(X_{eh}, LΩ_{O^{eh}/ℤ}^{∗}/F^{n})

is a quasi-isomorphism.

2. The complex RΓ_{c}(X_{eh}, LΩ_{O^{eh}/ℤ}^{∗}/F^{n}) is bounded with finite cohomology groups.

3. We have

∏_{i∈ℤ} | H_{c}^{i}(X_{eh}, LΩ_{O^{eh}/ℤ}^{∗}/F^{n}) |^{(−1)^{i}} = q^{χ_{c}^{eh}(X/𝔽_{q},O,n)}.  (2)

4. Assume moreover that for any smooth projective variety Y of dimension ≤ d, the usual Weil-étale cohomology groups H_{W}^{i} (Y, ℤ(n)) are finitely generated for all i. Then one has

Δ(X/ℤ, n) = ℤ · λ_{X} (ζ^{∗}(X, n)^{−1}) .

In particular, Theorem 1.1(1)–(3) holds (unconditionally) for dim(X) ≤ 2 and Theorem 1.1(4) holds for dim(X) ≤ 1. This note is organized as follows. We fix some notations and definitions in Section 2. In Section 3, we give the proof of Theorem 1.1, which is based on the following computation of the cohomology sheaves of the complex LΛ_{O^{eh}}^{n}L_{O^{eh}/ℤ}: we define an isomorphism (see Proposition 3.6)

H^{i−n}(LΛ_{O^{eh}}^{n}L_{O^{eh}/ℤ}) ≃ Ω_{O^{eh}/𝔽_{q}}^{i≤n}

where Ω^{i≤n} := Ω^{i} for i ≤ n and Ω^{i≤n} := 0 for i > n. This argument also gives a slightly different proof of the main result of [6], see Remark 3.5.

2 Preliminaries

2.1 The derived de Rham complex

Given a ring A and an A-module M, we denote by Λ_{A}(M) (resp. Γ_{A}(M)) the exterior A-algebra of M (resp. the divided power algebra of M, see [1] App. A), and by Λ_{A}^{i}(M) (resp. Γ_{A}^{i}(M)) its submodule of homogeneous elements of degree i. If (S, A) is a ringed topos and M an A-module, one defines Λ_{A}(M), Γ_{A}(M), Λ_{A}^{i}(M) and Γ_{A}^{i}(M) as above, internally in S. Then Λ_{A}(M) (resp. Γ_{A}(M)) coincides with the sheafification of U ↦ Λ_{A(U)}(M(U)) (resp. U ↦ Γ_{A(U)}(M(U))). We denote by LΛ_{A}^{i} the left derived functor of the (non-additive) exterior power functor Λ_{A}^{i} (see [4] I.4.2). We often omit the subscript A and simply write Λ^{i}M, Γ^{i}M and LΛ^{i}M. Let A → B be a morphism of rings in S. We denote by Ω_{B/A}^{1} the B-module of Kähler differe ntials, we set Ω_{B/A}^{i} := Λ_{B}^{i}Ω_{B/A}^{1} and we denote by Ω_{B/A}^{<n} the complex of A-modules [Ω_{B/A}^{0} → Ω_{B/A}^{1} → ⋯ → Ω_{B/A}^{n−1}] put in degrees [0, n−1]. Let P_{A}(B) be the standard simplicial free resolution of the A-algebra B (see [4] I.1.5.5.6), and let L_{B/A} be the cotangent complex ([4] II.1). By definition L_{B/A} is the complex of B-modules associated with the simplicial B-module Ω_{P_{A}(B)/A}^{1}⊗_{P_{A}(B)}B. Similarly we define LΛ_{B}^{i}L_{B/A} as the (actual) complex of B-modules associated with the simplicial B-module Ω_{P_{A}(B)/A}^{i} ⊗_{P_{A}(B)} B. The derived de Rham complex modulo F^{n} is defined as the total complex (see [5] VIII.2.1)

LΩ_{B/A}^{∗}/F^{n} := Tot(Ω_{P_{A}(B)/A}^{<n})

which we simply see in this paper as a complex of A-modules. The Hodge filtration on LΩ_{B/A}^{∗}/F^{n} satisfies gr^{p}(LΩ_{B/A}^{∗}/F^{n}) ≃ LΛ_{B}^{p}L_{B/A}[−p] for p < n and gr^{p}(LΩ_{B/A}^{∗}/F^{n}) = 0 otherwise. For example, if (X, O_{X}) is a scheme, then P_{ℤ}(O_{X}) denotes the standard simplicial free resolution of ℤ → O_{X} in the small Zariski topos of the scheme X, and L_{X/ℤ} := L_{O_{X}/ℤ} is the cotangent complex associated with the morphism of schemes X → Spec(ℤ). If f : S′ → S is a morphism of topoi, we write f^{−1} : S → S′ for the set-theoretic inverse image functor of f. Let f : (S′, A′) → (S, A) be a morphism of ringed topoi, i.e. a morphism of topoi f : S′ → S together with a morphism of rings f^{−1}A → A′ in S′. One defines

LΩ_{f}^{∗}/F^{n} = LΩ_{(S′,A′)/(S,A)}^{∗}/F^{n} := LΩ_{A′/f^{−1}A}^{∗}/F^{n}

which is a complex of f^{−1}A-modules in S′. We denote by f^{∗} : Mod(A) → Mod(A′) the inverse image functor for modules, i.e. f^{∗}M := f^{−1}M ⊗_{f^{−1}A} A′, where Mod(A) (resp. Mod(A′)) is the category of A-modules in S (resp. of A′-modules in S′).

Lemma 2.1. Let f : S′ → S be a morphism of topoi and let A → B be a morphism of rings in S. Then we have f^{−1}(P_{A}(B)) ≃ P_{f^{−1}A}(f^{−1}B),

f^{−1} (LΩ_{B/A}^{∗}/F^{n}) ≃ LΩ_{f^{−1}B/f^{−1}A}^{∗}/F^{n}, an isomorphism of f^{−1}B-modules

f^{−1}(Ω_{B/A}^{i}) ≃ Ω_{f^{−1}B/f^{−1}A}^{i} and an isomorphism of complexes of f^{−1}B-modules f^{−1}(LΛ_{B}^{i}L_{B/A}) ≃ LΛ_{f^{−1}B}^{i}L_{f^{−1}B/f^{−1}A}.

Proof. The identifications f^{−1}(P_{A}(B)) ≃ P_{f^{−1}A}(f^{−1}B) and f^{−1}(Ω_{B/A}^{1}) ≃ Ω_{f^{−1}B/f^{−1}A}^{1} follow from the definitions (see [4] II.1.2.1.4 and [4] II.1.1.4.1). Moreover we have f^{−1}(Λ_{R}^{i}(M)) ≃ Λ_{f^{−1}R}^{i}(f^{−1}M) for any ring R in S and any R-module M. The result follows easily.

2.2 Derived de Rham cohomology with compact support

The following definition is due to Thomas Geisser [2]. Let Sch^{d}/𝔽_{q} be the category of separated schemes of finite type over 𝔽_{q} of dimension ≤ d.

Definition 2.2. The eh-topology on Sch^{d}/𝔽_{q} is the Grothendieck topology generated by the following coverings:

• étale coverings

• abstract blow-ups: If we have a cartesian square

\xymatrix{Z′ \ar[r]^{i′} \ar[d]^{f′} & X′ \ar[d]^{f} \\ Z \ar[r]^{i} & X}

where f is proper, i a closed embedding, and f induces an isomorphism

X′ − Z′ ⟶^{∼} X − Z, then (X′ →^{f} X, Z →^{i} X) is a covering.

We denote by PSh(Sch^{d}/𝔽_{q}) the category of presheaves of sets on Sch^{d}/𝔽_{q} and by Sh_{eh}(Sch^{d}/𝔽_{q}) the topos of eh-sheaves of sets on Sch^{d}/𝔽_{q}. Note that the functor

y : Sch^{d}/𝔽_{q} ↪ PSh(Sch^{d}/𝔽_{q}) → Sh_{eh}(Sch^{d}/𝔽_{q}),

given by composing the Yoneda embedding and eh-sheafification, is not fully faithful. Hence the eh-topology is not subcanonical. For example, if X^{red} denotes the maximal reduced closed subscheme of X ∈ Sch^{d}/𝔽_{q}, then the induced map yX^{red} → yX is an isomorphism. If U is an object of Sch^{d}/𝔽_{q} and F an eh-sheaf on Sch^{d}/𝔽_{q}, we choose a Nagata compactification U ↪ X with closed complement Z ↪ X (so that X is proper over 𝔽_{q} and U is open and dense in X), and we define

RΓ_{c}(U_{eh}, F) := Cone (RΓ(X_{eh}, F) → RΓ(Z_{eh}, F)) [−1].

Here RΓ(X_{eh}, F) denotes the cohomology of the slice topos Sh_{eh}(Sch^{d}/𝔽_{q})/yX with coefficients in F×yX → yX. Equivalently, RΓ(X_{eh}, −) is the total derived functor of the functor F ↦ F(X). It can be shown that RΓ_{c}(U_{eh}, F) does not depend on the compactification (see [2] Proposition 3.2). Then RΓ_{c}(U_{eh}, F) is contravariant for proper maps and covariant for open immersions. For an open-closed decomposition (U →^{j} X ←^{i} Z), there is an exact triangle

RΓ_{c}(U_{eh}, F) → RΓ_{c}(X_{eh}, F) → RΓ_{c}(Z_{eh}, F) →

Notation 2.3. The structure ring O^{eh} on Sh_{eh}(Sch^{d}/𝔽_{q}) is the eh-sheaf associated with the presheaf of rings

R : (Sch^{d}/𝔽_{q})^{op} ⟶ Rings X ⟼ O_{X}(X) .

Consider the morphism of ringed topoi

ψ : (Sh_{eh}(Sch^{d}/𝔽_{q}), O^{eh}) ⟶ (Spec(ℤ), O_{Spec(ℤ)}) induced by the evident morphism of sites. Let

LΩ_{O^{eh}/ℤ}^{∗}/F^{n} := LΩ_{ψ}^{∗}/F^{n}

be the corresponding derived de Rham complex modulo the nth-step of the Hodge filtration. Derived de Rham cohomology modulo F^{n} with compact support is given by

X ↦ RΓ_{c}(X_{eh}, LΩ_{O^{eh}/ℤ}^{∗}/F^{n})

for X ∈ Sch^{d}/𝔽_{q}. It is covariantly functorial for open immersions and contravariantly functorial for proper maps.

We now explain our notation LΩ_{O^{eh}/ℤ}^{∗}/F^{n}. There is a unique morphism of rings ℤ^{eh} → O^{eh}, where ℤ^{eh} denotes the constant sheaf of rings associated with ℤ on Sh_{eh}(Sch^{d}/𝔽_{q}). Let LΩ_{O^{eh}/ℤ^{eh}}^{∗}/F^{n} be the corresponding derived de Rham complex modulo F^{n}. Then we have

LΩ_{ψ}^{∗}/F^{n} ≃ LΩ_{O^{eh}/ℤ^{eh}}^{∗}/F^{n}.  (3)

Indeed, consider the structure sheaf O_{Spec(ℤ)} and the constant sheaf ℤ over the small Zariski topos of Spec(ℤ). We have L_{O_{Spec(ℤ)}/ℤ} = 0 (see [4] II.2.3.1 and II.2.3.6), hence L_{O^{eh}/ψ^{−1}O_{Spec(ℤ)}} ≃ L_{O^{eh}/ψ^{−1}ℤ} = L_{O^{eh}/ℤ^{eh}}. We obtain LΛ^{∗}L_{O^{eh}/ℤ^{eh}} ≃ LΛ^{∗}L_{O^{eh}/ψ^{−1}O_{Spec(ℤ)}} hence

LΩ_{O^{eh}/ℤ^{eh}}^{∗}/F^{n} ≃ LΩ_{O^{eh}/ψ^{−1}O_{Spec(ℤ)}}^{∗}/F^{n} := LΩ_{ψ}^{∗}/F^{n}

by the Hodge filtration. Finally, we note that L_{O^{eh}/ℤ} and LΩ_{O^{eh}/ℤ}^{∗}/F^{n} could be left-unbounded (see however Corollary 3.8).

2.3 The fundamental line

For an object C in the derived category of abelian groups such that H^{i}(C) is finitely generated for all i and H^{i}(C) = 0 for almost all i, we set

det_{ℤ}(C) := ⨂_{i∈ℤ} det_{ℤ}^{(−1)^{i}}H^{i}(C).

If H^{i}(C) is moreover finite for all i, then we call the following isomorphism

det_{ℤ}(C) ⊗_{ℤ} ℚ →^{∼} ⨂_{i∈ℤ} det_{ℚ}^{(−1)^{i}} (H^{i}(C) ⊗_{ℤ} ℚ) →^{∼} ⨂_{i∈ℤ} det_{ℚ}^{(−1)^{i}}(0) →^{∼} ℚ

the canonical ℚ-trivialization of det_{ℤ}(C). In this situation, the canonical ℚtrivialization det_{ℤ}(C) ⊗_{ℤ} ℚ ≃ ℚ identifies det_{ℤ}(C) with

ℤ · (∏_{i∈ℤ} |H^{i}(C)|^{(−1)^{i+1}}) ⊂ ℚ.

For X ∈ Sch^{d}/𝔽_{q}, one defines [2]

RΓ_{W,c}(X, ℤ(n)) := RΓ(W_{𝔽_{q}}, RΓ_{c}(X_{\overline{𝔽}_{q},eh}, ρ^{−1}ℤ(n)))

where \overline{𝔽}_{q} is an algebraic closure, W_{𝔽_{q}} is the Weil group, ρ is the morphism defined in Lemma 3.1, and the ℤ(n) on the right hand side is the motivic complex on Sm^{d}/𝔽_{q}. Assuming that RΓ_{W,c}(X, ℤ(n)) and RΓ_{c}(X_{eh}, LΩ_{O^{eh}/ℤ}^{∗}/F^{n}) are both well defined and perfect, the fundamental line is defined as follows:

Δ(X/ℤ, n) := det_{ℤ}RΓ_{W,c}(X, ℤ(n)) ⊗_{ℤ} det_{ℤ}RΓ_{c}(X_{eh}, LΩ_{O^{eh}/ℤ}^{∗}/F^{n}).

Recall that W_{𝔽_{q}} ≃ ℤ with generator given by the Frobenius F. Consider the map 𝔣 : W_{𝔽_{q}} → W_{𝔽_{1}} := ℝ satisfying 𝔣(F) = log(q), and define θ = Id_{ℝ} ∈ H^{1}(ℝ, ℝ). Then 𝔣^{∗}θ ∈ H^{1}(W_{𝔽_{q}}, ℝ) maps the Frobenius F ∈ W_{𝔽_{q}} to log(q) ∈ ℝ, whereas e ∈ H^{1}(W_{𝔽_{q}}, ℝ) maps the Frobenius F to 1 ∈ ℝ. We have

RΓ_{W,c}(X, ℤ(n))_{ℝ} ≃ RΓ(W_{𝔽_{q}}, RΓ_{c}(X_{\overline{𝔽}_{q},eh}, ρ^{−1}ℤ(n))_{ℝ}).

So cup-product with the class 𝔣^{∗}θ ∈ H^{1}(W_{𝔽_{q}}, ℝ) defines a map

H_{W,c}^{i}(X, ℤ(n))_{ℝ} ⟶^{∪θ} H_{W,c}^{i+1}(X, ℤ(n))_{ℝ}

which differs from

H_{W,c}^{i}(X, ℤ(n))_{ℝ} ⟶^{∪e} H_{W,c}^{i+1}(X, ℤ(n))_{ℝ}

by the factor log(q). The complex

⋯ ⟶^{∪θ} H_{W,c}^{i}(X, ℤ(n))_{ℝ} ⟶^{∪θ} H_{W,c}^{i+1}(X, ℤ(n))_{ℝ} ⟶^{∪θ} ⋯

is acyclic [2] hence gives a trivialization

λ_{X} : ℝ ⟶^{∼} det_{ℝ}RΓ_{W,c}(X, ℤ(n))_{ℝ} ⟶^{∼} Δ(X/ℤ, n) ⊗_{ℤ} ℝ.

where the second isomorphism is induced by the canonical ℚtrivialization of det_{ℤ}RΓ_{c}(X_{eh}, LΩ_{O^{eh}/ℤ}^{∗}/F^{n}), whose existence requires that RΓ_{c}(X_{eh}, LΩ_{O^{eh}/ℤ}^{∗}/F^{n}) is bounded with finite cohomology groups.

3 Proof of Theorem 1.1

We denote by Sm^{d}/𝔽_{q} the full subcategory of Sch^{d}/𝔽_{q} consisting of smooth 𝔽_{q}-schemes. We endow Sm^{d}/𝔽_{q} with the Zariski topology and we denote by Sh_{Zar}(Sm^{d}/𝔽_{q}) the corresponding topos. Recall the following description of the topos Sh_{Zar}(Sm^{d}/𝔽_{q}) (see [3] IV.4.10.6). A sheaf F on Sh_{Zar}(Sm^{d}/𝔽_{q}) can be seen as a family of sheaves F_{X} on the small Zariski topos X_{Zar} for any X ∈ Sm^{d}/𝔽_{q} together with transition maps α_{f} : f^{−1}F_{X} → F_{Y} for any map f : Y → X satisfying α_{f∘g} = α_{g} ∘ g^{−1}α_{f} and such that α_{f} is an isomorphism whenever f is an open immersion. A morphism F → G in Sh_{Zar}(Sm^{d}/𝔽_{q}) is a given by a family of morphisms F_{X} → G_{X} compatible with the transition maps. For any X ∈ Sm^{d}/𝔽_{q}, the functor

\begin{aligned}res_{X} : & Sh_{Zar}(Sm^{d}/𝔽_{q}) & ⟶ & X_{Zar} \\ & F & ⟼ & F_{X} \end{aligned} ,

mapping the big Zariski sheaf F to its restriction F_{X} to the small Zariski site of X, commutes with arbitrary small limits and colimits. It is therefore the inverse image of a morphism of topoi

s_{X} : X_{Zar} ⟶ Sh_{Zar}(Sm^{d}/𝔽_{q})/X ⟶ Sh_{Zar}(Sm^{d}/𝔽_{q}).

In fact the morphism X_{Zar} ⟶ Sh_{Zar}(Sm^{d}/𝔽_{q})/X is a section of the morphism

Sh_{Zar}(Sm^{d}/𝔽_{q})/X ≃ Sh_{Zar}((Sm^{d}/𝔽_{q})/X) ⟶ X_{Zar}  (4)

which is induced by the evident morphism of sites. The same description of abelian sheaves on Sh_{Zar}(Sm^{d}/𝔽_{q}) is valid. We denote by O the canonical structure ring on Sh_{Zar}(Sm^{d}/𝔽_{q}), i.e. O(X) := O_{X}(X) for any X ∈ Sm^{d}/𝔽_{q}. We have res_{X}(O) = O_{X} where O_{X} denotes the usual structure sheaf on the smooth scheme X. As above, a complex of O-modules F can be seen as family of complexes of O_{X}-modules F_{X} in the small Zariski topos X_{Zar} together with transition maps of complexes of O_{Y} -modules α_{f} : f^{∗}F_{X} := f^{−1}F_{X} ⊗_{f^{−1}O_{X}} O_{Y} → F_{Y} for any map f : Y → X satisfying α_{f∘g} = α_{g} ∘ g^{∗}α_{f}, and such that α_{f} is an isomorphism whenever f is an open immer sion. We denote by R(d) the condition given in ([2] Definition 2.4). The morphism ρ of the next lemma was defined in ([2] Lemma 2.5), see also ([7] Proposition 5.11).

Lemma 3.1. Assume that R(d) holds. Then we have a composite morphism of topoi

ρ : Sh_{eh}(Sch^{d}/𝔽_{q}) ⟶^{∼} Sh_{eh}(Sm^{d}/𝔽_{q}) ⟶ Sh_{Zar}(Sm^{d}/𝔽_{q})

where the first morphism is an equivalence. Moreover we have

ρ^{−1}O ≃ O^{eh}.  (5)

Proof. We consider the topology on Sm^{d}/𝔽_{q} induced by the eh-topology on Sch^{d}/𝔽_{q} (see [3] III.3), and we define Sh_{eh}(Sm^{d}/𝔽_{q}) as the topos of sheaves on this site. It follows from R(d) and ([2] Lemma 2.2.b) that Sm^{d}/𝔽_{q} is a topologically generating full subcategory of Sch^{d}/𝔽_{q} with respect to the ehtopology. By ([3] III.4.1), the first morphism is an equivalence. The inclusion functor (Sm^{d}/𝔽_{q}, Zar) → (Sch^{d}/𝔽_{q}, eh) is continuous, i.e. if F is an eh-sheaf on Sch^{d}/𝔽_{q} then its restriction to Sm^{d}/𝔽_{q} is a Zariski sheaf. In other words, the induced eh-topology on Sm^{d}/𝔽_{q} is stronger than the Zariski topology; hence the second morphism is well defined.

Let u : Sm^{d}/𝔽_{q} → Sch^{d}/𝔽_{q} be the inclusion functor, and let f : Sh_{eh}(Sch^{d}/𝔽_{q}) →^{∼} Sh_{eh}(Sm^{d}/𝔽_{q}) be the induced equivalence. We have a commutative square (see [3] III.1.3)

\xymatrix{Sh_{eh}(Sch^{d}/𝔽_{q}) & Sh_{eh}(Sm^{d}/𝔽_{q}) \ar[l]_{f^{−1}} \\ PSh(Sch^{d}/𝔽_{q}) \ar[u]^{a} & PSh(Sm^{d}/𝔽_{q}) \ar[l]_{u_{!}} \ar[u]^{a^{sm}}}

where the vertical arrows are the associated sheaf functors. Let F ∈ PSh(Sch^{d}/𝔽_{q}) be a presheaf of sets and let u^{∗} be the right adjoint of u_{!}. The adjunction morphism u_{!}u^{∗}F → F is ”bicouvrant” (see [3] III.4.1.1) hence a(u_{!}u^{∗}F) →^{∼} a(F) is an isomorphism (see [3] II.5.3). Since the square above is commutative, we obtain

f^{−1} ∘ a^{sm} ∘ u^{∗}(F) ≃ a ∘ u_{!} ∘ u^{∗}(F) ≃ a(F).

So we have an isomorphism of left exact functors f^{−1} ∘ a^{sm} ∘ u^{∗} ≃ a, hence a similar isomorphism of functors between the categories of ring objects. Let R (resp. O) be the presheaf of rings on Sch^{d}/𝔽_{q} (resp. on Sm^{d}/𝔽_{q}) mapping X to O_{X}(X). By definition we have O = u^{∗}R, O^{eh} = a(R) and g^{−1}O = a^{sm}(O), where g : Sh_{eh}(Sm^{d}/𝔽_{q}) → Sh_{Zar}(Sm^{d}/𝔽_{q}) is the morphism of topoi defined above. We obtain

ρ^{−1}(O) ≃ f^{−1} ∘ g^{−1}(O) ≃ f^{−1} ∘ a^{sm}(O) ≃ f^{−1} ∘ a^{sm} ∘ u^{∗}(R) ≃ a(R) =: O^{eh}.

We may therefore promote ρ into a morphism of ringed topoi

ρ : (Sh_{eh}(Sch^{d}/𝔽_{q}), O^{eh}) ⟶ (Sh_{Zar}(Sm^{d}/𝔽_{q}), O).

For any X ∈ Sm^{d}/𝔽_{q}, we shall also consider the morphism of ringed topoi obtained by localisation over X:

ρ_{/X} : (Sh_{eh}(Sch^{d}/𝔽_{q}), O^{eh})/yX ⟶ (Sh_{Zar}(Sm^{d}/𝔽_{q}), O)/X.

We denote by ℤ the constant sheaf on either Sh_{Zar}(Sm^{d}/𝔽_{q}) or Sh_{eh}(Sch^{d}/𝔽_{q}), and we apply the constructions of Section 2.1 to the unique morphism of rings ℤ → O^{eh} (respectively ℤ → O) in the topos Sh_{eh}(Sch^{d}/𝔽_{q}) (respectively in the topos Sh_{Zar}(Sm^{d}/𝔽_{q})); see (3) and its proof.

Lemma 3.2. Assume R(d). We have

LΩ_{O^{eh}/ℤ}^{∗}/F^{n} ≃ ρ^{−1} (LΩ_{O/ℤ}^{∗}/F^{n})  (6)

and the complex of abelian sheaves LΩ_{O/ℤ}^{∗}/F^{n} on Sh_{Zar}(Sm^{d}/𝔽_{q}) is given by the complexes of abelian sheaves on X_{Zar}

res_{X} (LΩ_{O/ℤ}^{∗}/F^{n}) = LΩ_{X/ℤ}^{∗}/F^{n} := Tot(Ω_{P_{ℤ}(O_{X})/ℤ}^{<n})  (7)

and obvious transition maps. Similarly, we have an isomorphism of complexes of O^{eh}-modules

LΛ_{O^{eh}}^{i}L_{O^{eh}/ℤ} ≃ ρ^{−1} (LΛ_{O}^{i}L_{O/ℤ})  (8)

and the complex of O-modules LΛ_{O}^{i}L_{O/ℤ} is given by the complexes of O_{X}- modules

res_{X} (LΛ_{O}^{i}L_{O/ℤ}) = LΛ_{O_{X}}^{i} L_{X/ℤ} := Tot(Ω_{P_{ℤ}(O_{X})/ℤ}^{i} ⊗_{P_{ℤ}(O_{X})} O_{X}) (9)

and obvious transition maps. Finally, we have an isomorphism of O^{eh}-modules

Ω_{O^{eh}/ℤ}^{i} ≃ ρ^{−1}Ω_{O/ℤ}^{i}  (10)

and the O-module Ω_{O/ℤ}^{i} is given by the O_{X}-modules

res_{X} (Ω_{O/ℤ}^{i}) = Ω_{X/ℤ}^{i}  (11)

and obvious transition maps.

Proof. The complex LΩ_{X/ℤ}^{∗}/F^{n} := Tot(Ω_{P_{ℤ}(O_{X})/ℤ}^{<n}) is functorial on the nose in X ∈ Sm^{d}/𝔽_{q}. Indeed, given a map f : Y → X, there is a canonical morphism of complexes of abelian sheaves

f^{−1}LΩ_{X/ℤ}^{∗}/F^{n} ≃ LΩ_{f^{−1}O_{X}/ℤ}^{∗}/F^{n} → LΩ_{Y/ℤ}^{∗}/F^{n},  (12)

where the first map is supplied by Lemma 2.1 and the second map is induced by the structural morphism f^{−1}O_{X} → O_{Y}. The map f^{−1}LΩ_{X/ℤ}^{∗}/F^{n} → LΩ_{Y/ℤ}^{∗}/F^{n} is an isomorphism of complexes of abelian sheaves if f : Y → X is an open immersion. Similarly, the map f induces a morphism of complexes of O_{Y} -modules

f^{∗}LΛ_{O_{X}}^{i} L_{X/ℤ} ≃ LΛ_{f^{−1}O_{X}}^{i} L_{f^{−1}O_{X}/ℤ} ⊗_{f^{−1}O_{X}} O_{Y} → LΛ_{O_{Y}}^{i} L_{O_{Y} /ℤ}  (13)

which is an isomorphism of complexes if f is an open immersion. We apply Lemma 2.1 to the morphism of topoi

s_{X} : X_{Zar} ⟶ Sh_{Zar}(Sm^{d}/𝔽_{q})

and we observe that the transition maps

f^{−1}res_{X}(LΩ_{O/ℤ}^{∗}/F^{n}) → res_{Y} (LΩ_{O/ℤ}^{∗}/F^{n}) and

f^{∗}res_{X}(LΛ_{O}^{i}L_{O/ℤ}/F^{n}) → res_{Y} (LΛ_{O}^{i}L_{O/ℤ}/F^{n})

can be identified with (12) and (13) respectively. This yields (7) and (9). We obtain (6) and (8) by applying Lemma 2.1 to the morphism

ρ : Sh_{eh}(Sch^{d}/𝔽_{q}) ⟶ Sh_{Zar}(Sm^{d}/𝔽_{q})

since we have O^{eh} ≃ ρ^{−1}O by (5). The proof of (10) and (11) is similar.

For a complex C of sheaves of modules on some topos, we denote by H^{i}(C) its i-th cohomology sheaf.

Lemma 3.3. Let X be a smooth separated scheme of finite type over 𝔽_{q}. Then there is a canonical isomorphism of sheaves of O_{X}-modules

H^{i}(LΛ^{n}L_{X/ℤ}[−n]) ≃ Ω_{X/𝔽_{q}}^{i≤n}

where Ω_{X/𝔽_{q}}^{i≤n} := Ω_{X/𝔽_{q}}^{i} for 0 ≤ i ≤ n and Ω_{X/𝔽_{q}}^{i≤n} = 0 otherwise. Moreover, for

f : Y → X a morphism in Sm^{d}/𝔽_{q}, the square of O_{Y} -modules

\xymatrix{f^{∗}H^{i}(LΛ^{n}L_{X/ℤ}[−n]) \ar[r]^{∼} \ar[d] & f^{∗}Ω_{X/𝔽_{q}}^{i≤n} \ar[d] \\ H^{i}(LΛ^{n}L_{Y/ℤ}[−n]) \ar[r]^{∼} & Ω_{Y/𝔽_{q}}^{i≤n}}

commutes, where the left vertical map is induced by (13) and the right vertical map is the evident one.

Proof. Let X be a scheme in Sm^{d}/𝔽_{q}. We have an exact triangle in the derived category D(O_{X}) of O_{X}-modules (see [6] for details):

O_{X}[1] → L_{X/ℤ} → Ω_{X/𝔽_{q}}^{1} →^{ω_{X}} O_{X}[2].

Let U ⊂ X be an affine open subscheme. Then ω_{U} ∈ Ext_{O_{U}}^{2} (Ω_{U/𝔽_{q}}^{1}, O_{U}) = 0 and there is a unique isomorphism

α_{U} : L_{U/ℤ} ⟶^{∼} O_{U}[1] ⊕ Ω_{U/𝔽_{q}}^{1}

in the derived category D(O_{U}) of O_{U}-modules, such that H^{−1}(α_{U}) : H^{−1}(L_{U/ℤ}) ≃ O_{U} and H^{0}(α_{U}) : H^{0}(L_{U/ℤ}) ≃ Ω_{X/𝔽_{q}}^{1} are the isomorphisms given by the triangle above. Indeed, the canonical map

Hom_{D(O_{U})}(L_{U/ℤ}, O_{U}[1] ⊕ Ω_{U/𝔽_{q}}^{1})
⟶ Hom_{O_{U}} (H^{−1}(L_{U/ℤ}), O_{U}) ⊕ Hom_{O_{U}} (H^{0}(L_{U/ℤ}), Ω_{U/𝔽_{q}}^{1})

is an isomorphism, as it follows from the spectral sequence

∏_{n∈ℤ} Ext^{p}(H^{n}(L_{U/ℤ}), H^{n+q}(O_{U}[1] ⊕ Ω_{U/𝔽_{q}}^{1})) ⇒

⇒ H^{p+q}(RHom(L_{U/ℤ}, O_{U}[1] ⊕ Ω_{U/𝔽_{q}}^{1}))

and from the fact higher Ext’s vanish since U is affine and Ω_{X/𝔽_{q}}^{1} is locally free of finite rank. Then α_{U} is functorial in the open affine subscheme U, in the sense that, if V ⊆ U is the inclusion of an open affine subscheme V, then α_{U} | V = α_{V} by uniqueness of α_{V}. We obtain the following isomorphism in D(O_{U}) (see [6] for details):

LΛ^{n}L_{U/ℤ}
≃ LΛ^{n}([O_{U} →^{0} Ω_{U/𝔽_{q}}^{1}][1])
≃ [Γ^{n}O_{U} ⊗ Λ^{0}Ω_{U/𝔽_{q}}^{1} →^{0} Γ^{n−1}O_{U} ⊗ Λ^{1}Ω_{U/𝔽_{q}}^{1} →^{0} ⋯ →^{0} Γ^{0}O_{U} ⊗ Λ^{n}Ω_{U/𝔽_{q}}^{1}][n]

where the differential maps are all trivial. This yields a canonical isomorphism of O_{U}-modules

a_{U} : H^{i}(LΛ^{n}L_{X/ℤ}[−n]) | U ≃ H^{i}(LΛ^{n}L_{U/ℤ}[−n]) ≃ Γ^{n−i}O_{U} ⊗_{O_{U}} Ω_{U/𝔽_{q}}^{i}

for any i ∈ ℤ, where Γ^{n−i}O_{U} := 0 for n − i < 0 and Ω_{U/𝔽_{q}}^{i} := 0 for i < 0. Moreover, the isomorphisms a_{U} are compatible with the restriction maps given by inclusions of affine open subsets V ⊆ U, in the sense that (a_{U}) | V = a_{V}. Covering X by open affine subschemes U (recall that X is separated so that the intersection of two affine open subschemes is affine), the identifications a_{U} therefore give an isomorphism of sheaves of O_{X}-modules

H^{i}(LΛ^{n}L_{X/ℤ}[−n]) ≃ Γ^{n−i}O_{X} ⊗_{O_{X}} Ω_{X/𝔽_{q}}^{i}.

For n−i ≥ 0, the O_{X}-module Γ^{n−i}O_{X} is free of rank one with generator γ_{n−i}(1) where 1 ∈ O_{X} is the unit section and γ_{n−i} : O_{X} → Γ^{n−i}O_{X} the canonical map. So we obtain an isomorphism

H^{i}(LΛ^{n}L_{X/ℤ}[−n]) ≃ Γ^{n−i}O_{X} ⊗_{O_{X}} Ω_{X/𝔽_{q}}^{i} ≃ Ω_{X/𝔽_{q}}^{i≤n}.  (14)

We now check that the isomorphism (14) is functorial in X ∈ Sm^{d}/𝔽_{q}. Let Y and X be schemes in Sm^{d}/𝔽_{q} and let f : Y → X be an arbitrary map. There is a morphism of exact triangles (see [4] II.2.1.5)

\xymatrix{Lf^{∗}O_{X}[1] \ar[r] \ar[d] & Lf^{∗}L_{X/ℤ} \ar[r] \ar[d] & Lf^{∗}Ω_{X/𝔽_{q}}^{1} \ar[r]^{Lf^{∗}ω_{X}} \ar[d] & Lf^{∗}O_{X}[2] \ar[d] \\ O_{Y} [1] \ar[r] & L_{Y/ℤ} \ar[r] & Ω_{Y/𝔽_{q}}^{1} \ar[r]^{ω_{Y}} & O_{Y} [2]}

Suppose first that X and Y are affine. Then ω_{X} = 0 and ω_{Y} = 0, and the square

\xymatrix{Lf^{∗}L_{X/ℤ} \ar[r]_{Lf^{∗}α_{X}}^{∼} \ar[d] & f^{∗}O_{X}[1] ⊕ f^{∗}Ω_{X/𝔽_{q}}^{1} \ar[d] \\ L_{Y/ℤ} \ar[r]_{α_{Y}}^{∼} & O_{Y} [1] ⊕ Ω_{Y/𝔽_{q}}^{1}}

commutes in D(O_{Y}), since a morphism Lf^{∗}L_{X/ℤ} → O_{Y} [1] ⊕ Ω_{Y/𝔽_{q}}^{1} is determined by the morphisms it induces on cohomology (as for α_{X} above). Hence the bottom square in the following diagram

\xymatrix{Lf^{∗}LΛ^{n}L_{X/ℤ} \ar[r]^{∼} \ar[d] & f^{∗}[Γ_{O_{X}}^{n} O_{X} ⊗ Λ_{O_{X}}^{0} Ω_{X/𝔽_{q}}^{1} →^{0} ⋯ →^{0} Γ_{O_{X}}^{0} O_{X} ⊗ Λ_{O_{X}}^{n} Ω_{X/𝔽_{q}}^{1} ][n] \ar[d] \\ LΛ^{n}Lf^{∗}L_{X/ℤ} \ar[r]^{∼} \ar[d] & [Γ_{O_{Y}}^{n} f^{∗}O_{X} ⊗ Λ_{O_{Y}}^{0} f^{∗}Ω_{X/𝔽_{q}}^{1} →^{0} ⋯ →^{0} Γ_{O_{Y}}^{0} f^{∗}O_{X} ⊗ Λ_{O_{Y}}^{n} f^{∗}Ω_{X/𝔽_{q}}^{1} ][n] \ar[d] \\ LΛ^{n}L_{Y/ℤ} \ar[r]^{∼} & [Γ^{n}O_{Y} ⊗ Ω_{Y/𝔽_{q}}^{0} →^{0} ⋯ →^{0} Γ^{0}O_{Y} ⊗ Ω_{Y/𝔽_{q}}^{n}][n]}

commutes as well (see [4] I.4.3.1.3). Here the top left vertical map is induced by the derived version Lf^{∗}LΛ_{O_{X}}^{n} → LΛ_{O_{Y}}^{n} Lf^{∗} of the natural transformation f^{∗}Λ_{O_{X}}^{n} → Λ_{O_{Y}}^{n} f^{∗}, and the top right vertical map is induced by f^{∗}Λ_{O_{X}}^{i} → Λ_{O_{Y}}^{i} f^{∗} and f^{∗}Γ_{O_{X}}^{n−i} → Γ_{O_{Y}}^{n−i}f^{∗}. It follows that the upper square in the previous diagram commutes. Since the cohomology sheaves of LΛ^{n}L_{X/ℤ} are flat O_{X}- modules, we have the isomorphism

f^{∗}H^{i}(LΛ^{n}L_{X/ℤ}[−n]) →^{∼} H^{i}(Lf^{∗}LΛ^{n}L_{X/ℤ}[−n]).

We obtain the following commutative square of O_{Y} -modules

\xymatrix{f^{∗}H^{i}(LΛ^{n}L_{X/ℤ}[−n]) \ar[r]^{∼} \ar[d] & f^{∗}(Γ^{n−i}O_{X} ⊗_{O_{X}} Ω_{X/𝔽_{q}}^{i}) \ar[d] \\ H^{i}(LΛ^{n}L_{Y/ℤ}[−n]) \ar[r]^{∼} & Γ^{n−i}O_{Y} ⊗_{O_{Y}} Ω_{Y/𝔽_{q}}^{i}}

where X and Y are affine schemes in Sm^{d}/𝔽_{q}. Let f : Y → X be a map between arbitrary X, Y in Sm^{d}/𝔽_{q}. Covering Y and X by affine open subschemes (compatibly with f) we see that the previous square commutes for arbitrary X and Y. The result follows because the identification of O_{X}-modules Γ_{O_{X}}^{n−i}O_{X} ≃ O_{X} is functorial in X. Indeed, the map f^{∗}(Γ_{O_{X}}^{n−i}O_{X}) → Γ_{O_{Y}}^{n−i}O_{Y} maps γ_{n−i}(1) to itself.

Remark 3.4. An isomorphism of the form

LΛ^{n}L_{X/ℤ} ≃ [O_{X} →^{0} Ω_{X/𝔽_{q}}^{1} →^{0} ⋯ →^{0} Ω_{X/𝔽_{q}}^{n}][n]

is false in general, e.g. take n = 1 and X such that α_{X} ≠ 0 (i.e. such that X has no lifting over ℤ/p^{2}ℤ).

Remark 3.5. In order to prove the main result of [6], one may use Lemma 3.3 above instead of ([6] Lemma 2).

Proposition 3.6. Assume R(d). There is a canonical isomorphism of sheaves of O^{eh}-modules

H^{i}(LΛ^{n}L_{O^{eh}/ℤ}[−n]) ≃ Ω_{O^{eh}/𝔽_{q}}^{i≤n}

where Ω_{O^{eh}/𝔽_{q}}^{i≤n} = Ω_{O^{eh}/𝔽_{q}}^{i} for 0 ≤ i ≤ n and Ω_{O^{eh}/𝔽_{q}}^{i≤n} = 0 otherwise.

Proof. We first work in the ringed topos (Sh_{Zar}(Sm^{d}/𝔽_{q}), O). By exactness of res_{X}, Lemma 3.2(9) and Lemma 3.3, we have

res_{X}(H^{i}(LΛ^{n}L_{O/ℤ}[−n])) ≃ H^{i}(res_{X}(LΛ^{n}L_{O/ℤ}[−n]))
≃ H^{i}(LΛ^{n}L_{X/ℤ}[−n]))
≃ Ω_{X/𝔽_{q}}^{i≤n}

for any X in Sm^{d}/𝔽_{q}. Moreover, for a morphism f : Y → X in Sm^{d}/𝔽_{q}, the transition map

α_{f} : f^{∗}res_{X}(H^{i}(LΛ^{n}L_{O/ℤ}[−n])) ⟶ res_{Y} (H^{i}(LΛ^{n}L_{O/ℤ}[−n]))

may be identified with the canonical map (see Lemma 3.2)

f^{∗}H^{i}(LΛ^{n}L_{X/ℤ}[−n]) ⟶ H^{i}(LΛ^{n}L_{Y/ℤ}[−n])

which in turn may be identified with the canonical map

f^{∗}Ω_{X/𝔽_{q}}^{i≤n} ⟶ Ω_{Y/𝔽_{q}}^{i≤n}

by Lemma 3.3. In view of (11), we obtain an isomorphism

H^{i}(LΛ^{n}L_{O/ℤ}[−n]) ≃ Ω_{O/𝔽_{q}}^{i≤n}  (15)

of O-modules in the topos Sh_{Zar}(Sm^{d}/𝔽_{q}). By Lemma 3.2 (8) and by exactness of ρ^{−1}, we have

H^{i}(LΛ^{n}L_{O^{eh}/ℤ}) ≃ H^{i}(ρ^{−1}LΛ^{n}L_{O/ℤ}) ≃ ρ^{−1}H^{i}(LΛ^{n}L_{O/ℤ}).  (16)

By (16), (15) and (10), we obtain

H^{i}(LΛ^{n}L_{O^{eh}/ℤ}[−n]) ≃ ρ^{−1}H^{i}(LΛ^{n}L_{O/ℤ}[−n]) ≃ ρ^{−1}Ω_{O/𝔽_{q}}^{i≤n} ≃ Ω_{O^{eh}/𝔽_{q}}^{i≤n}.

Remark 3.7. One may think of trying to prove Proposition 3.6 more directly using the exact triangle

O^{eh}[1] → L_{O^{eh}/ℤ} → Ω_{O^{eh}/𝔽_{q}}^{1} →^{ω^{eh}} O^{eh}[2],

which is the image by ρ^{−1} of the exact triangle

O[1] → L_{O/ℤ} → Ω_{O/𝔽_{q}}^{1} →^{ω} O[2]

in the derived category of O-modules on Sh_{Zar}(Sm^{d}/𝔽_{q}). A direct computation of LΛ^{n}L_{O^{eh}/ℤ} as in (14) would not work since the extension ω is non-trivial by Remark 3.4 and Lemma 3.2.

The following corollary follows immediately from Proposition 3.6.

Corollary 3.8. If R(d) holds then LΛ_{O^{eh}}^{p}L_{O^{eh}/ℤ} is concentrated in degrees [−p, 0] and LΩ_{O^{eh}/ℤ}^{∗}/F^{n} is concentrated in degrees [0, n − 1].

Corollary 3.9. Let X be a smooth projective scheme over 𝔽_{q} of dimension d and let n ∈ ℤ be an integer. If R(d) holds then the canonical maps

RΓ(X_{Zar}, LΛ_{O_{X}}^{p} L_{X/ℤ}) → RΓ(X_{eh}, LΛ_{O^{eh}}^{p}L_{O^{eh}/ℤ})

and

RΓ(X_{Zar}, LΩ_{X/ℤ}^{∗}/F^{n}) → RΓ(X_{eh}, LΩ_{O^{eh}/ℤ}^{∗}/F^{n})

are quasi-isomorphisms.

Proof. The morphism of ringed topoi

(Sh_{eh}(Sch^{d}/𝔽_{q}), O^{eh})/y(X) ⟶^{ρ_{/X}} (Sh_{Zar}(Sm^{d}/𝔽_{q}), O)/X ⟶^{(4)} (X, O_{X}). (17)

induces a morphism of (derived Hodge to de Rham) spectral sequences from

E_{1}^{p,q} = H^{q}(X_{Zar}, LΛ^{p<n}L_{X/ℤ}) ⟹ H^{p+q}(X_{Zar}, LΩ_{X/ℤ}^{∗}/F^{n})  (18)

to

′E_{1}^{p,q} = H^{q}(X_{eh}, LΛ^{p<n}L_{O^{eh}/ℤ}) ⟹ H^{p+q}(X_{eh}, LΩ_{O^{eh}/ℤ}^{∗}/F^{n}).  (19)

Here the convergent spectral sequences (18) and (19) are obtained (using Corollary 3.8) as spectral sequences for the hypercohomology of filtered bounded below complexes. One is therefore reduced to showing that the maps

H^{q}(X_{Zar}, LΛ^{p}L_{X/ℤ}) → H^{q}(X_{eh}, LΛ^{p}L_{O^{eh}/ℤ})

are isomorphisms. By Lemma 3.3, Proposition 3.6 and Corollary 3.8, the morphism (17) induces a morphism of hypercohomology spectral sequences from

E_{2}^{i,j} = H^{i}(X_{Zar}, Ω_{X/𝔽_{q}}^{j≤p}) ⟹ H^{i+j}(X_{Zar}, LΛ^{p}L_{X/ℤ}[−p])

to

′E_{2}^{i,j} = H^{i}(X_{eh}, Ω_{O^{eh}/𝔽_{q}}^{j≤p}) ⟹ H^{i+j}(X_{eh}, LΛ^{p}L_{O^{eh}/ℤ}[−p]).

One is therefore reduced to showing that the map

H^{i}(X_{Zar}, Ω_{X/𝔽_{q}}^{j}) → H^{i}(X_{eh}, Ω_{O^{eh}/𝔽_{q}}^{j})

is an isomorphism for any i, j. Assuming R(d), this follows from ([2] Theorem 4.7) since Ω_{O^{eh}/𝔽_{q}}^{j} ≃ ρ^{−1}Ω_{O/𝔽_{q}}^{j}.

Recall from the introduction that one defines

χ_{c}^{eh}(X/𝔽_{q}, O, n) := ∑_{i≤n,j∈ℤ}(−1)^{i+j} · (n − i) · dim_{𝔽_{q}}H_{c}^{j}(X_{eh}, Ω_{O^{eh}/𝔽_{q}}^{i}).

Corollary 3.10. Let X be a separated scheme of finite type over 𝔽_{q} of dimension d and let n ∈ ℤ be an integer. If R(d) holds then the complex RΓ_{c}(X_{eh}, LΩ_{O^{eh}/ℤ}^{∗}/F^{n}) is bounded with finite cohomology groups, and we have

∏_{i∈ℤ} | H_{c}^{i}(X_{eh}, LΩ_{O^{eh}/ℤ}^{∗}/F^{n}) |^{(−1)^{i}} = q^{χ_{c}^{eh}(X/𝔽_{q},O,n)}.

Proof. We consider the convergent spectral sequences

H_{c}^{q}(X_{eh}, LΛ^{p<n}L_{O^{eh}/ℤ}) ⟹ H_{c}^{p+q}(X_{eh}, LΩ_{O^{eh}/ℤ}^{∗}/F^{n})  (20)

and

H_{c}^{i}(X_{eh}, Ω_{O^{eh}/𝔽_{q}}^{j≤p}) ⟹ H_{c}^{i+j}(X_{eh}, LΛ^{p}L_{O^{eh}/ℤ}[−p]).  (21)

In view of Corollary 3.8 and the isomorphism (see [2] Remark before Lemma 3.5)

RΓ_{c}(X_{eh}, −) ≃ RHom(ℤ_{eh}^{c}(X), −)

(20) and (21) may be obtained as spectral sequences for the hypercohomology of filtered bounded below complexes. The complex RΓ_{c}(X_{eh}, Ω_{O^{eh}/𝔽_{q}}^{j}) ≃

RΓ_{c}(X_{eh}, ρ^{−1}Ω_{O/𝔽_{q}}^{j}) is bounded with finite cohomology groups by ([2] Corollary 4.8). In view of (20) and (21), the complexes RΓ_{c}(X_{eh}, LΛ^{p}L_{O^{eh}/ℤ}/F^{n}) and RΓ_{c}(X_{eh}, LΩ_{O^{eh}/ℤ}^{∗}/F^{n}) are also bounded with finite cohomology groups. By ([6] Lemma 1), the spectral sequences (20) and (21) give isomorphisms

det_{ℤ}RΓ_{c}(X_{eh}, LΩ_{O^{eh}/ℤ}^{∗}/F^{n})  (22)
⟶^{∼} ⨂_{p<n} det_{ℤ}^{(−1)^{p}}RΓ_{c}(X_{eh}, LΛ^{p}L_{O^{eh}/ℤ})  (23)
⟶^{∼} ⨂_{p<n} det_{ℤ}RΓ_{c}(X_{eh}, LΛ^{p}L_{O^{eh}/ℤ}[−p])  (24)

⟶^{∼} ⨂_{p<n} ( ⨂_{i≤p,j} det_{ℤ}^{(−1)^{i+j}}H_{c}^{j}(X_{eh}, Ω_{O^{eh}/𝔽_{p}}^{i}))  (25)

such that the square of isomorphisms

\xymatrix{(det_{ℤ}RΓ_{c}(X_{eh}, LΩ_{O^{eh}/ℤ}^{∗}/F^{n}))_{ℚ} \ar[r] \ar[d]^{γ} & (⨂_{p<n} ⨂_{i≤p,j} det_{ℤ}^{(−1)^{i+j}} H_{c}^{j}(X_{eh}, Ω_{O^{eh}/𝔽_{q}}^{i}))_{ℚ} \ar[d]^{γ′} \\ ℚ \ar[r]^{Id} & ℚ}

commutes, where the top horizontal map is induced by (25), and the vertical isomorphisms are the canonical trivializations (see Section 2.3). The result follows:

ℤ · (∏_{i∈ℤ} | H_{c}^{i}(X_{eh}, LΩ_{O^{eh}/ℤ}^{∗}/F^{n}) |^{(−1)^{i}})^{−1}

= γ (det_{ℤ}RΓ_{c}(X_{eh}, LΩ_{O^{eh}/ℤ}^{∗}/F^{n}))

= γ′ (⨂_{p<n} ⨂_{i≤p,j} det_{ℤ}^{(−1)^{i+j}}H_{c}^{j}(X_{eh}, Ω_{O^{eh}/𝔽_{q}}^{i}))

= ℤ · q^{−χ_{c}^{eh}(X/𝔽_{q},O,n)}.

Recall from Section 2.3 the definitions of Δ(X/ℤ, n) and λ_{X}.

Corollary 3.11. Let X be a separated scheme of finite type over 𝔽_{q} of dimension d and let n ∈ ℤ be an integer. Assume that for any smooth projective variety Y of dimension ≤ d the Weil-étale cohomology groups H_{W}^{i} (Y, ℤ(n)) are finitely generated for all i. If R(d) holds, then one has

Δ(X/ℤ, n) = ℤ · λ_{X} (ζ^{∗}(X, n)^{−1}) .

Proof. All the schemes we consider in this proof are in Sch^{d}/𝔽_{q}. For an open-

closed decomposition (U →^{j} X ←^{i} Z), we have exact triangles

RΓ_{c}(U_{eh}, LΩ_{O^{eh}/ℤ}^{∗}/F^{n}) → RΓ_{c}(X_{eh}, LΩ_{O^{eh}/ℤ}^{∗}/F^{n}) → RΓ_{c}(Z_{eh}, LΩ_{O^{eh}/ℤ}^{∗}/F^{n})

and

RΓ_{W,c}(U, ℤ(n)) → RΓ_{W,c}(X, ℤ(n)) → RΓ_{W,c}(Z, ℤ(n)).  (26)

Moreover, the triangle (26) is compatible (in the obvious sense) with ∪θ. This gives an isomorphism

Δ(X/ℤ, n) ≃ Δ(U/ℤ, n) ⊗_{ℤ} Δ(Z/ℤ, n)  (27)

such that the square of isomorphisms

\xymatrix{ℝ \ar[r]^{Id} \ar[d]^{λ_{X}} & ℝ \ar[d]^{λ_{U} ⊗λ_{Z}} \\ Δ(X/ℤ, n)_{ℝ} \ar[r]^{(27)_{ℝ}} & Δ(U/ℤ, n)_{ℝ} ⊗_{ℝ} Δ(Z/ℤ, n)_{ℝ}}

commutes. Similarly, one has

ζ^{∗}(X, n) = ζ^{∗}(U, n) · ζ^{∗}(Z, n).

It follows that if the result is true for two out of the three schemes (X, U, Z) then it is true for the third. Moreover, the result is true for X smooth projective by [6], Corollary 3.9 and ([2] Theorem 4.3). It follows for arbitrary X ∈ Sch^{d}/𝔽_{q} by ([2] Lemma 2.7).

Acknowlegments. I am grateful to Denis-Charles Cisinski and Matthias Flach for interesting comments. The author was supported by ANR-12-JS01-0007 and ANR-15-CE40-0002.

References

[1] Berthelot, P.; Ogus, A.: Notes on crystalline cohomology. Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1978.

[2] Geisser, T.: Arithmetic cohomology over finite fields and special valyues of ζ-functions. Duke Math. J. 133 (1) (2006), 27–57.

[3] Artin, M.; Grothendieck, A.; Verdier, J.L.: Theorie des Topos et Cohomologie Etale des Schemas (SGA 4), Springer, 1972, Lecture Notes in Math 269, 270, 271.

[4] Illusie, L.: Complexe cotangent et déformations. I. Lecture Notes in Mathematics, Vol. 239. Springer-Verlag, Berlin-New York, 1971.

[5] Illusie, L.: Complexe cotangent et déformations. II. Lecture Notes in Mathematics, Vol. 283. Springer-Verlag, Berlin-New York, 1972.

[6] Morin, B.: Milne’s correcting factor and derived de Rham cohomology. Documenta Math. 21 (2016) 39–48.

[7] Suslin, A.; Voevodsky, V.: Bloch-Kato conjecture and motivic cohomology with finite coefficients. The arithmetic and geometry of algebraic cycles (Banff, AB, 1998), 117–189, NATO Sci. Ser. C Math. Phys. Sci., 548, Kluwer Acad. Publ., Dordrecht, 2000.

Baptiste Morin CNRS, IMB Université de Bordeaux 351, cours de la Libération F 33405 Talence cedex France Baptiste.Morin@math.ubordeaux.fr 1322

Documenta Mathematica 22 (2017)