Milne’s Correcting Factor

and Derived De Rham Cohomology

Baptiste Morin

Received: June 15, 2015 Revised: November 13, 2015

Communicated by Stephen Lichtenbaum

Abstract. Milne’s correcting factor is a numerical invariant playing an important role in formulas for special values of zeta functions of varieties over finite fields. We show that Milne’s factor is simply the Euler characteristic of the derived de Rham complex (relative to ℤ) modulo the Hodge filtration.

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

A result of Milne ([9] Theorem 0.1) describes the special values of the zeta function of a smooth projective variety X over a finite field satisfying the Tate conjecture. A very natural reformulation of this result was given by Lichtenbaum and Geisser (see [2], [7], [8] and [10]) using Weil-étale cohomology of motivic complexes. They conjecture that

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

and show that (1) holds whenever the groups H_{W}^{i} (X, ℤ(n)) are finitely generated. Here H_{W}^{∗} (X, ℤ(n)) denotes Weil-étale motivic cohomology, e ∈ H^{1}(W_{𝔽_{q}}, ℤ) is a fundamental class and χ(H_{W}^{∗} (X, ℤ(n)), e) is the Euler characteristic of the complex

(2)  ⋯ ⟶^{∪e} H_{W}^{i} (X, ℤ(n)) ⟶^{∪e} H_{W}^{i+1}(X, ℤ(n)) ⟶^{∪e} ⋯

More precisely, the cohomology groups of the complex (2) are finite and χ(H_{W}^{∗} (X, ℤ(n)), ∪e) is the alternating product of their orders. Finally, Milne’s correcting factor q^{χ(X/𝔽_{q},O,n)} was defined in [9] by the formula

χ(X/𝔽_{q}, O_{X}, n) = ∑_{i≤n,j}(−1)^{i+j} · (n − i) · dim_{𝔽_{q}}H^{j}(X, Ω_{X/𝔽_{q}}^{i}).

The author was supported by ANR-12-BS01-0002 and ANR-12-JS01-0007.

It is possible to generalize (1) in order to give a conjectural description of special values of zeta functions of all separated schemes of finite type over 𝔽_{q} (see [3] Conjecture 1.4), and even of all motivic complexes over 𝔽_{q} (see [11] Conjecture 1.2). The statement of those more general conjectures is in any case very similar to formula (1). The present note is motivated by the hope for a further generalization, which would apply to zeta functions of all algebraic schemes over Spec(ℤ). As briefly explained below, the special-value conjecture for (flat) schemes over Spec(ℤ) must take a rather different form than formula (1). Going back to the special case of smooth projective varieties over finite fields, this leads to a slightly different restatement of formula (1). Let X be a regular scheme proper over Spec(ℤ). The "fundamental line"

Δ(X /ℤ, n) := det_{ℤ}RΓ_{W,c}(X , ℤ(n)) ⊗_{ℤ} det_{ℤ}RΓ_{dR}(X /ℤ)/F^{n}

should be a well defined invertible ℤ-module endowed with a canonical trivialization

ℝ ⟶^{∼} Δ(X /ℤ, n) ⊗_{ℤ} ℝ.

involving a fundamental class θ ∈ H^{1}(ℝ, ℝ) = ”H^{1}(W_{𝔽_{1}}, ℝ)” analogous to e ∈ H^{1}(W_{𝔽_{q}}, ℤ). Here RΓ_{W,c}(X , ℤ(n)) denotes Weil-étale cohomology with compact support. However, there is no natural trivialization ℝ →^{∼} det_{ℤ}RΓ_{W,c}(X , ℤ(n)) ⊗_{ℤ} ℝ. Consequently, it is not possible to define an Euler characteristic generalizing χ(H_{W}^{∗} (X, ℤ(n)), ∪e), neither to define a correcting factor generalizing Milne’s correcting factor: one is forced to consider the fundamental line as a whole. Let us go back to the case of smooth projective varieties X/𝔽_{q}, which we now see as schemes over ℤ. Accordingly, we replace Z(X, t) with ζ(X, s) = Z(X, q^{−s}), the fundamental class e with θ and the cotangent sheaf Ω_{X/𝔽_{q}}^{1} ≃ L_{X/𝔽_{q}} with the cotangent complex L_{X/ℤ}. Assuming that H_{W}^{i} (X, ℤ(n)) is finitely generated for all i, the fundamental line

(3)  Δ(X/ℤ, n) := det_{ℤ}RΓ_{W} (X, ℤ(n)) ⊗_{ℤ} det_{ℤ}RΓ(X, LΩ_{X/ℤ}^{∗}/F^{n})

is well defined and cup-product with θ gives a trivialization

λ : ℝ ⟶^{∼} Δ(X/ℤ, n) ⊗_{ℤ} ℝ.

Here LΩ_{X/ℤ}^{∗}/F^{n} is Illusie’s derived de Rham complex modulo the Hodge filtration (see [6] VIII.2.1). The aim of this note is to show that the Euler characteristic of RΓ(X, LΩ_{X/ℤ}^{∗}/F^{n}) equals q^{χ(X/𝔽_{q},O_{X},n)}, hence that Milne’s correcting factor is naturally part of the fundamental line. We denote by ζ^{∗}(X, n) the leading coefficient in the Taylor development of ζ(X, s) near s = n.

Theorem. Let X be a smooth proper scheme over 𝔽_{q} and let n ∈ ℤ be an integer. Then we have

∏_{i∈ℤ} | H^{i}(X, LΩ_{X/ℤ}^{∗}/F^{n}) |^{(−1)^{i}} = q^{χ(X/𝔽_{q},O_{X},n)}.

Assume moreover that X is projective and that the groups H_{W}^{i} (X, ℤ(n)) are finitely generated for all i. Then one has

Δ(X/ℤ, n) = ℤ · λ (log(q)^{ρ_{n}} · χ(H_{W}^{∗} (X, ℤ(n)), ∪e)^{−1} · q^{−χ(X/𝔽_{q},O_{X},n)})
= ℤ · λ (ζ^{∗}(X, n)^{−1})

where ρ_{n} := −ord_{s=n}ζ(X, s) is the order of the pole of ζ(X, s) at s = n.

Before giving the proof, we need to fix some notations. 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). Let A be a finite abelian group, which we see as a complex concentrated in degree 0. Then the canonical ℚtrivialization det_{ℤ}(A) ⊗_{ℤ} ℚ ≃ ℚ identifies det_{ℤ}(A) with |A|^{−1} · ℤ ⊂ ℚ, where |A| denotes the order of A. Given a ring R and an R-module M, we denote by Γ_{R}(M) the universal divided power R-algebra of M, and by Γ_{R}^{i}(M) its submodule of homogeneous elements of degree i. We refer to ([1] Appendix A) for the definition of Γ_{R}(M) and its main properties. There is a canonical map γ^{i} : M → Γ_{R}^{i}(M), such that composition with γ^{i} induces a bijection Hom_{R}(Γ_{R}^{i}(M), N) →^{∼} P^{i}(M, N), where P^{i}(M, N) is the set of "homogeneous polynomial functions of degree i". The functor Γ_{R}^{i} sends free modules of finite type to free modules of finite type. Moreover Γ_{R}^{i} commutes with filtered colimits, hence sends flat modules to flat modules. If M is free of rank one, then so is Γ_{R}^{i}(M). If (T, R) is a ringed topos and M an R-module, then Γ_{R}(M) is the sheafification of U ↦ Γ_{R(U)}(M(U)). We also denote by Λ_{R}^{i} the (non-additive) exterior power functor and by LΛ_{R}^{i} its left derived functor (see [5] I.4.2). We often omit the subscript R and simply write Γ^{i}M, Λ^{i}M and LΛ^{i}M. Let X be a scheme. The notation RΓ(X, −) refers to hypercohomology with respect to the Zariski topology.

Proof. Since Milne’s correcting factor is insensitive to restriction of scalars (i.e. q^{χ(X/𝔽_{q},O_{X},n)} = p^{χ(X/𝔽_{p},O_{X},n)}), we may consider X over 𝔽_{p}. We need the following

Lemma 1. Let E_{∗}^{∗,∗} = (E_{r}^{p,q}, d_{r}^{p,q})_{r}^{p,q} be a cohomological spectral sequence of abelian groups with abutment H^{∗}. Assume that there exists an index r_{0} such that E_{r_{0}}^{p,q} is finite for all (p, q) ∈ ℤ^{2} and E_{r_{0}}^{p,q} = 0 for all but finitely many (p, q). Then we have a canonical isomorphism

ι : ⨂_{p,q} det_{ℤ}^{(−1)^{p+q}}E_{r_{0}}^{p,q} ⟶^{∼} ⨂_{n} det_{ℤ}^{(−1)^{n}}H^{n}

such that the square of isomorphisms

\xymatrix{(⨂_{p,q} det_{ℤ}^{(−1)^{p+q}}E_{r_{0}}^{p,q}) ⊗ ℚ \ar[r]^{ι⊗ℚ} \ar[d] & (⨂_{n} det_{ℤ}^{(−1)^{n}}H^{n}) ⊗ ℚ \ar[d] \\ ℚ \ar[r]^{Id} & ℚ}

commutes, where the vertical maps are the canonical ℚ-trivializations.

Proof. For any t ≥ r_{0}, consider the bounded cochain complex C_{t}^{∗} of finite abelian groups:

⋯ ⟶ ⨁_{p+q=n−1} E_{t}^{p,q} ⟶ ⨁_{p+q=n} E_{t}^{p,q} ⟶^{⊕d_{t}^{p,q}} ⨁_{p+q=n+1} E_{t}^{p,q} ⟶ ⋯

The fact that the cohomology of C_{t}^{∗} is given by H^{n}(C_{t}^{∗}) = ⨁_{p+q=n} E_{t+1}^{p,q} gives an isomorphism

⨂_{p,q} det_{ℤ}^{(−1)^{p+q}}E_{t}^{p,q} ⟶^{∼} ⨂_{p,q} det_{ℤ}^{(−1)^{p+q}}E_{t+1}^{p,q}

compatible with the canonical ℚ-trivializations. By assumption, there exists an index r_{1} ≥ r_{0} such that the spectral sequence degenerates at the r_{1}-page, i.e. E_{r_{1}}^{∗,∗} = E_{∞}^{∗,∗}. The induced filtration on each H^{n} is such that gr^{p}H^{n} = E_{∞}^{p,n−p}. We obtain isomorphisms

⨂_{p,q} det_{ℤ}^{(−1)^{p+q}}E_{r_{0}}^{p,q} →^{∼} ⨂_{p,q} det_{ℤ}^{(−1)^{p+q}}E_{∞}^{p,q} →^{∼}

→^{∼} ⨂_{n} ⨂_{p} det_{ℤ}^{(−1)^{n}}E_{∞}^{p,n−p} →^{∼} ⨂_{n} det_{ℤ}^{(−1)^{n}}H^{n}

compatible with the canonical ℚ-trivializations. □

Consider the Hodge filtration F^{∗} on the derived de Rham complex LΩ_{X/ℤ}^{∗}. By ([6] VIII.2.1.1.5) we have

gr(LΩ_{X/ℤ}^{∗}) ≃ ⨁_{p≥0} LΛ^{p}L_{X/ℤ}[−p].

This gives a (convergent) spectral sequence

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

where LΛ^{p<n}L_{X/ℤ} := LΛ^{p}L_{X/ℤ} for p < n and LΛ^{p<n}L_{X/ℤ} := 0 otherwise. The scheme X is proper and LΛ^{p}L_{X/ℤ} is isomorphic, in the derived category D(O_{X}) of O_{X}-modules, to a bounded complex of coherent sheaves (see (6) below). It follows that E_{1}^{p,q} is a finite dimensional 𝔽_{p}-vector space for all (p, q) vanishing for almost all (p, q). By Lemma 1, this yields isomorphisms

det_{ℤ}RΓ(X, LΩ_{X/ℤ}^{∗}/F^{n}) ⟶^{∼} ⨂_{i} det_{ℤ}^{(−1)^{i}}H^{i}(X, LΩ_{X/ℤ}^{∗}/F^{n})
⟶^{∼} ⨂_{p<n,q} det_{ℤ}^{(−1)^{p+q}}H^{q}(X, LΛ^{p}L_{X/ℤ})
⟶^{∼} ⨂_{p<n} det_{ℤ}^{(−1)^{p}}RΓ(X, LΛ^{p}L_{X/ℤ})

which are compatible with the canonical ℚ-trivializations. The transitivity triangle (see [5] II.2.1) for the composite map X →^{f} Spec(𝔽_{p}) → Spec(ℤ) reads as follows (using [5] III.3.1.2 and [5] III.3.2.4(iii)):

(4)  Lf^{∗}(pℤ/p^{2}ℤ)[1] → L_{X/ℤ} → Ω_{X/𝔽_{p}}^{1}[0] →^{ω} Lf^{∗}(pℤ/p^{2}ℤ)[2].

We set L := Lf^{∗}(pℤ/p^{2}ℤ), a trivial invertible O_{X}-module. By ([5] Théorème III.2.1.7), the class

ω ∈ Ext_{O_{X}}^{2} (Ω_{X/𝔽_{p}}^{1}, L) ≃ H^{2}(X, T_{X/𝔽_{p}})

is the obstruction to the existence of a lifting of X over ℤ/p^{2}ℤ. If such a lifting does exist then we have ω = 0, in which case the following lemma is superfluous. For an object C of D(O_{X}) with bounded cohomology, we set

gr_{τ} C := ⨁_{i∈ℤ} H^{i}(C)[−i].

Lemma 2. We have an isomorphism

det_{ℤ}RΓ(X, LΛ^{p}L_{X/ℤ}) ≃ det_{ℤ}RΓ(X, LΛ^{p}(gr_{τ} L_{X/ℤ}))

compatible with the canonical ℚ-trivializations.

Proof. The map X → Spec(ℤ) is a local complete intersection, hence the complex L_{X/ℤ} has perfect amplitude ⊂ [−1, 0] (see [5] III.3.2.6). In other words, L_{X/ℤ} is locally isomorphic in D(O_{X}) to a complex of free modules of finite type concentrated in degrees −1 and 0. By ([4] 2.2.7.1) and ([4] 2.2.8), L_{X/ℤ} is globally isomorphic to such a complex, i.e. there exists an isomorphism L_{X/ℤ} ≃ [M → N] in D(O_{X}), where M and N are finitely generated locally free O_{X}-modules put in degrees −1 and 0 respectively. Consider the exact sequences

(5)  0 → L → M → F → 0 and 0 → F → N → Ω → 0

where L := Lf^{∗}(pℤ/p^{2}ℤ) and Ω := Ω_{X/𝔽_{p}}^{1} are finitely generated and locally free. It follows that F is also finitely generated and locally free. One has an isomorphism in D(O_{X})

(6) LΛ^{p}L_{X/ℤ} ≃ [Γ^{p}M → Γ^{p−1}M ⊗ N → ⋯ → M ⊗ Λ^{p−1}N → Λ^{p}N]

where the right hand side sits in degrees [−p, 0] (see [6] VIII.2.1.2 and [5] I.4.3.2.1). Moreover, in view of (4) we may choose an isomorphism

gr_{τ} L_{X/ℤ} ≃ [L →^{0} Ω]

in D(O_{X}), the right hand side being concentrated in degrees [−1, 0]. Hence the complex LΛ^{p}(gr_{τ} L_{X/ℤ}) ∈ D(O_{X}) is represented by a complex of the form

(7) LΛ^{p}(gr_{τ} L_{X/ℤ}) ≃ LΛ^{p}([L → Ω]) ≃
≃ [Γ^{p}L → Γ^{p−1}L ⊗ Ω → ⋯ → L ⊗ Λ^{p−1}Ω → Λ^{p}Ω]

where the right hand side sits in degrees [−p, 0]. Lemma 1 and (6) give an isomorphism

(8) det_{ℤ}RΓ(X, LΛ^{p}L_{X/ℤ}) ≃ ⨂_{0≤q≤p} det_{ℤ}^{(−1)^{p−q}}RΓ(X, Γ^{p−q}M ⊗ Λ^{q}N)

compatible with the ℚ-trivializations. The second exact sequence in (5) endows Λ^{q}N with a finite decreasing filtration Fil^{∗} such that gr_{Fil}^{i}(Λ^{q}N) = Λ^{i}F ⊗ Λ^{q−i}Ω. Since Γ^{p−q}M is flat, Fil^{∗} induces a similar filtration on Γ^{p−q}M ⊗ Λ^{q}N such that

gr_{Fil}^{i}(Γ^{p−q}M ⊗ Λ^{q}N) = Γ^{p−q}M ⊗ Λ^{i}F ⊗ Λ^{q−i}Ω.

This filtration induces an isomorphism

(9) det_{ℤ}RΓ(X, Γ^{p−q}M ⊗ Λ^{q}N) ≃ ⨂_{0≤i≤q} det_{ℤ}RΓ(X, Γ^{p−q}M ⊗ Λ^{i}F ⊗ Λ^{q−i}Ω)

compatible with the ℚ-trivializations. Lemma 1 and (7) give an isomorphism

(10) det_{ℤ}RΓ(X, LΛ^{p}(gr_{τ} L_{X/ℤ})) ≃ ⨂_{0≤i≤p} det_{ℤ}^{(−1)^{p−i}}RΓ(X, Γ^{p−i}L ⊗ Λ^{i}Ω)

compatible with the ℚ-trivializations. Moreover, we have an isomorphism (see [5] I.4.3.1.7)

Γ^{p−i}L ≃ [Γ^{p−i}M → Γ^{p−i−1}M ⊗ F → ⋯ → M ⊗ Λ^{p−i−1}F → Λ^{p−i}F]

where the right hand side sits in degrees [0, p − i]. Since Λ^{i}Ω is flat, we have an isomorphism between Γ^{p−i}L ⊗ Λ^{i}Ω and

[Γ^{p−i}M ⊗ Λ^{i}Ω → Γ^{p−i−1}M ⊗ F ⊗ Λ^{i}Ω → ⋯ ⋯ → M ⊗ Λ^{p−i−1}F ⊗ Λ^{i}Ω → Λ^{p−i}F ⊗ Λ^{i}Ω].

By Lemma 1, we have (11)

det_{ℤ}RΓ(X, Γ^{p−i}L ⊗ Λ^{i}Ω) ≃ ⨂_{0≤j≤p−i} det_{ℤ}^{(−1)^{j}}RΓ(X, Γ^{p−i−j}M ⊗ Λ^{j}F ⊗ Λ^{i}Ω).

Putting (10), (11), (9) and (8) together, we obtain isomorphisms

det_{ℤ}RΓ(X, LΛ^{p}(gr_{τ} L_{X/ℤ})) ≃
≃ ⨂_{0≤i≤p} det_{ℤ}^{(−1)^{p−i}}RΓ(X, Γ^{p−i}L ⊗ Λ^{i}Ω)

≃ ⨂_{0≤i≤p} ( ⨂_{0≤j≤p−i} det_{ℤ}^{(−1)^{p−i−j}}RΓ(X, Γ^{p−i−j}M ⊗ Λ^{j}F ⊗ Λ^{i}Ω))

= ⨂_{0≤q≤p} ( ⨂_{0≤i,j ; i+j=q} det_{ℤ}^{(−1)^{p−q}}RΓ(X, Γ^{p−q}M ⊗ Λ^{j}F ⊗ Λ^{i}Ω))

≃ ⨂_{0≤q≤p} det_{ℤ}^{(−1)^{p−q}}RΓ(X, Γ^{p−q}M ⊗ Λ^{q}N)
≃ det_{ℤ}RΓ(X, LΛ^{p}L_{X/ℤ})

compatible with the canonical ℚ-trivializations. □

Recall from (7) that the complex LΛ^{p}(gr_{τ} L_{X/ℤ}) is isomorphic in D(O_{X}) to a complex of the form

0 → Γ^{p}L → Γ^{p−1}L ⊗ Ω_{X/𝔽_{p}}^{1} → ⋯ → Γ^{1}L ⊗ Ω_{X/𝔽_{p}}^{p−1} → Ω_{X/𝔽_{p}}^{p} → 0

put in degrees [−p, 0]. An isomorphism of 𝔽_{p}-vector spaces 𝔽_{p} ≃ pℤ/p^{2}ℤ induces an identification O_{X} ≃ L, and more generally O_{X} ≃ Γ^{i}L for any i ≥ 0. Hence (LΛ^{p}(gr_{τ} L_{X/ℤ}))[−p] ∈ D(O_{X}) is represented by a complex of the form

(12)  0 → O_{X} → Ω_{X/𝔽_{p}}^{1} → ⋯ → Ω_{X/𝔽_{p}}^{p} → 0

sitting in degrees [0, p]. We obtain a spectral sequence

E_{1}^{i,j} = H^{j}(X, Ω_{X/𝔽_{p}}^{i≤p}) ⟹ H^{i+j}(X, (LΛ^{p}(gr_{τ} L_{X/ℤ}))[−p])

where Ω^{i≤p} := Ω^{i} for i ≤ p and Ω^{i≤p} := 0 for i > p. By Lemma 1 again, we get an identification

⨂_{i≤p,j} det_{ℤ}^{(−1)^{i+j}}H^{j}(X, Ω_{X/𝔽_{p}}^{i}) ⟶^{∼} det_{ℤ}RΓ(X, (LΛ^{p}(gr_{τ} L_{X/ℤ}))[−p])
⟶^{∼} det_{ℤ}^{(−1)^{p}}RΓ(X, LΛ^{p}(gr_{τ} L_{X/ℤ})).

In summary, we have the following isomorphisms

(13) det_{ℤ}RΓ(X, LΩ_{X/ℤ}^{∗}/F^{n}) ⟶^{∼} ⨂_{p<n} det_{ℤ}^{(−1)^{p}}RΓ(X, LΛ^{p}L_{X/ℤ})
(14)  ⟶^{∼} ⨂_{p<n} det_{ℤ}^{(−1)^{p}}RΓ(X, LΛ^{p}(gr_{τ} L_{X/ℤ}))

(15)  ⟶^{∼} ⨂_{p<n} ( ⨂_{i≤p,j} det_{ℤ}^{(−1)^{i+j}}H^{j}(X, Ω_{X/𝔽_{p}}^{i}))

such that the square

\xymatrix{(det_{ℤ}RΓ(X, LΩ_{X/ℤ}^{∗}/F^{n}))⊗ ℚ \ar[r] \ar[d]^{γ} & (⨂_{p<n} ⨂_{i≤p,j} det_{ℤ}^{(−1)^{i+j}}H^{j}(X, Ω_{X/𝔽_{p}}^{i}))⊗ ℚ \ar[d]^{γ′} \\ ℚ \ar[r]^{Id} & ℚ}

commutes, where the top horizontal map is induced by (15), and the vertical isomorphisms are the canonical trivializations. The first assertion of the theorem follows:

ℤ · (∏_{i∈ℤ} | H^{i}(X, LΩ_{X/ℤ}^{∗}/F^{n}) |^{(−1)^{i}})^{−1} =

= γ (det_{ℤ}RΓ(X, LΩ_{X/ℤ}^{∗}/F^{n}))

= γ′ (⨂_{p<n} ⨂_{i≤p,j} det_{ℤ}^{(−1)^{i+j}}H^{j}(X, Ω_{X/𝔽_{p}}^{i}))
= ℤ · p^{−χ(X/𝔽_{p},O_{X},n)}.

We now explain why the second assertion of the theorem is a restatement of ([2] Theorem 1.3). We assume that H_{W}^{i} (X, ℤ(n)) is finitely generated for all i ∈ ℤ (X and n being fixed). Recall from [2] that this assumption implies the following: H_{W}^{i} (X, ℤ(n)) is in fact finite for i ≠ 2n, 2n + 1, the complex (2) has finite cohomology groups and one has

ρ_{n} := −ord_{s=n}ζ(X, s) = rank_{ℤ}H_{W}^{2n}(X, ℤ(n)).

In particular the complex

(16)  ⋯ ⟶^{∪e} H_{W}^{i} (X, ℤ(n)) ⊗ ℚ ⟶^{∪e} H_{W}^{i+1}(X, ℤ(n)) ⊗ ℚ ⟶^{∪e} ⋯

is acyclic. This gives a trivialization

β : ℚ ⟶^{∼} ⨂_{i} det_{ℚ}^{(−1)^{i}} (H_{W}^{i} (X, ℤ(n)) ⊗ ℚ) ⟶^{∼}

⟶^{∼} (⨂_{i} det_{ℤ}^{(−1)^{i}}H_{W}^{i} (X, ℤ(n))) ⊗ ℚ

such that

ℤ · β (χ(H_{W}^{∗} (X, ℤ(n)), ∪e)^{−1}) = ⨂_{i} det_{ℤ}^{(−1)^{i}}H_{W}^{i} (X, ℤ(n)).

The class e ∈ H^{1}(W_{𝔽_{q}}, ℤ) = Hom(W_{𝔽_{q}}, ℤ) maps the Frobenius Frob ∈ W_{𝔽_{q}} to 1 ∈ ℤ. We define the map

W_{𝔽_{q}} = ℤ · Frob ⟶ ℝ =: W_{𝔽_{1}}

as the map sending Frob to log(q), while θ ∈ H^{1}(W_{𝔽_{1}}, ℝ) = Hom(ℝ, ℝ) is the identity map. It follows that the acyclic complex

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

induces a trivialization

α : ℝ ⟶^{∼} ⨂_{i} det_{ℝ}^{(−1)^{i}} (H_{W}^{i} (X, ℤ(n)) ⊗ ℝ) ⟶^{∼}(⨂_{i} det_{ℤ}^{(−1)^{i}}H_{W}^{i} (X, ℤ(n)))⊗ℝ

such that

ℤ · α (χ(H_{W}^{∗} (X, ℤ(n)), ∪e)^{−1} · log(q)^{ρ_{n}}) = ⨂_{i} det_{ℤ}^{(−1)^{i}}H_{W}^{i} (X, ℤ(n)).

The trivialization λ is the product of α with the canonical trivialization

ℝ ⟶^{∼} det_{ℤ}RΓ(X, LΩ_{X/ℤ}^{∗}/F^{n}) ⊗_{ℤ} ℝ.

Hence we have

ℤ · λ (log(q)^{ρ_{n}} · χ(H_{W}^{∗} (X, ℤ(n)), ∪e)^{−1} · q^{−χ(X,O_{X},n)}) = Δ(X/ℤ, n).

Moreover, formula (1) gives

ζ^{∗}(X, s) = ±log(q)^{−ρ_{n}} · χ(H_{W}^{∗} (X, ℤ(n)), ∪e) · q^{χ(X,O_{X},n)}

hence the result follows from ([2] Theorem 1.3). □

Acknowledgments. I would like to thank Matthias Flach, Stephen Lichtenbaum and Niranjan Ramachandran for their interest and comments.

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Baptiste Morin CNRS, IMB Université de Bordeaux 351, cours de la Libération F 33405 Talence cedex, France Baptiste.Morin@math.ubordeaux1.fr