Duality for ℤ-Constructible Sheaves

on Curves over Finite Fields

Thomas Geisser^{1}

Received: January 27, 2012 Revised: October 10, 2012

Communicated by Stephen Lichtenbaum

Abstract. We prove a duality theorem for Weil-etale cohomology of ℤ-constructible sheaves on curves over finite fields.

2010 Mathematics Subject Classification: 14F20, 14F42, 11G20 Keywords and Phrases: Finite fields, curves, duality, ℤ-construcible sheaves.

1 Introduction

Let X be a (possibly singular) curve over a finite field and F be a ℤconstructible sheaf on X. In [3], Deninger defined a dualizing complex 𝔾, and in [2] he proves a duality between the groups H_{c}^{i}(X_{et}, F) and Ext_{et}^{3−i}(F, 𝔾). The cohomology groups are finitely generated for i = 0, 1, of cofinite type for i = 2, 3, and conversely for the extension groups). If X is smooth, Lichtenbaum [9] proved a duality of finitely generated groups between Weil-etale cohomology groups with compact support H_{c}^{i}(X_{W} , ℤ) and Weil-etale cohomology groups H^{i}(X_{W} , 𝔾_{m}). In this paper, we generalize and unify these results. We define, for any curve X over a finite field and any ℤ-constructible sheaf F, Weil-etale Borel-Moore homology groups H_{i}^{c}(X_{ar}, F), and construct a pairing of finitely generated abelian groups between Weil-etale cohomology with compact supports H_{c}^{i}(X_{W} , F) and H_{i}^{c}(X_{ar}, F). More precisely, for 0 ≤ i ≤ 2, there are pairings of finitely generated free groups

H_{i}^{c}(X_{ar}, F)/tor × H_{c}^{i}(X_{W} , F)/tor → ℤ;

1Supported by JSPS Grant-in-aid (B) #30571963

Documenta Mathematica 17 (2012) 989–1002

and for 0 ≤ i ≤ 3 there are pairings of finite groups

H_{i−1}^{c}(X_{ar}, F)_{tor} × H_{c}^{i}(X_{W} , F)_{tor} → ℚ/ℤ.

All other cohomology and homology groups vanish. If X is smooth, then the groups H_{i}^{c}(X_{ar}, ℤ) are isomorphic to Weil-etale cohomology with 𝔾_{m}-coefficients, and we recover Lichtenbaum’s result. In fact, the author’s original motivation was to understand the asymmetry between the coefficients ℤ = ℤ(0) and 𝔾_{m} = ℤ(1)[1] in Lichtenbaum’s result: the groups H^{i}(X_{W} , 𝔾_{m}) are finitely generated, but in general H^{i}(X_{W} , ℤ) is not. For this reason, one also cannot expect duality results for positive weights. The relationship to Deninger’s result is given by long exact sequences relating etale and Weil-etale cohomology and extension groups [4], see below. The strategy of proof of our duality result is to first show finite generation of the groups involved, and then reduce to the case of torsion coefficients, which was treated in [6] for arbitrary schemes over finite fields. We note that for higher dimensional schemes, Weil-etale cohomology groups are not well-behaved. Instead one has to use the eh-topology, see the discussion and example in [5]. But in this case, the standard methods to construct the pairing fail. Moreover, a duality result as the above would for F = ℤ imply that CH_{0}(X, i)_{ℚ} = 0 for i > 0 and X smooth and projective. This is a special case of Parshin’s conjecture that K_{i}(X)_{ℚ} = 0 for i > 0 and X smooth and projective over a finite field (Parshin’s conjecture is known if X is a curve). We thank the referee for his careful reading and helpful comments.

2 Arithmetic homology and cohomology

2.1 The dualizing complex

We recall some properties of Bloch’s higher Chow complex [1]. For a scheme X essentially of finite type over a fixed field k, z_{0}(X, i) is defined as the free abelian group generated by closed integral subschemes of dimension i on X ×_{k} Δ^{i} which meet all faces properly. If z_{0}(X, ∗) is the (homological) complex of abelian groups obtained by taking the alternating sum of intersection with face maps as differentials, then varying X we obtain a complex of sheaves z_{0}(−, ∗) for the etale topology. By definition, the higher Chow group CH_{0}(X, i) is the homology in degree i of z_{0}(X, ∗). We define ℤ_{X}^{c} = z_{0}(−, ∗) to be the (cohomological) complex which is the etale sheaf z_{0}(−, i) in degree −i, and omit X if there is no ambiguity. For a quasi-finite, flat map f : X → Y, there is a pullback f^{∗}ℤ_{Y}^{c} → ℤ_{X}^{c}, and for a proper map f : X → Y there is a push-forward f_{∗}ℤ_{X}^{c} → ℤ_{Y}^{c}. For a closed embedding i : Z → X over k, the isomorphism ℤ_{Z}^{c} ≅ Ri^{!}ℤ_{X}^{c} on the Zariski-site is called purity or localization property. It implies an isomorphism [1] between cohomology and hypercohomology

CH_{0}(X, i) ≅ H_{i}^{c}(X_{Zar}, ℤ^{c}) := H^{−i}(X_{Zar}, ℤ^{c}).  (1)

The following result is proved in [6]:

Duality for ℤ-Constructible Sheaves … 991

Theorem 2.1 a) Over an algebraically closed base field, the complex ℤ_{X}^{c} has etale hypercohomological descent, i.e. if ℤ^{c} → I^{·} is an injective resolution of etale sheaves, then ℤ^{c}(U) → I^{·}(U) is a quasi-isomorphism for every U → X etale.

b) If f : X → Y is a proper map over a perfect field k, then there is a functorial push-forward f_{∗} : Rf_{∗}ℤ_{X}^{c} → ℤ_{Y}^{c} in the derived category of etale sheaves.

c) If i : Z → X is a closed embedding over a perfect field k, then we have a quasi-isomorphism ℤ_{Z}^{c} ⟶^{∼} Ri^{!}ℤ_{X}^{c} of etale sheaves on Z.

If f : X → k is proper over a perfect field, then the trace map agrees on the stalk Spec k̄ with the map sending a complex to its highest cohomology group, composed with the degree map,

tr : Rf_{∗}ℤ_{X}^{c}(k̄) ←^{∼} ℤ_{X}^{c}(Xk̄) → CH_{0}(Xk̄) ⟶^{degk̄} ℤ.

By a result of Nart [11], ℤ_{X}^{c} it is quasi-isomorphic to a shift of Deninger’s complex 𝔾, but we prefer to work with Bloch’s complex, as it yields the correct dualizing complex in higher dimensions.

2.2 Arithmetic Borel-Moore homology and cohomology

We fix a finite field 𝔽_{q} with Galois group Ĝ = Gal(𝔽̄_{q}/𝔽_{q}) and let G ⊂ Ĝ be its Weil group, the free abelian subgroup of rank 1 generated by the Frobenius endomorphism φ. Given a separated scheme of finite type X over 𝔽_{q}, let X̄ = X ×_{𝔽_{q}} 𝔽̄_{q}. For a sheaf F on the small etale site Et_{X}^{∼} of X, we define arithmetic Borel-Moore homology groups as the cohomology groups of the complex

R Hom_{ar}(F, ℤ_{X}^{c}) := RΓ_{G}R Hom_{et}(F|X̄, ℤc ; X̄)[1]

so that H_{i}^{c}(X_{ar}, F) := Ext_{ar}^{−i}(F, ℤ^{c}). This generalizes the groups H_{i}^{c}(X_{ar}, ℤ) considered in [7]. The Leray spectral sequence degenerates to

0 → Ext_{et}^{−i}(F|X̄, ℤc ; X̄)_{G} → H_{i}^{c}(X_{ar}, F) → Ext_{et}^{1−i}(F|X̄, ℤc ; X̄)^{G} → 0. (2)

Let f : X → Y be a map, F ∈ Et_{Y}^{∼} and G ∈ Et_{X}^{∼}. If f is proper, then the map Rf_{∗}ℤ_{X}^{c} → ℤ_{Y}^{c} induces by adjunction covariant functoriality H_{i}^{c}(X_{ar}, f^{∗}F) → H_{i}^{c}(Y_{ar}, F). If f is a closed embedding, then the quasi-isomorphism Rf^{!}ℤ_{Y}^{c} ≅ ℤ_{X}^{c} induces by adjunction an isomorphism H_{i}^{c}(X_{ar}, G) ⟶^{∼} H_{i}^{c}(Y_{ar}, f_{∗}G). If f is flat and quasi-finite, then the map f^{∗}ℤ_{Y}^{c} → ℤ_{X}^{c} induces contravariant functoriality H_{i}^{c}(Y_{ar}, F) → H_{i}^{c}(X_{ar}, f^{∗}F). If f is etale, then since f^{∗}ℤ_{Y}^{c} ≅ ℤ_{X}^{c} we obtain an isomorphism

H_{i}^{c}(X_{ar}, G) = Ext_{ar}^{−i}(G, ℤ_{X}^{c}) ≅ Ext_{ar}^{−i}(f_{!}G, ℤ_{Y}^{c}) = H_{i}^{c}(Y_{ar}, f_{!}G).  (3)

For a closed subscheme i : Z → X with open complement j : U → X and F ∈ Et_{X}^{∼}, the short exact sequence 0 → j_{!}j^{∗}F → F → i_{∗}i^{∗}F → 0 induces a localization sequence

→ H_{i+1}^{c}(U_{ar}, F|_{U}) → H_{i}^{c}(Z_{ar}, F|_{Z}) → H_{i}^{c}(X_{ar}, F) → H_{i}^{c}(U_{ar}, F|_{U}) → . (4)

Theorem 2.2 For every etale sheaf F ∈ Et_{X}^{∼}, there is a long exact sequence

→ Ext_{et}^{1−i}(F, ℤ_{X}^{c}) → H_{i}^{c}(X_{ar}, F) → Ext_{et}^{−i}(F, ℤ_{X}^{c})_{ℚ} ⟶^{δ} Ext_{et}^{2−i}(F, ℤ_{X}^{c}) →,

and the map δ has torsion image. In particular, for any torsion sheaf F,

Ext_{et}^{1−i}(F, ℤ_{X}^{c}) ≅ H_{i}^{c}(X_{ar}, F).

If F is a sheaf of ℚ-vector spaces, then the long exact sequence is split by (2), and we obtain

H_{i}^{c}(X_{ar}, F)_{ℚ} ≅ Ext_{et}^{1−i}(F, ℤ_{X}^{c})_{ℚ} ⊕ Ext_{et}^{−i}(F, ℤ_{X}^{c})_{ℚ}.

Proof. By [4], any complex of Ĝ-modules gives rise to a distinguished triangle RΓĜC^{·} → RΓ_{G}C^{·} → RΓĜC^{·} ⊗ ℚ[−1]. We apply this to the complex of Ĝ-modules R Hom_{et}(F|X̄, ℤc ; X̄), and it suffices to show that RΓĜR Hom_{et}(F|X̄, ℤc ; X̄) ≅ R Hom_{et}(F, ℤ^{c}). But if ℤ^{c} → I^{·} is an injective resolution, then HomX̄(F, I^{j}) is acyclic for (−)Ĝ [10, III Cor. 2.13c)], hence the claim follows from HomX̄(F, G)Ĝ ≅ Hom_{X}(F, G) for etale sheaves. □

If j : U → X is a compactification of the curve U and F in Et_{U}^{∼}, then Weil-etale cohomology is defined as H^{i}(U_{W} , F) := H^{i}RΓ_{G}RΓ(Ū_{et}, F), and Weil-etale cohomology with compact support as

H_{c}^{i}(U_{W} , F) := H^{i}RΓ_{G}RΓ_{c}(X̄_{et}, j_{!}F).  (5)

Lemma 2.3 Consider a cartesian diagram

Z′ ⟶ X′
↓  ^{f}↓
Z ⟶^{i} X.

with i a closed embedding, f finite, and such that f induces an isomorphism of dense open subsets X′ − Z′ → X − Z. Then for any etale sheaf F on X, there is an exact triangle

RΓ(X_{et}, F) → RΓ(Z_{et}, F|_{Z}) ⊕ RΓ(X′_{et}, F|_{X′}) → RΓ(Z′_{et}, F|_{Z′}).

Proof. This follows because for finite maps f, the functor f_{∗} is exact. Concretely, if d is the map Z′ → X, then R^{s}f_{∗}F|_{X′} = R^{s}d_{∗}F|_{Z′} for s > 0. Hence the triangle is induced by the short exact sequence of sheaves on X, 0 → F → i_{∗}F|_{Z} ⊕ f_{∗}F|_{X′} → d_{∗}F|_{Z′} → 0, which is easily checked on stalks. □

Proposition 2.4 The complex RΓ_{c}(X̄_{et}, j_{!}F), hence definition (5), is independent of the choice of the compactification j : U → X.

Proof. Given two compactifications : U → X and j′ : U′ → X′, we consider the closure C of U in X × X′ and by comparing the compactifications p_{1} : C → X and p_{2} : C → X′, we can assume that there is a finite map f : X′ → X with fj′ = j. Then by the Lemma

RΓ(X̄_{et}j_{!}F) = coneRΓ(X̄_{et}, j_{∗}F) → RΓ(Z̄_{et}, j_{∗}F|_{Z}) = coneRΓ(X̄_{et}, f_{∗}j′_{∗}F) → RΓ(Z̄_{et}, f′_{∗}(j′_{∗}F|_{Z′})) = coneRΓ(X̄′_{et}, j′_{∗}F) → RΓ(Z̄′_{et}, (j′_{∗}F)|_{Z′}) = RΓ(X̄′_{et}, j′_{!}F)

□

For a closed subscheme Z of the curve X with open complement U, we obtain a localization sequence

→ H_{c}^{i−1}(Z_{W} , F|_{Z}) → H_{c}^{i}(U_{W} , F|_{U}) → H_{c}^{i}(X_{W} , F) → H_{c}^{i}(Z_{W} , F|_{Z}) →, (6)

and the Leray spectral sequence degenerates to

0 → H_{c}^{i−1}(X̄_{et}, F)_{G} → H_{c}^{i}(X_{W} , F) → H_{c}^{i}(X̄_{et}, F)^{G} → 0.  (7)

The analog of Theorem 2.2 holds, in particular, there is a long exact sequence

→ H_{c}^{i}(X_{et}, F) → H_{c}^{i}(X_{W} , F) → H_{c}^{i−1}(X_{et}, F)_{ℚ} ⟶^{δ} H_{c}^{i+1}(X_{et}, F) →, (8)

and we have

H_{c}^{i}(X_{W} , F)_{ℚ} ≅ H_{c}^{i}(X_{et}, F)_{ℚ} ⊕ H_{c}^{i−1}(X_{et}, F)_{ℚ}.  (9)

Remark. The definition (5) given here agrees with the definition in [5] only for curves. For schemes of higher dimension Proposition 2.4 does not hold, see [5]. Thus the etale topology has to be replaced by the eh-topology in order to obtain good properties.

2.3 Finite generation

Lemma 2.5 Let F be a ℤ-constructible etale sheaf on a zero-dimensional scheme P.

a) The groups H_{i}^{c}(P_{ar}, F) are finite for i = −1, finitely generated for i = 0, finitely generated free for i = 1, and trivial otherwise.

b) The groups H_{c}^{i}(P_{W} , F) are finitely generated for i = 0, 1, and trivial otherwise.

Proof. We may assume that P = Spec 𝔽_{q^{r}}.

a) Since ℤ_{P}^{c} ≅ ℤ, the group Ext_{et}^{0}(F, ℤc ; P̄) = Hom_{Ab}(FP̄/tor, ℤ) is free, the group Ext_{et}^{1}(F, ℤc ; P̄) = (_{tor}FP̄)^{∗} is finite, and the other extension groups vanish. The Lemma follows with (2).

b) This follows from (7) because H_{c}^{i}(P̄_{et}, F) is finitely generated for i = 0, and trivial for i ≠ 0. □

Proposition 2.6 Let F be a ℤ-constructible etale sheaf on a curve X. Then the groups H_{c}^{i}(X_{W} , F) and H_{i}^{c}(X_{ar}, F) are finitely generated.

Proof. Using (6) and (4) we can (by shrinking X) assume that F is locally constant. Shrinking X further, we can assume that there is a finite etale Galois covering f : Y → X with Galois group A such that F|_{Y} is constant. Since RΓ_{X} ≅ RΓ_{A}RΓ_{Y} and since group cohomology of finite groups with finitely generated coefficients is finitely generated, we can assume that F is constant. Increasing X again we can assume that X is smooth and proper. The groups H^{i}(X_{W} , ℤ) are finitely generated by [4, Prop.7.4], and the groups H_{i}^{c}(X_{ar}, ℤ) are finitely generated by [7]. This implies finite generation for finitely generated constant coefficients. □

Lemma 2.7 Let X̄ be a curve over an algebraically closed field and F a sheaf on X̄. Then the groups H^{i}(X̄_{et}, F) and H_{c}^{i}(X̄_{et}, F) are torsion for i > 1 and vanish for i > 3.. In particular, for X a curve over a finite field, H_{c}^{i}(X_{W} , F) is torsion for i > 2 and vanishes for i > 4.

Proof. Clearly the result for cohomology implies the result for cohomology with compact support. Writing F as a colimit of ℤ-constructible sheaves we can assume that F is ℤ-constructible. For a closed embedding i : Z → X with open complement j : U → X, we have a short exact sequence of sheaves 0 → j_{!}j^{∗}F → F → i_{∗}i^{∗}F → 0. Since a zero-dimensional scheme over an algebraically closed field has cohomological dimension 0, we can assume that X is normal and connected, and F = M is locally constant and finitely generated. Let g : η → X be the embedding of the generic point. Then j_{∗}M_{η} ≅ M, and k(η) has cohomological dimension 1. We consider the case M torsion and M torsion free separately. If M is torsion, then R^{1}g_{∗}M_{η} is torsion, the higher derived images vanish, and H^{i}(k(η)_{et}, M_{η}) = 0 for i > 0. If M is torsion free, then R^{1}g_{∗}M_{η} = 0, R^{2}g_{∗}M_{η} is torsion, the higher derived image are zero, and H^{i}(k(η)_{et}, M_{η}) = 0 for i > 2. Now the claim follows by analyzing the spectral sequence H^{s}(X̄_{et}, R^{t}g_{∗}M_{η}) ⇒ H^{s+t}(k(η)_{et}, M_{η}). □

Lemma 2.8 If X̄ is a curve over an algebraically closed field, then the groups Ext_{et}^{i}(F|X̄, ℤc ; X̄) vanish unless −1 ≤ i ≤ 4. In particular, the groups H_{i}^{c}(X̄_{ar}, F) vanish unless −4 ≤ i ≤ 2.

Proof. The lower bound is obtained by observing that ℤX̄ is quasi-isomorphic to a complex concentrated in degrees −1, 0. By the analog of (4) and Lemma 2.5 for etale extension groups, we can assume that F is locally constant. Consider the spectral sequence

H^{s}(X̄, E𝔵t^{t}(F|X̄, ℤc ; X̄)) ⇒ Ext_{et}^{s+t}(F|X̄, ℤc ; X̄).

Since F locally constant, the E𝔵t^{t}-sheaf can be calculated at stalks [10, III 1.31], and since ℤc ; X̄ is concentrated in non-positive degrees, the stalks vanish for t > 1. On the other hand, we just saw that H^{s}(X̄_{et}, F) = 0 for s > 3. The final statement follows from (2). □

3 The pairing over an algebraically closed field

Let X be a proper curve over an algebraically closed field k, and F^{·} a bounded complex of etale sheaves. Choose injective resolutions ℤ_{X}^{c} → I_{X}^{·}, and F → J^{·}. We can assume that J^{·} is bounded, because X has strict cohomological dimension 3. If I = (ℚ → ℚ/ℤ) is the injective resolution of ℤ as an abelian group, we obtain by Theorem 2.1 a map

I_{X}^{·}(X) ←^{∼} ℤ_{X}^{c}(X) → ℤ ≅ I

in the derived category of bounded above complexes of abelian groups. Hence we obtain a natural transformation of derived functors D^{b}(Et_{X}^{∼}) → D^{−}(Ab),

τ_{X}(F^{·}) : R Hom_{et}(F^{·}, ℤ_{X}^{c}) ≅ Hom_{et}(J^{·}, I_{X}^{·}) ⟶^{Γ} Hom_{Ab}(J^{·}(X), I_{X}^{·}(X))

⟶^{tr} Hom_{Ab}(Γ(X_{et}, J^{·}), I) ⟶^{∼} R Hom_{Ab}(RΓ(X_{et}, F^{·}), ℤ).

For an arbitrary separated curve X over k, we choose a compactification j : X → X′ of X, and define

τ_{X}(F) : R Hom_{et}(j_{!}F^{·}, ℤ_{X′}^{c}) ⟶^{τ_{X′} (j_{!}F)}

R Hom_{Ab}(RΓ(X′_{et}, j_{!}F^{·}), ℤ) ≅ R Hom_{Ab}(RΓ_{c}(X_{et}, F^{·}), ℤ). (10)

By compatibility of the trace map with proper push-forward, the usual argument comparing compactifications shows that this is independent of the compactification. Let RT_{m} = R Hom_{ℤ}(ℤ/m, −) be the right derived functor of the m-torsion functor. Then for every bounded complex F^{·}, there is a distinguished triangle

RT_{m}F^{·} → F^{·} ⟶^{×m} F^{·} → RT_{m}F^{·}[1].  (11)

Proposition 3.1 Let F^{·} be a bounded complex of ℤ-constructible sheaves on a curve X over an algebraically closed field k. Then the map induced by (10),

f_{i} : Ext_{et}^{−i}(F^{·}, ℤ_{X}^{c}) → R Hom_{Ab}^{−i}(RΓ_{c}(X_{et}, F^{·}), ℤ)

has divisible kernel, torsion free cokernel, and coker f_{i}/m ≅ _{m} ker f_{i−1}.

Proof. If G^{·} is a bounded below complex of constructible sheaves, then τ_{X}(G^{·}) is a quasi-isomorphism by the main theorem of [6]. In particular, τ_{X}(RT_{m}F^{·}) is a quasi-isomorphism for every m and F^{·}. We apply the map τ_{X} to the triangle (11) and obtain a map of long exact sequences

Ext_{X}^{−i−1}(RT_{m}F^{·}, ℤ_{X}^{c}) ⟶^{∼} R Hom_{Ab}^{−i−1}(RΓ_{c}(X_{et}, RT_{m}F^{·}), ℤ)
↓  ↓
Ext_{X}^{−i}(F^{·}, ℤ_{X}^{c})  ⟶^{f_{i}}  R Hom_{Ab}^{−i}(RΓ_{c}(X_{et}, F^{·}), ℤ)
×m↓  ^{×m}↓
Ext_{X}^{−i}(F^{·}, ℤ_{X}^{c})  ⟶^{f_{i}}  R Hom_{Ab}^{−i}(RΓ_{c}(X_{et}, F^{·}), ℤ)
↓  ↓
Ext_{X}^{−i}(RT_{m}F^{·}, ℤ_{X}^{c}) ⟶^{∼} R Hom_{Ab}^{−i}(RΓ_{c}(X_{et}, RT_{m}^{·}), ℤ)

By the previous discussion, the upper and lower map are isomorphisms. If K_{i} and C_{i} are the kernel and cokernel of f_{i}, respectively, then a diagram chase shows that we get, for every integer m, an exact sequence

0 → C_{i} ⟶^{×m} C_{i} → K_{i−1} ⟶^{×m} K_{i−1} → 0.

This implies that C_{i} is torsion free, K_{i−1} is divisible, and _{m}K_{i−1} = C_{i}/m. □

4 The main Theorem

We are going to descend the pairing τ_{X}(F) to a pairing of arithmetic cohomology groups.

Lemma 4.1 If M^{·} is complex of ℤ[G]-modules and N^{·} a complex of abelian groups (viewed as a complex of trivial ℤ[G]-modules), then there is a quasiisomorphism

RΓ_{G}R Hom_{Ab}(M^{·}, N^{·}) = R Hom_{G}(M^{·}, N^{·}) ≅ R Hom_{Ab}(RΓ_{G}M^{·}, N^{·})[−1].

Proof. For any ℤ[G]-module M and abelian group N we have Hom_{Ab}(M_{G}, N) ≅ Hom_{G}(M, N). Since the total left derived functor L(−)_{G} of the coinvariant functor agrees with the shift of the total right derived functor R(−)^{G}[−1] of the invariant functor, we get the adjunction R Hom_{Ab}(RΓ_{G}M, N) ≅ R Hom_{G}(M, N[1]). The lemma follows. □

Restricting the action of the Galois group Ĝ on the source and target of τ_{X}(F) to the Weil group G, and applying RΓ_{G}, we get by Lemma 4.1 the duality homomorphism

δ_{X}(F) : R Hom_{ar}(F, ℤ^{c}) = RΓ_{G}R Hom_{et}(F, ℤ^{c}|X̄)[1] ⟶^{RΓ_{G}τ_{X}(F)}

RΓ_{G}R Hom_{Ab}(RΓ_{c}(X̄_{et}, F), ℤ)[1] ≅

R Hom_{Ab}(RΓ_{G}RΓ_{c}(X̄_{et}, F), ℤ) = R Hom_{Ab}(RΓ_{c}(X_{W} , F), ℤ). (12)

Theorem 4.2 Let F^{·} be a bounded complex of ℤ-constructible sheaves on the curve X over 𝔽_{q}. Then the pairing (12) induces a quasi-ismorphism

R Hom_{ar}(F, ℤ^{c}) ⟶^{∼} R Hom_{Ab}(RΓ_{c}(X_{W} , F), ℤ).

Proof. We apply δ to the triangle (11) note that the map δ_{X}(RT_{m}F) ≅ RΓ_{G}τ_{X}(RT_{m}F) is a quasi-isomorphism for every m by the main theorem of [6]. If K_{i} and C_{i} are the kernel and cokernel of H_{i}^{c}(X_{ar}, F) → R Hom_{Ab}^{−i}(RΓ_{c}(X_{ar}, F), ℤ), respectively, then the argument of Proposition 3.1 gives, for every integer m, an exact sequence

0 → C_{i+1} ⟶^{×m} C_{i+1} → K_{i} ⟶^{×m} K_{i} → 0.

Since K_{i} is divisible, finite generation implies that K_{i} is trivial, and then C_{i+1} is divisible hence trivial. □

Corollary 4.3 We have perfect pairings of finitely generated groups

H_{i}^{c}(X_{ar}, F)/tor × H_{c}^{i}(X_{W} , F)/tor → ℤ;  (13)
torH_{i−1}^{c}(X_{ar}, F) ×_{tor}H_{c}^{i}(X_{W} , F) → ℚ/ℤ.  (14)

The torsion free groups vanish unless 0 ≤ i ≤ 2, and the torsion groups vanish unless 0 ≤ i ≤ 3.

Proof. Taking the map induced on −ith cohomology groups by δ_{X}(F), we get an isomorphism

H_{i}^{c}(X_{ar}, F) ≅ H^{−i}R Hom_{Ab}(RΓ_{c}(X_{W} , F), ℤ).

The Leray spectral sequence degenerates into

0 → Ext_{Ab}^{1}(H_{c}^{i+1}(X_{W} , F), ℤ) → H^{−i}R Hom_{Ab}(RΓ_{c}(X_{W} , F), ℤ)

→ Hom_{Ab}(H_{c}^{i}(X_{W} , F), ℤ) → 0.

For a finitely generated abelian group M, the surjection Hom_{Ab}(M, ℚ/ℤ) → Ext_{Ab}^{1}(M, ℤ) induces an isomorphism Hom_{Ab}(_{tor}M, ℚ/ℤ) ≅ Ext_{Ab}^{1}(M, ℤ). Since this is torsion and Hom_{Ab}(−, ℤ) is torsion free, we have

H^{−i}R Hom_{Ab}(RΓ_{c}(X_{W} , F))/tor = Hom_{Ab}(H_{c}^{i}(X_{W} , F), ℤ)

and

torH^{−i}R Hom_{Ab}(RΓ_{c}(X_{W} , F)) = Ext_{Ab}^{1}(H_{c}^{i+1}(X_{W} , F), ℤ)

= Hom_{Ab}(_{tor}H_{c}^{i+1}(X_{W} , F), ℚ/ℤ).

The vanishing follows by Lemma’s 2.7 and 2.8. □

Corollary 4.4 The pairing

δ_{X}(F) : R Hom_{ar}(F, ℤ^{c}) → R Hom_{Ab}(RΓ_{c}(X_{W} , F), ℤ)

is a quasi-isomorphism for every sheaf F ∈ Et_{X}^{∼}.

Proof. Given F on Et_{X}^{∼}, we can write F as the filtered colimit of ℤ-constructible sheaves, F = colim F_{i}. Since the etale site is noetherian, RΓ(X_{et}, −) commutes with colimits, the duality pairing can be identified with

R Hom_{ar}(F, ℤ_{X}^{c})  R Hom_{ar}(colim F_{i}, ℤ_{X}^{c})
↓  ↓
R Hom_{Ab}(RΓ_{c}(X_{W} , F), ℤ)  R Hom_{Ab}(colim RΓ_{c}(X_{W} , F_{i}), ℤ)

The right hand map induces a map between the spectral sequences of [12, Theorem 1]

E_{2}^{s,t} = lim^{s} Ext_{ar}^{t}(F_{i}, ℤ_{X}^{c})  ^{⇒}  Ext_{ar}^{s+t}(colim F_{i}, ℤ^{c})
↓  ↓
E_{2}^{s,t} = lim^{s} Ext_{Ab}^{t}(RΓ_{c}(X_{W} , F_{i}), ℤ) ^{⇒} Ext_{Ab}^{s+t}(colim RΓ_{c}(X_{W} , F_{i}), ℤ).

The map on E_{2}-terms is an isomorphism, and the spectral sequences converge, by Lemma’s 2.7 and 2.8. Hence we get an isomorphism on the abutment. □

5 Constant coefficients

In this section we connect our results to the result of Lichtenbaum [9]. Recall that if X is smooth, then ℤ_{X}^{c} ≅ 𝔾_{m}[1], hence H_{i}^{c}(X_{ar}, ℤ) ≅ H^{2−i}(X_{W} , 𝔾_{m}). For an abelian group A, let A^{∗} = Hom(A, ℚ/ℤ).

Proposition 5.1 (Lichtenbaum [9, Thm.6.1c,d]) Let U be the complement of s > 0 points of a connected smooth and proper curve X. Then

H_{c}^{i}(U_{W} , ℤ) = \begin{cases}0 & i = 0; \\ ℤ^{s}/ℤ & i = 1; \\ Hom(Γ(U, O_{U})^{×}, ℤ) ⊕ Pic(U)^{∗} & i = 2; \\ Γ(X, O_{X}^{×})^{∗} & i = 3. \end{cases}

H_{i}^{c}(U_{ar}, ℤ) = \begin{cases}0 & i = 0; \\ Pic(U) ⊕ ker(ℤ^{s} → ℤ) & i = 1; \\ Γ(U, O_{U}^{×}) & i = 2. \end{cases}

Proof. For cohomology, this follows from the long exact sequence (6) comparing the cohomology and homology of U to its compactification X. For homology we have H_{2}(X_{ar}, ℤ) = H^{0}(X_{W} , 𝔾_{m}) = H^{0}(X_{et}, 𝔾_{m}) by (7), and the long exact sequence (4) gives the other two groups. Note that Γ(U, O_{U}^{×}) ≅ Γ(X, O_{X}^{×}) ⊕ ker(ℤ^{⊕s} → ℤ). □

The dual graph of a proper curve X is defined as follows. Let X′ be the normalization of X, S be the set of singular points of X and S′ = S ×_{X} X′. Then the dual graph is a bipartite graph with vertices the points of S′ and the connected components of X′, and an edge for each point in S′ connecting its image in S with the component it is contained it. If X is connected, then Γ is connected, hence H^{0}(Γ, ℤ) = ℤ and H^{1}(Γ, ℤ) is free of rank |π_{0}(S)| − |π_{0}(S′)| + |π_{0}(X′)|. The following proposition generalizes Lichtenbaum [9, Thm.6.1a,b], since for smooth and proper curves, CH_{0}(X) ≅ Pic(X), CH_{0}(X, 1) = Γ(X, O_{X}^{×}) and H^{1}(Γ, ℤ) = 0.

Proposition 5.2 Let X be a connected proper curve with normalization X′ = ∐_{i} X_{i} and dual graph Γ. Then the non-vanishing cohomology and homology groups are

H_{c}^{i}(X_{W} , ℤ) = \begin{cases}H^{0}(Γ, ℤ) & i = 0; \\ H^{0}(Γ, ℤ) ⊕ H^{1}(Γ, ℤ) & i = 1; \\ H^{1}(Γ, ℤ) ⊕ ∐ Pic^{0}(X_{i})^{∗} & i = 2; \\ ∐ Γ(X_{i}, O_{X_{i}}^{×})^{∗} & i = 3. \end{cases}

H_{i}^{c}(X_{ar}, ℤ) = \begin{cases}H_{0}(Γ, ℤ) & i = 0; \\ CH_{0}(X) ⊕ H_{1}(Γ, ℤ) & i = 1; \\ CH_{0}(X, 1) & i = 2. \end{cases}

Proof. In the smooth, proper case, the result for cohomology follows from (7) and H^{i}(X_{W} , ℤ) ≅ H^{i−1}(X_{et}, ℚ/ℤ) for i ≥ 2. The calculation for homology can also be found in [7]. In the general case, one uses the long exact sequences (4) and (6) arising from the cartesian square

S′ ⟶ ∐ X_{i}
↓  ↓
S ⟶ X.
□

6 Comparison to Deninger’s results

By Nart [11], Deninger’s [3] dualizing complex 𝔾 is quasi-isomorphic to the shift ℤ_{X}^{c}[−1]. According to Deninger [2], if X is smooth and proper, then the groups H^{i}(X_{et}, F) and Ext_{et}^{i}(F, 𝔾) = Ext_{et}^{i−1}(F, ℤ_{X}^{c}) are finitely generated for i = 0, 1 and of cofinite type for i = 2, 3. We are going to recover and improve Deninger’s result in this section. Let

r = rank H_{c}^{0}(X_{W} , F)
s = rank H_{c}^{2}(X_{W} , F).

Lemma 6.1 Let X be a separated curve over a finite field.

a) The groups H_{c}^{i}(X_{et}, F) and Ext_{et}^{i−1}(F, ℤ^{c}) are torsion for i ≠ 0, 1.

b) We have

rank H_{c}^{1}(X_{W} , F) = r + s
rank H_{c}^{0}(X_{et}, F) = r
rank H_{c}^{1}(X_{et}, F) = s.

Proof. a) The first statement is Lemma 2.7. By Theorem 2.2, we have Ext_{et}^{i−1}(F, ℤ^{c})_{ℚ} ⊆ H_{2−i}^{c}(X_{ar}, F)_{ℚ} as well as Ext_{et}^{i−1}(F, ℤ^{c})_{ℚ} ⊆ H_{1−i}^{c}(X_{ar}, F)_{ℚ} and we can conclude with Corollary 4.3

b) This follows from a) and (9). □

Now consider the long exact sequence (8)

0 → H_{c}^{1}(X_{et}, F) → H_{c}^{1}(X_{W} , F) ⟶^{α} H_{c}^{0}(X_{et}, F)_{ℚ} →^{δ_{0}} H_{c}^{2}(X_{et}, F) →

H_{c}^{2}(X_{W} , F) → H_{c}^{1}(X_{et}, F)_{ℚ} →^{δ_{1}} H_{c}^{3}(X_{et}, F) → H_{c}^{3}(X_{W} , F) → 0.

Some easy considerations together with the fact that Weil-etale cohomology is finitely generated, and that H_{c}^{2}(X_{et}, F) and H_{c}^{3}(X_{et}, F) are torsion, gives

Theorem 6.2 Let X be a separated curve over a finite field.

a) The groups H_{c}^{0}(X_{et}, F), H_{c}^{1}(X_{et}, F) are finitely generated, the groups H_{c}^{2}(X_{et}, F), H_{c}^{3}(X_{et}, F) are cofinitely generated of corank r and s, respectively, and all other groups vanish.

b) We have a decomposition into finite and cofree groups

torH_{c}^{1}(X_{et}, F) ≅ _{tor}H_{c}^{1}(X_{W} , F);
H_{c}^{2}(X_{et}, F) ≅ _{tor}H_{c}^{2}(X_{W} , F) ⊕ (H_{c}^{1}(X_{W} , F)/H_{c}^{1}(X_{et}, F)) ⊗ ℚ/ℤ;
H_{c}^{3}(X_{et}, F) ≅ H_{c}^{3}(X_{W} , F) ⊕ H_{c}^{2}(X_{W} , F) ⊗ ℚ/ℤ.

For example, the image of δ_{0} is H_{c}^{0}(X_{et}, F) ⊗ ℚ/ℤ because it is torsion and H_{c}^{1}(X_{W} , F)/tor is a finitely generated abelian group. By the Lemma, Theorem 2.6 and (8), H_{c}^{i}(X_{et}, F) ≅ H_{c}^{i}(X_{W} , F) = 0 for i > 3. By Lemma 6.1a), the exact sequence from Theorem 2.2 becomes

0 → Ext_{et}^{0}(F, ℤ_{X}^{c}) → H_{1}^{c}(X_{ar}, F) ⟶^{β} Ext_{et}^{−1}(F, ℤ_{X}^{c})_{ℚ} → Ext_{et}^{1}(F, ℤ_{X}^{c}) →

H_{0}^{c}(X_{ar}, F) → Ext_{et}^{0}(F, ℤ_{X}^{c})_{ℚ} → Ext_{et}^{2}(F, ℤ_{X}^{c}) → H_{−1}^{c}(X_{ar}, F) → 0 together with an isomorphism Ext_{et}^{−1}(F, ℤ^{c}) = H_{2}^{c}(X_{ar}, F). A similar argument as above gives

Theorem 6.3 Let X be a separated curve over a finite field.

a) Then the groups Ext_{et}^{−1}(F, ℤ_{X}^{c}) and Ext_{et}^{0}(F, ℤ_{X}^{c}) are finitely generated of rank s and r, respectively, the groups Ext_{et}^{1}(F, ℤ_{X}^{c}) and Ext_{et}^{2}(F, ℤ_{X}^{c}) are cofinitely generated of corank s and r, respectively, and all other groups vanish.

b) We have a decomposition into finite and cofree groups

tor Ext_{et}^{0}(F, ℤ_{X}^{c}) ≅ _{tor}H_{1}(X_{ar}, F),
Ext_{et}^{1}(F, ℤ_{X}^{c}) ≅ _{tor}H_{0}^{c}(X_{ar}, F) ⊕ (H_{1}^{c}(X_{ar}, F)/ Ext_{et}^{0}(F, ℤ_{X}^{c})) ⊗ ℚ/ℤ,
Ext_{et}^{2}(F, ℤ_{X}^{c}) ≅ H_{−1}^{c}(X_{ar}, F) ⊕ H_{0}(X_{ar}, F) ⊗ ℚ/ℤ.

Corollary 6.4 (Deninger) There are isomorphisms of discrete torsion groups of cofinite type:

Ext_{et}^{−1}(F, ℤ_{X}^{c})^{∗} ≅H_{c}^{3}(X_{et}, F);
H_{c}^{0}(X_{et}, F)^{∗} ≅ Ext_{et}^{2}(F, ℤ_{X}^{c})
tor Ext_{et}^{0}(F, ℤ^{c})^{∗} ≅_{cotor}H_{c}^{2}(X_{et}, F)
torH_{c}^{1}(X_{et}, F)^{∗} ≅_{cotor} Ext_{et}^{1}(F, ℤ^{c})

Proof. This follows by splicing together the torsion and cotorsion part of Corollary 4.3. For example,

Ext_{et}^{−1}(F, ℤ^{c})^{∗} ≅ (_{tor}H_{2}(X_{ar}, F))^{∗} ⊕ (H_{2}(X_{ar}, F)/tor)^{∗}.

The first term is H_{c}^{3}(X_{W} , F), and the second term is

Hom(Hom(H_{c}^{2}(X_{W} , F)/tor, ℤ), ℚ/ℤ) ≅ H_{c}^{2}(X_{W} , F) ⊗ ℚ/ℤ

because H_{c}^{2}(X_{W} , F) is a finitely generated abelian group. The last two isomorphisms are

tor Ext_{et}^{0}(F, ℤ^{c})^{∗} = _{tor}H_{1}^{c}(X_{ar}, F)^{∗} = _{tor}H_{c}^{2}(X_{W} , F) = _{cotor}H_{c}^{2}(X_{et}, F),

and

torH_{c}^{1}(X_{et}, F)^{∗} = _{tor}H_{c}^{1}(X_{W} , F)^{∗} = _{tor}H_{0}^{c}(X_{ar}, F) = _{cotor} Ext_{et}^{1}(F, ℤ^{c}).
□

We leave it as an open problem to derive the remaining isomorphisms

(Ext_{et}^{0}(F, ℤ^{c})/tor)^{∗} ≅H_{c}^{2}(X_{et}, F)/cotor
(H_{c}^{1}(X_{et}, F)/tor)^{∗} ≅ Ext_{et}^{1}(F, ℤ^{c})/cotor

from our results. In higher dimension, the etale cohomology and extension groups will be mixed in the sense that they contain both a finitely generated free subgroup as well as a cofinitely generated torsion divisible subgroup.

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Thomas Geisser Nagoya University Nagoya Japan geisser@math.nagoya-u.ac.jp