ZETA FUNCTIONS OF REGULAR ARITHMETIC SCHEMES AT s= 0

arXiv:1103.6061v1 [math.NT] 30 Mar 2011

ZETA FUNCTIONS OF REGULAR ARITHMETIC SCHEMES AT s = 0 BAPTISTE MORIN

Abstract. Lichtenbaum conjectured in [22] the existence of a Weil-étale cohomology in order to describe the vanishing order and the special value of the Zeta function of an arithmetic scheme X at s = 0 in terms of EulerPoincaré characteristics. Assuming the (conjectured) finite generation of some motivic cohomology groups we construct such a cohomology theory for regular schemes proper over Spec(Z). In particular, we compute (unconditionally) the right Weil-étale cohomology of number rings and projective spaces over number rings. We state a precise version of Lichtenbaum’s conjecture, which expresses the vanishing order (resp. the special value) of the Zeta function ζ(X , s) at s = 0 as the rank (resp. the determinant) of a single perfect complex of abelian groups RΓW,c (X , Z). Then we relate this conjecture to Soulé’s conjecture and to the Tamagawa Number Conjecture. Lichtenbaum’s conjecture for projective spaces over the ring of integers of an abelian number field follows.

1. Introduction In [22] Lichtenbaum conjectured the existence of a Weil-étale cohomology in order to describe the vanishing order and the special value of the Zeta function of an arithmetic scheme i.e. a separated scheme of finite type over Z at a negative integer s = n in terms of Euler-Poincaré characteristics. More precisely, we have the following Conjecture 1.1. (Lichtenbaum) Let X be an arithmetic scheme. There exist i (X , Z) and H i (X , R) such that the following holds. cohomology groups HW,c W,c i (X , Z) are finitely generated and zero for i large. (1) The groups HW,c (2) The natural map from Z to R-coefficients induces isomorphisms i i HW,c (X , Z) ⊗ R ≃ HW,c (X , R). 1 (X , R) such that cup-product with (3) There exists a canonical class θ ∈ HW θ turns the sequence ∪θ

∪θ

∪θ

i ˜ −→ ... ˜ −→ H i+1 (X , R) ... −→ HW,c (X , R) W,c

into a bounded acyclic complex of finite dimensional vector spaces. Mathematics subject classification (2010): Primary 14F20, 11S40, Secondary: 11G40. Keywords: Weil-étale cohomology, Zeta-functions 1

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(4) The vanishing order of the zeta function ζ(X , s) at s = 0 is given by the formula X i ords=0 ζ(X , s) = rankZ RΓW,c (X , Z) := (−1)i .i.rankZ HW,c (X , Z) i≥0

(5) The leading coefficient in the Taylor development of ζ(X , s) at s = 0 is given up to sign by Z · λ(ζ ∗ (X , 0)) = detZ RΓW,c(X , Z). ∼

where λ : R → detR RΓW,c(X , Z)R is induced by (3). Lichtenbaum defined such cohomology groups for smooth varieties over finite fields in [21], but similar attempts to define such a cohomology for flat arithmetic schemes failed. In [22] Lichtenbaum gave the first construction for number rings. He defined a Weil-étale topology which bears the same relation to the usual étale topology as the Weil group does to the Galois group. Under a vanishing statement, he was able to show that his cohomology miraculously yields the value of Dedekind zeta functions at s = 0. But this cohomology with coefficients in Z was then shown to be infinitely generated in even degrees i ≥ 4 [6]. Consequently, Lichtenbaum’s complex computing the Z-cohomology needs to be artificially truncated in the case of number rings, and is not helpful for flat schemes of dimension greater than 1. Moreover, it seems impossible to construct relevant sheaves Z(n) for n 6= 0 on the Weil-étale topology. However, Lichtenbaum’s definition does work with R-coefficients, and this fact extends to higher dimensional arithmetic schemes [7]. Moreover, the geometric intuition given by Lichtenbaum’s Weil-étale topos provides a guiding light for any other definition, as it is the case in this paper. Here we propose an alternative approach, whose basic idea can be explained as follows. The Weil group is (non-canonically) defined as an extension of the Galois group by the idèle class group "corresponding" to the fundamental class of class field theory. In this paper we use étale duality for arithmetic schemes rather than class field theory, in order to obtain a (canonical) extension of the étale Z-cohomology by the dual of motivic Q(d)-cohomology. Here d is the dimension of the scheme we consider. This idea was suggested by [5], [9] and [24]. This method allows us to define the Weil-étale cohomology groups of an arithmetic scheme X satisfying the following conjecture. We fix a regular scheme X proper over Z of Krull dimension d, and we denote by Z(d) Bloch’s cycle complex. Conjecture 1.2. (Lichtenbaum) The étale motivic cohomology groups H i (Xet , Z(d)) are finitely generated for 0 ≤ i ≤ 2d. This conjecture holds for projective spaces over number rings. If X is defined over a finite field Fq , a strong version of Conjecture 1.2 is equivalent to a previous conjecture of Lichtenbaum denoted by L(X , d) in [9]. Conjecture L(X , d) is known for the class A(Fq ) of projective smooth varieties over Fq which can be constructed out of products of smooth projective curves by union, base extension

ZETA FUNCTIONS AT s = 0

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and blow ups (this includes abelian varities, unirational varieties of dimension at most 3 and Fermat hypersurfaces). In [7] the Weil-étale topos X∞,W is naturally associated to the trivial action of the topological group R on the topological space X∞ := X (C)/GR , where X (C) is given with the complex topology. We set X := (X , X∞ ). Our first main result is the following Theorem 1.3. If X satisfies Conjecture 1.2 then there exist complexes RΓW (X , Z) and RΓW,c(X , Z) such that the following properties hold. • The complex RΓW (X , Z) is contravariantly functorial. • There is a canonical morphism RΓW (X , Z) → RΓ(X∞,W , Z) (see Proposition 9.2 for a precise statement) and an exact triangle RΓW,c (X , Z) → RΓW (X , Z) → RΓ(X∞,W , Z) → RΓW,c (X , Z)[1] i (X , Z) and H i (X , Z) are finitely generated and zero • The groups HW W,c for i large. i (X , Z) form an integral model for • The Weil-étale cohomology groups HW l-adic cohomology: for any prime number l and any i ∈ Z there is a canonical isomorphism i HW (X , Z) ⊗ Zl ≃ H i (X et , Zl )

• If X has characteristic p then there is a canonical isomorphism ∼

RΓ(XW , Z) −→ RΓW (X , Z) where the left hand side is the cohomology of the Weil-étale topos [21] and the right hand side is the complex defined in this paper. Moreover, Conjecture 1.1 holds for X . • If X = Spec(OF ) is the spectrum of a (totally imaginary) number ring, then X satisfies Conjecture 1.2 and there is a canonical isomorphism ∼

τ≤3 RΓ(XW , Z) −→ RΓW (X , Z) where τ≤3 RΓ(XW , Z) is the truncation of Lichtenbaum’s complex [22] and the right hand side is the complex defined in this paper. Moreover, Conjecture 1.1 holds for X . Notice that the same formalism is used to treat flat arithmetic schemes and schemes over finite fields (see [22] Question 1 in the Introduction). To go further we need to assume the following conjecture, which is a special case of a natural refinement for arithmetic schemes of the classical conjecture of Beilinson relating motivic cohomology to Deligne cohomology. Let X be a regular, proper and flat arithmetic scheme. Assume that X is connected of dimension d. Conjecture 1.4. (Beilinson-Flach) The Beilinson regulator 2d−1−i (X/R , R(d)) H 2d−1−i (X , Q(d))R → HD

is an isomorphism for 1 ≤ i ≤ 2d − 1 and there is an exact sequence 2d−1 (X/R , R(d)) → CH 0 (XQ )∗R → 0 0 → H 2d−1 (X , Q(d))R → HD

Now we can state our second main result.

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Theorem 1.5. Assume that X satisfies Conjectures 1.2 and 1.4. • Then Statements (1), (2) and (3) of Conjecture 1.1 hold. • Statement (4) of Conjecture 1.1, i.e. the identity ords=0 ζ(X , s) = rankZ RΓW,c (X , Z) is equivalent to Soulé’s Conjecture [26] for the vanishing order of ζ(X , s) at s = 0. • (Flach-Morin) If the L-functions of the motives hi (XQ ) satisfy the expected functional equation then Statement (4) holds. • (Flach-Morin) Assume that X is smooth proper over a number ring. Then Statement (5) of Conjecture 1.1, i.e. the identity Z · λ(ζ ∗ (X , 0)) = detZ RΓW,c (X , Z) is equivalent to the Tamagawa Number Conjecture for the motive h(XQ ). This result shows that the Weil-étale point of view is compatible with the Bloch-Kato Conjecture, answering a question of Lichtenbaum (see [22] Question 2 in the Introduction). We obtain the first cases where Conjecture 1.1 is known for flat arithmetic schemes: Corollary 1.6. Conjecture 1.1 holds in the following cases: • For X = Spec(OF ) the spectrum of a number ring. • For X = PnOF the projective n-space over the ring of integers in an abelian number field F/Q. ` • For X = PnOF − Y where F/Q is abelian and Y = i=s i=1 Yi is the disjoint n union in POF of varieties Yi over finite fields Fqi lying in A(Fqi ). Acknowledgments. I am very grateful to C. Deninger for comments and advices and to M. Flach for many discussions related to Weil-étale cohomology. I would also like to thank T. Chinburg, T. Geisser and S. Lichtenbaum for comments and discussions. This work is based on ideas of S. Lichtenbaum. Contents 1. Introduction 2. The Artin-Verdier étale topology 2.1. The Artin-Verdier étale topos 2.2. Bloch’s cycle complex 3. The conjectures L(X et , d) and L(X et , d)≥0 3.1. Projective spaces over number rings 3.2. Varieties over finite fields 4. The morphism αX 4.1. The case X (R) = ∅ 4.2. The case X (R) 6= ∅ 5. Cohomology with Z-coefficients 6. Cohomology with finite coefficients 7. Relationship with Lichtenbaum’s definition over finite fields 8. Relationship with Lichtenbaum’s definition for number rings

1 5 5 5 6 6 6 8 8 12 14 17 18 21

ZETA FUNCTIONS AT s = 0

9. Cohomology with compact support ˜ 9.1. Cohomology with R-coefficients 9.2. Cohomology with Z-coefficients 10. The conjecture B(X , d) 11. The regulator map 11.1. Flat arithmetic schemes 11.2. Schemes over finite fields 12. Zeta functions at s = 0 13. Relation to Soulé’s Conjecture 14. Relation to the Tamagawa Number Conjecture 15. Proven cases 15.1. Varieties over finite fields 15.2. Number rings 15.3. Projective spaces over number rings 15.4. Some open sub-schemes of projective spaces 15.5. Cohomology of projective spaces in terms of K-theory References

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21 22 23 28 29 29 30 31 32 33 34 34 35 35 39 39 40

2. The Artin-Verdier étale topology 2.1. The Artin-Verdier étale topos. Let X be a proper, regular and connected arithmetic scheme of dimension d := dim(X ). Let X∞ := X (C)/GR be the quotient topological space of X (C) by GR , where X (C) is given with the complex topology. We consider the Artin-Verdier étale topos X et given with the open-closed decomposition of topoi ϕ : Xet → X et ← Sh(X∞ ) : u∞ where Xet is the usual étale topos of the scheme X (i.e. the category of sheaves of sets on the small étale site of X ) and Sh(X∞ ) is the category of sheaves of sets on the topological space X∞ (see [7] Section 4). For any abelian sheaf A on Xet and any n > 0 the sheaf Rn ϕ∗ A is is 2-torsion and concentrated on X (R). In particular, if X (R) = ∅ then φ∗ ≃ Rφ∗ For X (C) = ∅ one has Xet = X et . 2.2. Bloch’s cycle complex. Let Z(n) := z n (−, 2n − ∗) be Bloch’s complex (see [8], [10], [20]), which we consider as a complex of abelian sheaves on the small étale site of X . We denote by H i (X , Q(n)) := H i (XZar , Q(n)) the Zariski hypercohomology of the cycle complex. For X (R) = ∅, we still denote by Z(n) = ϕ∗ Z(n) and Q(n) = ϕ∗ Q(n) the push-forward of the cycle complex on X et , and by H i (X et , Z(n)) the étale hypercohomology of Z(n).

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3. The conjectures L(X et , d) and L(X et , d)≥0 The following conjecture is an étale version of (a special case of) the motivic Bass Conjecture (see [15] Conjecture 37). The kernel and the cokernel of the restriction map H i (X et , Z(d)) → H i (Xet , Z(d)) are both finite groups killed by 2. Here the complex Z(d) on X denotes the dualizing complex for the ArtinVerdier étale cohomology. Conjecture 3.1. L(X et , d) Let X be a proper, regular and connected arithmetic scheme of dimension d. Then the group H i (X et , Z(d)), or equivalently H i (Xet , Z(d)), is finitely generated for any i ≤ 2d. In order to define the Weil-étale cohomology we shall assume a weak version of the previous conjecture. Conjecture 3.2. L(X et , d)≥0 Let X be a proper, regular and connected arithmetic scheme of dimension d. Then the group H i (X et , Z(d)), or equivalently H i (Xet , Z(d)), is finitely generated for 0 ≤ i ≤ 2d. The following is a special case of the Beilinson-Soulé vanishing conjecture. Conjecture 3.3. (Beilinson-Soulé) For any i < 0 we have H i (X , Q(d)) = 0. Lemma 4.1 shows that H i (Xet , Z(d)) = H i (X , Q(d)) for any i < 0, hence we have the Proposition 3.4. A scheme X as above satisfies Conjecture 3.1 if and only if it satisfies both Conjecture 3.2 and Conjecture 3.3 3.1. Projective spaces over number rings. Proposition 3.5. Any number ring X = Spec(OF ) satisfies Conjecture 3.1. Proof. In view of the quasi-isomorphism of complexes of étale sheaves Z(1) ≃ Gm [−1] the result follows from the finite generation of the unit group and finiteness of the class group of a number field.  Proposition 3.6. The projective space PnOF over a number ring satisfies Conjecture 3.1. Proof. This is proven in Theorem 15.7 (1).



3.2. Varieties over finite fields. Let Y = X be a smooth projective scheme over k = Fq of dimension d = dim(Y ). The Weil-étale topos YW (see [21] and [9]) gives rise to the right arithmetic cohomology H i (YW , Z(n)). The following conjecture is due to Lichtenbaum. Conjecture 3.7. L(YW , n) The Weil-étale cohomology group H i (YW , Z(n)) is finitely generated for every i. The following conjecture is due to B. Kahn (see [16] and [9]).

ZETA FUNCTIONS AT s = 0

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Conjecture 3.8. K(YW , n) For every prime l, and any i, the natural map induces an isomorphism i H i (YW , Z(n)) ⊗ Zl ≃ Hcont (Y, Zl (n))

Consider the full subcategory A(k) of the category of smooth projective varieties over the finite field k generated by product of curves and the following operations: ` (1) If X and Y are in A(k) then X Y is in A(k). (2) If Y is in A(k) and there are morphisms c : X → Y and c′ : Y → X in the category of Chow motives, such that c′ ◦ c : X → X is multiplication by a constant, then X is in A(k). (3) If k′ /k is a finite extension and X ×k k′ is in A(k), then X is in A(k). (4) If Y is a closed subscheme of X with X and Y in A(k), then the blow-up X ′ of X is in A(k). The following result is due to T. Geisser [9]. Theorem 3.9. (Geisser) Let Y be a smooth projective variety of dimension d. - One has K(YW , n) + K(YW , d − n) ⇒ L(YW , n) ⇒ K(YW , n). - Moreover, if Y belongs to A(k) then L(YW , n) holds for n ≤ 1 and n ≥ d−1. Proposition 3.10. Let Y be a connected smooth projective scheme over a finite field of dimension d. Then we have L(YW , d) ⇔ L(Yet , d) Proof. By [9], one has an exact sequence (1) ... → H i (Yet , Z(n)) → H i (YW , Z(n)) → H i−1 (Yet , Q(n)) → H i+1 (Yet , Z(n)) → ... With rational coefficients, this exact sequence yields isomorphisms (2)

H i (YW , Z(n)) ⊗ Q ≃ H i (YW , Q(n)) ≃ H i (Yet , Q(n)) ⊕ H i−1 (Yet , Q(n)).

For n = d the following is known (see [16] proof of Corollaire 3.10). • One has H i (Yet , Z(d)) = 0 for i > 2d + 2. • There is an isomorphism H 2d+2 (Yet , Z(d)) ≃ Q/Z. • The group H 2d (Yet , Z(d)) ≃ CH d (Y ) is finitely generated of rank 1. Assume now that Conjecture L(YW , d) holds. Let us first show that H i (YW , Z(d)) is finite for i 6= 2d, 2d + 1. By Theorem 3.9, Conjecture K(YW , d) holds, i.e. we have an isomorphism i H i (YW , Z(d)) ⊗ Zl ≃ Hcont (Y, Zl (d)) i (Y, Zl (d)) is finite for any l and any i. But for i 6= 2d, 2d+1, it is known that Hcont for any l and zero for almost all l (see [16] Proof of Corollaire 3.8 for references). Hence H i (YW , Z(d)) is finite for i 6= 2d, 2d+ 1. Then (2) gives H i (Yet , Q(d)) = 0 for i < 2d, hence H i (Yet , Q(d)) = 0 for i 6= 2d. The exact sequence (1) then shows that H i (Yet , Z(d)) → H i (YW , Z(d)) is injective for i ≤ 2d + 1, hence that H i (Yet , Z(d)) is finitely generated for i ≤ 2d + 1. Assume now that Conjecture L(Yet , d) holds. By Proposition 4.3 the natural map H i (Yet , Z) → Hom(H 2d+2−i (Yet , Z(d)), Q/Z)

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is an isomorphism of abelian groups for i ≥ 1. But H i (Yet , Z) is known to be finite for i 6= 0, 2 and zero for i ≥ 2d + 2 (except for d = 0). It follows that H j (Yet , Z(d)) is finite for j 6= 2d, 2d + 2. We get H i (Yet , Q(d)) = 0 for i 6= 2d. Moreover we have H 2d (Yet , Q(d)) = CH d (Y )Q ≃ Q. Finally, one has H 2d+1 (Yet , Z(d)) = 0 since H 1 (Yet , Z) = 0 and H 2d (Yet , Z(d)) is finitely generated as mentioned above. Hence L(YW , d) follows from (1) and (2) since the map Q ≃ H 2d (Yet , Q(d)) → H 2d+2 (Yet , Z(d)) ≃ Q/Z in the sequence (1) is the obvious surjection.  Corollary 3.11. Any variety Y in A(k) satisfies L(Yet , d). Proof. This follows from Theorem 3.9 and Proposition 3.10.



4. The morphism αX Let X be a proper regular connected arithmetic scheme of dimension d. The purpose of this section is to show Theorem 4.4. We need a version of duality for arithmetic schemes with Z-coefficients which requires Conjecture 3.2. 4.1. The case X (R) = ∅. Recall that in this case we have φ∗ = Rφ∗ . Lemma 4.1. Let X be a proper, regular and connected arithmetic scheme of dimension d such that X (R) is empty. We have H i (X et , Z(d)) = ≃ = ≃

0 for i > 2d + 2 Q/Z for i = 2d + 2 0 for i = 2d + 1 H i (X , Q(d)) for i < 0

Proof. We have H i (X et , Q(d)) ≃ H i (X , Q(d)) for any i (see [8] Proposition 3.6) and H i (X , Q(d)) = 0 for any i > 2d. The exact triangle (3)

Z(d) → Q(d) → Q/Z(d)

gives i−1 H i (X et , Z(d)) ≃ H i−1 (X et , Q/Z(d)) ≃ lim −→ H (X et , Z/nZ(d))

for i ≥ 2d + 2. The result for i ≥ 2d + 2 follows from (4)

(Z/nZ, Z/nZ(d)) H i−1 (X et , Z/nZ(d)) = Exti−1 X ,Z/nZ

(5)

i ≃ ExtX (Z/nZ, Z(d))

(6)

≃ H 2d+2−i (X et , Z/nZ)D

where (5) and (6) are given by [11] Lemma 2.4 and [11] Theorem 7.8 respectively, and H 2d+2−i (X et , Z/nZ)D denotes the Pontryagin dual of the finite group H 2d+2−i (X et , Z/nZ). The exact triangle (3) and the fact that H i (X et , Q/Z(d)) = 0 for i < 0 yield H i (X et , Z(d)) ≃ H i (X et , Q(d)) for i < 0. Hence the result for i < 0 follows from H i (X et , Q(d)) ≃ H i (X , Q(d)). It remains to treat the case i = 2d + 1. Assume that X is flat over Z. Then H 2d (X , Q(d)) ≃ CH d (X ) ⊗ Q = 0

ZETA FUNCTIONS AT s = 0

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by class field theory, hence we have H 2d+1 (X et , Z(d)) ≃ H 2d (X et , Q/Z(d)) 1 D ≃ lim −→ (H (X et , Z/nZ) ) D ≃ (lim ←− Hom(π1 (X et ), Z/nZ))

b D = Hom(π1 (X et )ab , Z) = 0 since the abelian fundamental group π1 (X et )ab is finite. Assume now that X is a smooth proper scheme over a finite field. Here the map CH 2d (X ) → H 2d (Xet , Z(d)) is an isomorphism (see for example [15] Theorem 88), and this group is finitely generated of rank one, again by class field theory. The exact triangle (3) yields an exact sequence → CH 2d (X ) → CH 2d (X ) ⊗ Q → H 2d (Xet , Q/Z(d)) → H 2d+1 (Xet , Z(d)) → 0 where the central map is surjective, as can be seen from 1 ∗ H 2d (Xet , Q/Z(d)) ≃ lim −→ (H (Xet , Z/nZ) ) ≃ Q/Z

 Definition 4.2. We denote by RΓ(X et , Z(d))≥0 the truncation of the complex RΓ(X et , Z(d)) so that we have H j (X et , Z(d))≥0 = H j (X et , Z(d)) for j ≥ 0 = 0 for j < 0. Note that the Beilinson-Soulé vanishing conjecture with Q-coefficients would imply that the natural map (7)

RΓ(X et , Z(d)) → RΓ(X et , Z(d))≥0

is a quasi-isomorphism, since one has H i (X et , Z(d)) ≃ H i (X , Q(d)) for i < 0. In particular Conjecture 3.1 implies that the map (7) is a quasi-isomorphism. Lemma 4.3. Assume that X satisfies L(X et , d)≥0 . Then the natural map H i (X et , Z) → Hom(H 2d+2−i (X et , Z(d))≥0 , Q/Z) is an isomorphism of abelian groups for i ≥ 1. Proof. The map of the lemma is given by the pairing 2d+2−i (Z, Z(d)) → H 2d+2 (X et , Z(d)) ≃ Q/Z. H i (X et , Z) × ExtX

For i = 1 the result follows from Lemma 4.1, since one has H 1 (X et , Z) = Homcont (π1 (X et , p), Z) = 0 because the fundamental group π1 (X et , p) of the Artin-Verdier étale topos is profinite.

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The scheme X is normal and connected, hence H i (X et , Q) = H i (Xet , Q) = Q, 0 for i = 0 and i ≥ 1 respectively. We obtain i−1 H i (X et , Z) = H i−1 (X et , Q/Z) = lim −→ H (X et , Z/nZ)

for i ≥ 2, since Xet is quasi-compact and quasi-separated. For any positive integer n the canonical map 2d+2−i ExtX (Z/nZ, Z(d)) × H i (X et , Z/nZ) → Q/Z

is a perfect pairing of finite groups (see [11] Theorem 7.8). The short exact sequence 0 → Z → Z → Z/nZ → 0 yields a long exact sequence H j−1 (X et , Z(d)) → H j−1 (X et , Z(d)) → ExtjX (Z/nZ, Z(d)) → H j (X et , Z(d)) → H j (X et , Z(d)) We obtain a short exact sequence j 0 → H j−1 (X et , Z(d))n → ExtX (Z/nZ, Z(d)) → n H j (X et , Z(d)) → 0

for any j. By left exactness of projective limits the sequence j j j−1 0 → lim ←− n H (X et , Z(d)) ←− H (X et , Z(d))n → lim ←− ExtX (Z/nZ, Z(d)) → lim j is exact. The module lim ←− n H (X et , Z(d)) vanishes for 0 ≤ j ≤ 2d + 1 since j H j (X et , Z(d)) is assumed to be finitely generated for such j. We have lim ←− n H (X et , Z(d)) = 0 for j < 0 since H j (X et , Z(d)) is uniquely divisible for j < 0. This yields an isomorphism of profinite groups ∼

∼

j j−1 2d+2−j lim (X et , Z/nZ))D , ←− H (X et , Z(d))n → lim ←− ExtX (Z/nZ, Z(d)) → (lim −→ H where the last isomorphism follows from the duality above (and from the fact j that ExtX (Z/nZ, Z(d)) is finite for any n). This gives isomorphisms of torsion groups ∼ 2d+2−j H 2d+2−(j−1) (X et , Z) → (lim (X et , Z/nZ))DD −→ H ∼

∼

j−1 D D j−1 → (lim ←− H (X et , Z(d))n ) → (H (X et , Z(d))≥0 )

for any j ≤ 2d + 1 (note that 2d + 2 − (j − 1) ≥ 2 ⇔ j ≤ 2d + 1). The last isomorphism above follows from the fact that H j (X et , Z(d)) is finitely generated for 0 ≤ j ≤ 2d and uniquely divisible for j < 0. Hence for any i ≥ 2 the natural map H i (X et , Z) → Hom(H 2d+2−i (X et , Z(d))≥0 , Q/Z) is an isomorphism.  Theorem 4.4. Assume that X satisfies L(X et , d)≥0 . There is a canonical morphism in D: αX : RHom(RΓ(X , Q(d))≥0 , Q[−2d − 2]) → RΓ(X et , Z) functorial in X and such that H i (αX ) factors as follows Hom(H 2d+2−i (X , Q(d))≥0 , Q) ։ H i (X et , Z)div ֒→ H i (X et , Z)

ZETA FUNCTIONS AT s = 0

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where H i (X et , Z)div denotes the maximal divisible subgroup of H i (X et , Z). Proof. We set δ := 2d + 2. There is a natural map in D: RHom(Z, Z(d)) → RHom(RΓ(X et , Z), RΓ(X et , Z(d))) → RHom(RΓ(X et , Z), Q/Z[−δ]) where the second map is induced by RΓ(X et , Z(d)) → RΓ(X et , Z(d))≥δ ≃ Q/Z[−δ]. Hence we have RΓ(X et , Z(d)) → RHom(RΓ(X et , Z), Q/Z[−δ])

(8)

The complex RHom(RΓ(X et , Z), Q/Z[−δ]) is acyclic in negative degrees hence (8) induces RΓ(X et , Z(d))≥0 → RHom(RΓ(X et , Z), Q/Z[−δ]) Applying the functor RHom(−, Q/Z[−δ]) we obtain RHom(RHom(RΓ(X et , Z), Q/Z[−δ]), Q/Z[−δ]) → RHom(RΓ(X et , Z(d))≥0 , Q/Z[−δ]). Composing the previous morphism with RΓ(X et , Z) → RHom(RHom(RΓ(X et , Z), Q/Z[−δ]), Q/Z[−δ]) we get (9)

RΓ(X et , Z) → RHom(RΓ(X et , Z(d))≥0 , Q/Z[−δ]).

Consider the following diagram RHom(RΓ(X , Q(d))≥0 , Q[−δ]) O

≃ a

RHom(RΓ(X et , Z(d))≥0 ⊗ Q, Q[−δ]) b ≃



RHom(RΓ(X et , Z(d))≥0 , Q[−δ]) c

RΓ(X et , Z)

 / RHom(RΓ(X , Z(d)) , Q/Z[−δ]) et ≥0

The map a is a quasi-isomorphism since étale motivic cohomology and motivic cohomology agree with rational coefficients. The map b is clearly a quasiisomorphism. The map c is induced by the quotient morphism Q[−δ] → Q/Z[−δ]. By Proposition 4.3 the map (9) sits in an exact triangle b RΓ(X et , Z) → RHom(RΓ(X et , Z(d))≥0 , Q/Z[−δ]) → Z/Z[0] → RΓ(X et , Z)[1]

We set DX := RHom(RΓ(X , Q(d))≥0 , Q[−δ]). We obtain an exact sequence of abelian groups b HomD (DX , Z/Z[−1]) → HomD (DX , RΓ(X et , Z))

b → HomD (DX , RHom(RΓ(X et , Z(d))≥0 , Q/Z[−δ])) → HomD (DX , Z/Z[0])

12

BAPTISTE MORIN

b := lim Z/nZ. But Z/Z[i] b where Z is a bounded complex of injective abelian ←− groups for i = −1, 0 so that b b HomD (DX , Z/Z[i]) = HomK (DX , Z/Z[i])

where K is the category of cochain complexes of abelian groups modulo hob motopy. Indeed, the surjective map V ։ Z/Z, where V = Hom(Q, S1 ) is the b solenoid, shows that Z/Z is divisible. The facts that DX is acyclic in degrees < 2 (since H j (X , Q(d)) = 0 for j > 2d) b and that Z/Z[i] is concentrated in degree −i = 0, 1 imply b b HomD (DX , Z/Z[i]) = HomK (DX , Z/Z[i]) = 0.

The natural map (given by composition with (9))

HomD (DX , RΓ(X et , Z)) → HomD (DX , RHom(RΓ(X et , Z(d))≥0 , Q/Z[−δ])) is therefore an isomorphism, hence there exists a unique morphism in D αX : DX = RHom(RΓ(X , Q(d))≥0 , Q[−2d − 2]) → RΓ(X et , Z) which makes the following diagram commutative: RHom(RΓ(X , Q(d))≥0 , Q[−δ]) ggg g g g ggggg  s gggg g / RHom(RΓ(X et , Z(d))≥0 , Q/Z[−δ]) RΓ(X et , Z) ∃! αXgggggg

Here the vertical map is the morphism defined by the previous commutative diagram. The horizontal map is the morphism (9). The functorial behavior of αX is given by Theorem 5.4.  4.2. The case X (R) 6= ∅. An Artin-Verdier duality theorem in the spirit of [1] would make the previous proof valid in the case X (R) 6= ∅. Such a result is not currently available in the literature so we use cohomology with compact support in the sense of [23]. Proposition 4.5. Theorem 4.4 holds for any proper regular connected arithmetic scheme X satisfying L(X et , d)≥0 . Proof. Let S = Spec(OX (X )) and let f : X → S be the canonical map. Then X is proper over the number ring S. We denote by ˆ c (Xet , F) := RΓ ˆ c (Set , Rf∗ F) RΓ ˆ c (Set , −) is defined as in the "cohomology with compact support", where RΓ [23]. We consider the map p

RHomX (Z, Z(d)) → RHomS (Rf∗ Z, Rf∗ Z(d)) → RHomS (Rf∗ Z, Z(1)[−2d + 2]) d ˆ c (Set , Rf∗ Z), Q/Z[−2d − 2]) = RHomD (RΓ ˆ c (Xet , Z), Q/Z[−δ]) → RHomD (RΓ where p is induced by the push-forward map of [11] Corollary 7.2 (b) and d is given by 1-dimensional Artin-Verdier duality [23]. This map induces

ˆ c (Xet , Z), Q/Z[−δ]). RΓ(Xet , Z(d))≥0 → RHom(RΓ

ZETA FUNCTIONS AT s = 0

13

ˆ c (Xet , Z) is acyclic in degrees > δ. We obtain the morphism since RΓ ˆ c (Xet , Z) → RHom(RΓ(Xet , Z(d))≥0 , Q/Z[−δ]). RΓ

(10)

Then we show that ˆ i (Xet , Z) → Hom(H 2d+2−i (Xet , Z(d))≥0 , Q/Z) H c is an isomorphism of abelian groups for i 6= 0 and that the map ˆ c0 (Xet , Z) → Hom(H 2d+2 (Xet , Z(d)), Q/Z) H is injective with divisible cokernel. The same argument as in the proof of Theorem 4.4 provides us with a commutative diagram RHom(RΓ(X , Q(d))≥0 , Q[−δ]) ggg g g ggggg  s gggg g / RHom(RΓ(Xet , Z(d))≥0 , Q/Z[−δ]) ˆ c (Xet , Z) RΓ ∃! α ˜ Xggggggg

Composing α ˜ X with ˆ c (Xet , Z) := RΓ ˆ c (Set , Rf∗ Z) → RΓ(Set , Rf∗ Z) ≃ RΓ(Xet , Z) RΓ we obtain the map DX = RHom(RΓ(X , Q(d))≥0 , Q[−δ]) → RΓ(Xet , Z) Consider now the exact triangle RΓX∞ (X et , Z) → RΓ(X et , Z) → RΓ(Xet , Z) Then for any n the group HXn∞ (X et , Z) := H n (RΓX∞ (X et , Z)) is killed by 2. It follows that HomD (DX , RΓX∞ (X et , Z)) = HomD (DX , RΓX∞ (X et , Z)[1]) = 0 since DX is divisible. Hence there exists a unique map αX : RHom(RΓ(X , Q(d))≥0 , Q[−δ]) → RΓ(X et , Z) such that the diagram RHom(RΓ(X , Q(d))≥0 , Q[−δ]) hh

∃! αXhhhhhhh

RΓ(X et , Z) commutes.

hh hhhhh h h h h s h

 / RΓ(Xet , Z)



14

BAPTISTE MORIN

5. Cohomology with Z-coefficients Throughout this section X denotes a proper regular connected arithmetic scheme of dimension d satisfying L(X et , d)≥0 . Definition 5.1. There exists an exact triangle α

X RHom(RΓ(X , Q(d))≥0 , Q[−δ]) → RΓ(X et , Z) → RΓW (X , Z)

→ RHom(RΓ(X , Q(d))≥0 , Q[−δ])[1] well defined up to a unique isomorphism in D. We define i HW (X , Z) := H i (RΓW (X , Z)).

The existence of such an exact triangle follows from axiom TR1 of triangulated categories. Its uniqueness is given by Corollary 5.5 and can be stated as follows. If we have two objects RΓW (X , Z) and RΓW (X , Z)′ given with exact triangles as in Definition 5.1, then there exists a unique map f : RΓW (X , Z) → RΓW (X , Z)′ in D which yields a morphism of exact triangles RHom(RΓ(X , Q(d))≥0 , Q[−δ])

αX

/ RΓ(X , Z) et

Id



RHom(RΓ(X , Q(d))≥0 , Q[−δ])

/ RΓ(X , Z) W f

Id αX



/ RΓ(X , Z) et

/



/ RΓ(X , Z) W

/

Lemma 5.2. Assume that X (R) = ∅. There is an exact sequence i (X , Z) → HomZ (H 2d+2−(i+1) (X et , Z(d))≥0 , Z) → 0 0 → H i (X et , Z)codiv → HW

for any i ∈ Z. Proof. From the exact triangle of Defintion 5.1, we obtain a long exact sequence i ... → Hom(H δ−i (X , Q(d))≥0 , Q) → H i (X et , Z) → HW (X , Z)

→ Hom(H δ−(i+1) (X , Q(d))≥0 , Q) → H i+1 (X et , Z) → ... which yields a short exact sequence (11)

i 0 → CokerH i (αX ) → HW (X , Z) → KerH i+1 (αX ) → 0

Assume for the moment that X (R) = ∅. For i ≥ 1, the morphism H i (αX ) : Hom(H δ−i (X , Q(d))≥0 , Q) → H i (X et , Z) is the following composite map: Hom(H δ−i (X , Q(d))≥0 , Q) ≃ Hom(H δ−i (X et , Z(d))≥0 , Q) → → Hom(H δ−i (X et , Z(d))≥0 , Q/Z) ≃ H i (X et , Z) Since H j (X et , Z(d))≥0 is assumed to be finitely generated, the image of the morphism H i (αX ) is the maximal divisible subgroup of H i (X et , Z), which we denote by H i (X et , Z)div . The same is true for i ≤ 0 since one has H 0 (αX ) : 0 = Hom(H δ (X , Q(d)), Q) → H i (X et , Z) = Z and H δ−i (X , Q(d)) = H i (X et , Z) = 0 for i < 0.

ZETA FUNCTIONS AT s = 0

15

We obtain CokerH i (αX ) = H i (X et , Z)/H i (X et , Z)div =: H i (X et , Z)codiv for any i ∈ Z. The same is true for X (R) 6= ∅. The kernel of H i (αX ) is given by the following commutative diagram with exact rows: 0

/ Hom(H δ−i (X , Q(d))≥0 , Q) O

/ KerH i (αX )

/ Hom(H δ−i (X et , Z(d))≥0 , Q/Z) O =

≃

0

/ Hom(H δ−i (X et , Z(d))≥0 , Z)

/ Hom(H δ−i (X et , Z(d))≥0 , Q)

/ Hom(H δ−i (X et , Z(d))≥0 , Q/Z)

The exact sequence (11) therefore takes the form i 0 → H i (X et , Z)codiv → HW (X , Z) → HomZ (H 2d+2−(i+1) (X et , Z(d)), Z) → 0

 i (X , Z) is finitely generated for any i ∈ Z, and Proposition 5.3. The group HW i (X , Z) = 0 for i < 0, H 0 (X , Z) = Z and H i (X , Z) = 0 for i large. one has HW W W

Proof. Assume that X (R) = ∅ and consider the exact sequence i 0 → H i (X et , Z)codiv → HW (X , Z) → HomZ (H 2d+2−(i+1) (X et , Z(d))≥0 , Z) → 0

One has H i (X et , Z) = 0 for i < 0, H 0 (X et , Z) = Z and H j (X et , Z(d)) is torsion for j > 2d. The result for i ≤ 0 follows. Let i ≥ 1. The isomorphism H i (X et , Z) ≃ Hom(H 2d+2−i (X et , Z(d))≥0 , Q/Z) and the fact that H 2d+2−i (X et , Z(d))≥0 is assumed to be finitely generated imply that H i (X et , Z)codiv is finite and that HomZ (H 2d+2−(i+1) (X et , Z(d))≥0 , Z) is finitely generated. For X (R) 6= ∅ we have an exact sequence i (X , Z) → KerH i+1 (αX ) → 0 0 → H i (X et , Z)codiv → HW

where KerH i+1 (αX ) is a lattice in Hom(H 2d+2−(i+1) (X , Q(d))≥0 , Q). It follows i (X , Z) is finitely generated for any i and vanishes for i < 0. Moreover that HW 0 (X , Z) = Z. For i large we have H 2d+2−(i+1) (X , Q(d)) one has HW ≥0 = 0 and i i (X , Z) = 0 for i large.  H (X et , Z) = 0. Hence HW Theorem 5.4. Let f : X → Y be a morphism of relative dimension c between proper regular connected arithmetic schemes, such that L(X et , dX )≥0 and L(Y et , dY )≥0 hold. We choose complexes RΓW (X , Z) and RΓW (Y, Z) as in Definition 5.1. Assume that either c ≥ 0 or L(X et , dX ) holds. Then there exists a unique map in D RΓW (Y, Z) → RΓW (X , Z) which sits in a morphism of exact triangles RHom(RΓ(X , Q(dX ))≥0 , Q[−δX ]) O

/ RΓ(X , Z) O et ∗ fet

RHom(RΓ(Y, Q(dY ))≥0 , Q[−δY ])

/ RΓ(Y , Z) et

/ RΓ (X , Z) W O ∗ fW

/

/ RΓ (Y, Z) W

/

16

BAPTISTE MORIN

Proof. Let X and Y be connected, proper and regular arithmetic schemes of dimension dX and dY respectively. We choose complexes RΓW (X , Z) and RΓW (Y, Z) as in Def. 5.1. Let f : X → Y be a morphism of relative dimension c = dX − dY . The morphism f is proper and the map z n (X , ∗) → z n−c (Y, ∗) induces a morphism f∗ Q(dX ) → Q(dY ))[−2c] of complexes of abelian Zariski sheaves on Y. We need to see that f∗ Q(dX ) ≃ Rf∗ Q(dX )

(12)

Localizing over the base, it is enough to show this fact for f a proper map over a discrete valuation ring. By [20], higher Chow groups satisfy localization for schemes of finite type over a discrete valuation ring, hence the Mayer-Vietoris property. It follows that, over a discrete valuation ring, the hypercohomology of the cycle complex coincides with its cohomology (as a complex of abelian group). This yields (12) and a morphism of complexes RΓ(X , Q(dX )) ≃ RΓ(Y, f∗ Q(dX )) → RΓ(Y, Q(dY ))[−2c] If c ≥ 0 this induces a morphism RΓ(X , Q(dX ))≥0 → RΓ(Y, Q(dY ))≥0 [−2c] If c < 0 then L(X et , dX ) holds by assumption, and we consider the map ∼

RΓ(X , Q(dX ))≥0 ← RΓ(X , Q(dX )) → RΓ(Y, Q(dY ))[−2c] → RΓ(Y, Q(dY ))≥0 [−2c] where the first isomorphism is given by Proposition 3.4. In both cases, we get a morphism RHom(RΓ(Y, Q(dY ))≥0 , Q[−δY ]) → RHom(RΓ(X , Q(dX ))≥0 , Q[−δX ]) such that the following square is commutative: RHom(RΓ(Y, Q(dY ))≥0 , Q[−δY ]) 

RHom(RΓ(X , Q(dX ))≥0 , Q[−δX ])

αY

αX

/ RΓ(Y , Z) et  / RΓ(X , Z) et

Hence there exists a morphism ∗ fW : RΓW (Y, Z) → RΓW (X , Z) ∗ inducing an morphism of exact triangles. We claim that such a morphism fW is unique. In order to ease the notations, we set

DX := RHom(RΓ(X , Q(dX ))≥0 , Q[−δX ]) and DY := RHom(RΓ(Y, Q(dY ))≥0 , Q[−δY ]). The complexes RΓW (X , Z) and RΓW (Y, Z) are both perfect complexes of abelian groups, since they are bounded complexes with finitely generated cohomology groups. Applying the functor HomD (−, RΓW (X , Z)) to the exact triangle DY → RΓ(Y et , Z) → RΓW (Y, Z) → DY [1]

ZETA FUNCTIONS AT s = 0

17

we obtain an exact sequence of abelian groups: HomD (DY [1], RΓW (X , Z)) → HomD (RΓW (Y, Z), RΓW (X , Z)) → HomD (RΓ(Y et , Z), RΓW (X , Z)) On the one hand, HomD (DY [1], RΓW (X , Z)) is uniquely divisible since DY [1] is a complex of Q-vector spaces. On the other hand, the abelian group HomD (RΓW (Y, Z), RΓW (X , Z)) is finitely generated as it follows from the spectral sequence Y q+i i (X , Z)) ⇒ H p+q (RHom(RΓW (X , Z), RΓW (X , Z))) Extp (HW (Y, Z), HW i∈Z

since RΓW (X , Z) and RΓW (Y, Z) are both perfect. Hence the morphism HomD (RΓW (Y, Z), RΓW (X , Z)) → HomD (RΓ(Y et , Z), RΓW (X , Z)) ∗ sitting in the is injective, which implies the uniqueness of the morphism fW commutative square below.

RΓ(Y et , Z)

/ RΓ (Y, Z) W ∗ fW

∗ fet



RΓ(X et , Z)

 / RΓ (X , Z) W

 Corollary 5.5. If X satisfies L(X et , d)≥0 then RΓW (X , Z) is well defined up to a unique isomorphism in D. Proof. Let RΓW (X , Z) and RΓW (X , Z)′ be two complexes as above. By Theorem 5.4, Id : X → X induces a unique isomorphism in D RΓW (X , Z) ≃ RΓW (X , Z)′  6. Cohomology with finite coefficients Again X denotes a proper regular connected arithmetic scheme of dimension d satisfying L(X et , d)≥0 . Theorem 6.1. For any positive integer n the given map between étale and Weil-étale cohomology induces an isomorphism RΓW (X , Z) ⊗L Z Z/nZ ≃ RΓ(X et , Z/nZ) Proof. Consider the exact triangle RHom(RΓ(X , Q(d)), Q[−δ]) → RΓ(X et , Z) → RΓW (X , Z) → ...

18

BAPTISTE MORIN

Taking derived tensor product − ⊗L Z Z/nZ we obtain the following diagram RHom(RΓ(X , Q(dX )), Q[−δ])

/ RΓ (X , Z) W

/ RΓ(X , Z) et

n

n



/

n





RHom(RΓ(X , Q(dX )), Q[−δ])

/ RΓ(X , Z) et

/ RΓ (X , Z) W



 / RΓ(X , Z/nZ) et

 / RΓW (X , Z) ⊗L Z/nZ Z

0

/

The upper and middle rows are both exact triangles. The colons are all exact as well: for the left colon we simply observe that multiplication by n gives an isomorphism of the complex of Q-vector spaces RHom(RΓ(X , Q(dX )), Q[−δ]); for the central colon we consider the exact sequence of sheaves 0 → Z → Z → Z/nZ → 0 on X et . It follows that the bottom row is exact, hence the map L RΓ(X et , Z/nZ) ≃ RΓ(X et , Z) ⊗L Z Z/nZ −→ RΓW (X , Z) ⊗Z Z/nZ

is an isomorphism in D.



Corollary 6.2. For any prime number l and any i ∈ Z there is natural isomorphism i i (X et , Zl ) HW (X , Z) ⊗ Zl ≃ Hcont Proof. The previous result yields an exact sequence i+1 i (X , Z) → 0 0 → HW (X , Z)ln → H i (X et , Z/ln Z) → ln HW

By left exactness of projective limits we get i i+1 i n n n 0 → lim ←− H (X et , Z/l Z) → lim ←− HW (X , Z)l → lim ←− l HW (X , Z) i+1 i+1 n But lim ←− l HW (X , Z) = 0 since HW (X , Z) is finitely generated. Moreover, we have i i n Hcont (X et , Zl ) ≃ lim ←− H (X et , Z/l Z).

since H i (X et , Z/ln Z) is finite.  7. Relationship with Lichtenbaum’s definition over finite fields Let Y be a scheme of finite type over a finite field k. We denote by Gk and Wk the Galois group and the Weil group of k respectively. The (small) Weil-étale sm is the category of W -equivariant sheaves of sets on the étale site of topos YW k ¯ The big Weil-étale topos is defined as the fiber product Y¯ = Y ×k k. YW := Yet × sm BG k

Spec(Z)et

Spec(Z)W ≃ Yet ×BGsm BWk k

(resp. BWk ) denotes the small classifying topos of Gk (resp. the where big classifying topos of Wk ). By [9] one has an exact triangle in the derived category of Ab(Yet ) Z → Rγ∗ Z → Q[−1] → Z[1]

/

ZETA FUNCTIONS AT s = 0

19

where γ : YW → Yet is the first projection. Applying RΓ(Yet , −) and rotating, we get RΓ(Yet , Q[−2]) → RΓ(Yet , Z) → RΓ(YW , Z) → RΓ(Yet , Q[−2])[1] Proposition 7.1. Let Y be a d-dimensional connected projective smooth scheme over k satisfying L(Yet , d)≥0 . Then there is a commutative diagram in D: / RΓ(Yet , Z)

RΓ(Yet , Q[−2])

Id



RHom(Γ(Y, Q(d))≥0 , Q[−2d − 2])

αY

 / RΓ(Yet , Z)

where the vertical maps are isomorphisms. Proof. For simplicity we assume that Y is geometrically connected over k. One has H 2d (Y, Q(d)) = CH d (Y )Q = Q and H i (Y, Q(d)) = 0 for i > 2d. The morphism RHomY (Q, Q(d)) → RHom(RΓ(Y, Q), RΓ(Y, Q(d))) → RHom(RΓ(Y, Q), Q[−2d]) induces a morphism (13)

RΓ(Yet , Q) ≃ Γ(Y, Q) → RHom(Γ(Y, Q(d))≥0 , Q[−2d])

It follows from Conjecture L(Yet , d)≥0 , Lemma 4.3 and the fact that H i (Yet , Z) is finite for i 6= 0, 2, that the group H i (Yet , Z(d))≥0 is finite for i 6= 2d, 2d + 2. This implies that the morphism (13) is a quasi-isomorphism. It remains to check the commutativity of the diagram. The complex DY := RHom(Γ(Y, Q(d))≥0 , Q[−2d − 2]) ≃ RΓ(Yet , Q[−2]) is concentrated in degree 2, in particular acyclic in degrees > 2. It follows that the horizontal maps in the diagram of the proposition uniquely factor through the truncated complex RΓ(Y et , Z)≤2 . It is therefore enough to show that the square / RΓ(Yet , Z)≤2 RΓ(Yet , Q[−2]) Id



RHom(Γ(Y, Q(d))≥0 , Q[−2d − 2])

αY

 / RΓ(Yet , Z)≤2

commutes. The exact triangle Z[0] → RΓ(Yet , Z)≤2 → π1 (Yet )D [−2] → Z[1] induces an exact sequence of abelian groups HomD (DY , Z[0]) → HomD (DY , τ≤2 RΓ(Yet , Z)) → HomD (DY , π1 (Yet )D [−2]) which shows that the map HomD (DY , τ≤2 RΓ(Yet , Z)) → HomD (DY , π1 (Yet )D [−2]) is injective. Indeed, Z[0] ≃ [Q → Q/Z] has an injective resolution acyclic in degrees ≥ 2, and DY is concentrated in degree 2. In view of the quasiisomorphism DY ≃ Hom(H 2d (Y, Q(d)), Q)[−2] ≃ H 0 (Y, Q)[−2]

20

BAPTISTE MORIN

one is reduced to show the commutativity of the following square (of abelian groups) H 0 (Y

d0,1 2

et , Q)

/ H 2 (Yet , Z) Id



Hom(H 2d (Y, Q(d)), Q)

 / H 2 (Yet , Z)

H 2 (αY )

By construction the map H 2 (αY ) is the following composition ∼

H 2 (αY ) : Hom(H 2d (Y, Q(d)), Q) ≃ Hom(CH d (Y ), Q) → CH d (Y )D ← π1 (Yet )D ≃ H 2 (Yet , Z) ∼

where the isomorphism CH d (Y )D ← π1 (Yet )D is the dual of the reciprocity law of class field theory (which yields an injective morphism with dense image CH d (Y ) → π1 (Yet )ab ). The upper horizontal map is the differential d0,1 2 in the spectral sequence H i (Yet , Rj (γ∗ )Z) ⇒ H i+j (YW , Z). There is a canonical isomorphism ′ H 0 (Yet , R1 (γ∗ )Z) = lim −→k′ /k Hom(Wk , Z) = Hom(Wk , Q),

k′ /k runs over the finite extensions of k, as it follows from the isomorphism of pro-discrete groups ′ π1 (YW , p) ≃ π1 (Yet′ , p) ×Gk Wk . which valid for any Y ′ connected étale over Y . Then the left vertical map in the square above is the map deg∗ : Hom(Wk , Q) → Hom(CH d (Y ), Q) induced by the degree map deg : CH d (Y ) → Z.F ≃ Wk and the differential d0,1 2 is the following map: Hom(Wk , Q) = Hom(π1 (YW , p), Q) → π1 (YW , p)D ≃ π1 (Yet , p)D One is therefore reduced to observe that the square Hom(Wk , Q)

d0,1 2

/ π1 (Yet )D

deg∗

Id



Hom(CH d (Y ), Q)

 / π (Y )D 1 et

H 2 (αY )

is commutative.  Theorem 7.2. If Y satisfies L(Yet , d)≥0 then there is a natural isomorphism in D ∼ RΓ(YW , Z) −→ RΓW (Y, Z) where the left hand side is the cohomology of the Weil-étale topos and the right hand side is the complex defined in this paper. Proof. This follows immediately from the previous proposition.



ZETA FUNCTIONS AT s = 0

21

8. Relationship with Lichtenbaum’s definition for number rings In this section we consider a totally imaginary number field F and let X = Spec(OF ). Recall that L(X et , Z(1)) holds, hence the complex RΓW (X , Z) is well defined. Theorem 8.1. There is a quasi-isomorphism ∼

RΓ(X W , Z)≤3 −→ RΓW (X , Z) where the left hand side is the truncation of Lichtenbaum’s complex [22] and the right hand side is the complex defined in this paper. Proof. By [24] Theorem 9.5, we have a quasi-isomorphism ∼

RΓ(X et , RZ) −→ RΓ(X W , Z) inducing ∼

RΓ(X et , RW Z) −→ RΓ(X W , Z)≤3 where RZ is the complex defined in [24] Theorem 8.5 and RW Z := RZ≤2 . We have an exact triangle 2 Z[0] → RW Z → RW Z[−2]

in the derived category of étale sheaves on X . Rotating and applying RΓ(X et , −) we get an exact triangle 2 Z)[−3] → RΓ(X et , Z) → RΓ(X W , Z)≤3 RΓ(X et , RW

We have canonical quasi-isomorphisms 2 RΓ(X et , RW Z)[−3] ≃ HomZ (OF× , Q)[−3] ≃ RHom(RΓ(X , Q(1), Q[−4]) 2 Z)[−3] → RΓ(X , Z) in the triangle It follows that the morphism RΓ(X et , RW et above is determined by the induced map 2 Z) → H 3 (X et , Z) = HomZ (OF× , Q/Z) HomZ (OF× , Q) = H 0 (X et , RW

and so is the morphism αX . In both cases this map is the obvious one. Hence the square 2 Z)[−3] RΓ(X et , RW ≃

/ RΓ(X , Z) et Id



RHom(RΓ(X , Q(1), Q[−4])

 / RΓ(X , Z) et

is commutative and the result follows.



9. Cohomology with compact support We recall below the definition given in [7] of the Weil-étale topos together ˜ with its cohomology with R-coefficients. Then we define Weil-étale cohomology with compact support and Z-coefficients.

22

BAPTISTE MORIN

˜ 9.1. Cohomology with R-coefficients. Let X be any proper regular connected arithmetic scheme. The Weil-étale topos is defined as a 2-fiber product of topoi X W := X et × Spec(Z)W Spec(Z)et There is a canonical morphism f : X W → Spec(Z)W → BR where BR is Grothendieck’s classifying topos of the topological group R (see [14] or [7]). Consider the sheaf yR on BR represented by R with the standard topology and trivial R-action. Then one defines the sheaf ˜ := f∗ (yR) R on X W . By [7], the following diagram consists of two pull-back squares of topoi, and the rows give open-closed decompositions: XW 

Xet

φ

ϕ

/X o W  /X o et

i∞

u∞

X∞,W 

Sh(X∞ )

Here the map X W := X et ×

Spec(Z)W −→ X et Spec(Z)et is the first projection, Sh(X∞ ) is the category of sheaves on the space X∞ and X∞,W = BR × Sh(X∞ ) where the product is taken over the final topos Sets. As shown in [7], the topos ˜ X W has the right R-cohomology with and without compact supports. We have ˜ := RΓ(X W , R) ˜ ≃ RΓ(BR , R) ˜ ≃ R[−1] ⊕ R RΓW (X , R) Concerning the cohomology with compact support, one has ˜ := RΓ(XW , φ! R) ˜ ≃ RΓ(Xet , ϕ! R)[−1] ⊕ RΓ(Xet , ϕ! R) (14) RΓW,c (X , R) Finally, the complex RΓ(Xet , ϕ! R) is quasi-isomorphic to Cone(R[0] → RΓ(X∞ , R))[−1]. Proposition 9.1. Cup product with the fundamental class θ = IdR ∈ H 1 (BR , R) yields a morphism ˜ → RΓW,c (X , R)[1] ˜ ∪θ : RΓW,c (X , R) such that the sequence i−1 ˜ → ˜ → H i+1 (X , R) ˜ → H i (X , R) (X , R) ... → HW,c W,c W,c

is an acyclic complex of finite dimensional R-vector spaces. ˜ Proof. In view of RΓW,c (X , R)[1] = RΓ(Xet , ϕ! R) ⊕ RΓ(Xet , ϕ! R)[1] the map ∪θ is given by projection and inclusion ˜ ։ RΓ(Xet , ϕ! R) ֒→ RΓW,c(X , R)[1]. ˜ RΓW,c (X , R) 

ZETA FUNCTIONS AT s = 0

23

9.2. Cohomology with Z-coefficients. In the remaining part of this section, X denotes a proper regular connected arithmetic scheme of dimension d satisfying L(X et , d)≥0 . If X has characteristic p then we set RΓW,c(X , Z) := i (X , Z) := RΓW (X , Z) and the cohomology with compact support is defined as HW,c i (X , Z). The case when X is flat over Z is the case of interest. HW The closed embedding u∞ : Sh(X∞ ) → Xet induces a morphism of complexes u∗∞ : RΓ(X et , Z) → RΓ(X∞ , Z) where RΓ(X et , Z) and RΓ(X∞ , Z) are defined using Godement resolutions (which in turn are defined by a conservative system of points of the topos X et ). Proposition 9.2. There exists a unique morphism i∗∞ : RΓW (X , Z) → RΓ(X∞,W , Z) which makes the following diagram commutative RHom(RΓ(X , Q(d))≥0 , Q[−δ])

/ RΓ(X , Z) et





/ RΓ (X , Z) W

u∗∞

/ RΓ(X∞ , Z)

0

/

∃ ! i∗∞

 / RΓ(X∞,W , Z) /

Proof. Recall from [7] that the second projection X∞,W := BR × Sh(X∞ ) → Sh(X∞ ) = X∞,et ∼

induces a quasi-isomorphism RΓ(X∞ , Z) −→ RΓ(X∞,W , Z). Hence the existence of the map i∗∞ will follow (Axiom TR3 of triangulated categories) from the fact that the map (15)

β : RHom(RΓ(X , Q(d))≥0 , Q[−δ]) → RΓ(X et , Z) → RΓ(X∞ , Z)

is the zero map. Again, we set DX = RHom(RΓ(X , Q(d))≥0 , Q[−δ]) for brevity. Then the uniqueness of i∗∞ follows from the exact sequence 0

HomD (DX [1], RΓ(X∞,W , Z)) → HomD (RΓW (X , Z), RΓ(X∞,W , Z)) → HomD (RΓ(X et , Z), RΓ(X∞,W , Z)) since HomD (DX [1], RΓ(X∞,W , Z)) is divisible while HomD (RΓW (X , Z), RΓ(X∞,W , Z)) is finitely generated. It remains to show that the morphism (15) is indeed trivial. Since DX is a complex of Q-vector spaces acyclic in degrees < 2, one can choose a (noncanonical) isomorphism M DX ≈ H k (DX )[−k] k≥2

Then β is identified with the collection of maps β ≈ (βk )k≥2 in D, with M βk : H k (DX )[−k] → H k (DX )[−k] ≈ DX → RΓ(X∞ , Z) k≥2

It is enough to show that βk = 0 for k ≥ 2. We fix such a k, and we consider the spectral sequence E2p,q = Extp (H k DX , H q+k (X∞ , Z)) ⇒ H p+q (RHom(H k (DX )[−k], RΓ(X∞ , Z)))

24

BAPTISTE MORIN

The group H k DX is uniquely divisible and H q+k (X∞ , Z) is finitely generated (hence has an injective resolution of length 1), so that Extp (H k (DX ), H q+k (X∞ , Z)) = 0 for p 6= 1. Hence the spectral sequence above degenerates and gives a canonical isomorphism HomD (H k DX [−k], RΓ(X∞ , Z)) ≃ Ext1 (H k DX , H k−1 (X∞ , Z)) Moreover, the long exact sequence for Ext∗ (H k DX , −) yields Ext1 (H k DX , H k−1 (X∞ , Z)) ≃ Ext1 (H k DX , H k−1 (X∞ , Z)cotor ) since the maximal torsion subgroup of H k−1 (X∞ , Z) is finite and H k DX is uniquely divisible. The short exact sequence 0 → H k−1 (X∞ , Z)cotor → H k−1 (X∞ , Q) → H k−1 (X∞ , Q/Z)div → 0 is an injective resolution of the Z-module H k−1 (X∞ , Z)cotor . This yields an exact sequence 0 → Hom(H k DX , H k−1 (X∞ , Q)) → Hom(H k DX , H k−1 (X∞ , Q/Z)div ) → Ext1 (H k DX , H k−1 (X∞ , Z)cotor ) → 0 Let us now define a natural lifting β˜k ∈ Hom(H k DX , H k−1 (X∞ , Q/Z)div ) of βk ∈ Ext1 (H k DX , H k−1 (X∞ , Z)cotor ) and show that this lifting β˜k is already zero. One can assume k ≥ 2. Recall that H k DX = Hom(H δ−k (X , Q(d))≥0 , Q). We have the following commutative diagram: Hom(H δ−k (X , Q(d))≥0 , Q) TTTT k TTH k) TTT(α TTTT TT*  ≃ k−1 / H k (X , Z) H (X et , Q/Z) et

H k−1 (X

H k−1 (X



Q, et , Q/Z)



Q, et , Q/Z)

≃

 / H k (X Q, et , Z)

≃

 / H k (X Q, et , Z)

≃

H k−1 (X (C), Q)



/ H k−1 (X (C), Q/Z)

 / H k (X (C), Z)

To make life easier, we assume that X (R) = ∅, so that H k−1 (X∞ , Q/Z) = H k−1 (X (C), Q/Z)GR The morphism given by the central colon of the diagram above factors through H k−1 (X∞ , Q/Z)div ⊆ H k−1 (X∞ , Q/Z) ⊆ H k−1 (X (C), Q/Z) and yields the desired lifting β˜k : Hom(H δ−k (X , Q(d)), Q) → H k−1 (X∞ , Q/Z)div

ZETA FUNCTIONS AT s = 0

25

The map H k−1 (XQ, et , Q/Z) → H k−1 (XQ, et , Q/Z) factors through H k−1 (XQ, et , Q/Z)GQ hence so does i ◦ β˜k : Hom(H δ−k (X , Q(d)), Q) → H k−1 (X∞ , Q/Z)div ⊂ H k−1 (X (C), Q/Z) where i is the obvious injection. In order to show that β˜k = 0 it is therefore enough to show that M (H k−1 (XQ, et , Q/Z)GQ )div = (H k−1 (XQ, et , Ql /Zl )GQ )div = 0. l

Let l be a fixed prime number. Let U ⊆ Spec(Z) on which l is invertible and such that XU → U is smooth. Let p ∈ U . By smooth and proper base change we have: H k−1 (XQ, et , Ql /Zl )Ip ≃ H k−1 (XFp , et , Ql /Zl ) Recall that H k−1 (XFp , et , Zl ) is a finitely generated Zl -module. We have an exact sequence 0 → H k−1 (XFp , et , Zl )cotor → H k−1 (XFp , et , Ql ) → H k−1 (XFp , et , Ql /Zl )div → 0 We get 0 → (H k−1 (XFp , et , Zl )cotor )GFp → H k−1 (XFp , et , Ql )GFp → (H k−1 (XFp , et , Ql /Zl )div )GFp → H 1 (GFp , H k−1 (XFp , et , Zl )cotor ) Again, H 1 (GFp , H k−1 (XFp , et , Zl )cotor ) is a finitely generated Zl -module, hence we get a surjective map H k−1 (XFp , et , Ql )GFp → ((H k−1 (XFp , et , Ql /Zl )div )GFp )div → 0 Note that ((H k−1 (XFp , et , Ql /Zl )div )GFp )div = (H k−1 (XFp , et , Ql /Zl )GFp )div By the Weil Conjectures, H k−1 (XFp , et , Ql ) is pure of weight k − 1 > 0. Hence there is no non-trivial element in H k−1 (XFp , et , Ql ) fixed by the Frobenius. This shows that (H k−1 (XFp , et , Ql /Zl )GFp )div = H k−1 (XFp , et , Ql )GFp = 0 hence that (H k−1 (XQ, et , Ql /Zl )GQp )div = ((H k−1 (XQ, et , Ql /Zl )Ip )GFp )div ≃ (H k−1 (XFp , et , Ql /Zl )GFp )div = 0 A fortiori, one has (H k−1 (XQ, et , Ql /Zl )GQ )div = 0 and the result follows. 

26

BAPTISTE MORIN

Definition 9.3. There exists an object RΓW,c (X , Z), well defined up to isomorphism in D, endowed with an exact triangle (16)

i∗

∞ RΓW,c (X , Z) → RΓW (X , Z) → RΓ(X∞,W , Z) → RΓW,c (X , Z)[1]

The determinant detZ RΓW,c (X , Z) is well defined up to a canonical isomorphism. The cohomology with compact support is defined (up to isomorphism only) as follows: i HW,c (X , Z) := H i (RΓW,c(Z)) Remark 9.4. There should be canonical lifting α ˜ X : DX → RΓc (Xet , Z) of the map αX . This would make the complex RΓW,c(X , Z) canonically defined. To see that detZ RΓW,c(X , Z) is indeed well defined, consider another object RΓW,c(X , Z)′ of D endowed with an exact triangle (16). There exists a (nonunique) morphism u : RΓW,c (X , Z) → RΓW,c (X , Z)′ lying in a morphism of exact triangles RΓ(X∞,W , Z)[−1]

/ RΓW,c (X , Z)

/ RΓ (X , Z) W

∃u ≃

Id



RΓ(X∞,W , Z)[−1]

/ RΓ(X∞,W , Z)

Id

Id





/ RΓW,c(X , Z)′

/ RΓ(X , Z)

 / RΓ(X∞,W , Z)

The map u induces ∼

detZ (u) : detZ RΓW,c (X , Z) −→ detZ RΓW,c(X , Z)′ By [17] p. 43 Corollary 2, detR (u) does not depend on the choice of u, since it coincides with the following canonical isomorphism detZ RΓW,c (X , Z) ≃ detZ RΓW (X , Z) ⊗ det−1 Z RΓ(X∞,W , Z) ≃ detZ RΓW,c (X , Z)′ Given a complex of abelian groups C we write CR for C ⊗ R. Proposition 9.5. There is a canonical and functorial direct sum decomposition in D: RΓW (X , Z)R ≃ RΓ(X et , R) ⊕ RHom(RΓ(X , Q(d))≥0 , R[−δ])[1] such that the following square commutes: RΓW (X , Z)R

i∗∞ ⊗R

/ RΓ(X∞,W , Z)R ≃

≃



RΓ(X et , R) ⊕ RHom(RΓ(X , Q(d))≥0 , R[−δ])[1]

 / RΓ(X∞ , R)

Proof. Applying (−) ⊗ R to the exact triangle of Definition 5.1, we obtain an exact triangle RHom(RΓ(X , Q(d))≥0 , R[−δ]) → RΓ(X et , R) → RΓW (X , Z)R →

ZETA FUNCTIONS AT s = 0

27

But the map RHom(RΓ(X , Q(d))≥0 , R[−δ]) → RΓ(X et , R) is trivial since RΓ(X et , R) ≃ R[0] is injective and RHom(RΓ(X , Q(d))≥0 , R[−δ]) is acyclic in degrees ≤ 1. This shows the existence of the direct sum decomposition. We write DR := RHom(RΓ(X , Q(d))≥0 , R[−δ]) and RΓ(X et , R) ≃ R[0] for brevity. The exact sequence HomD (DR [1], R[0]) → HomD (RΓW (X , Z), R[0]) → HomD (R[0], R[0]) → HomD (DR , R[0]) ∼

yields an isomorphism HomD (RΓW (X , Z), R[0]) → HomD (R[0], R[0]). Hence there exists a unique map sX : RΓW (X , Z)R → R[0] such that sX ◦ γX∗ = IdR[0] where γX∗ : R[0] → RΓW (X , Z) is the given map. The functorial behavior of sX follows from the fact that it is the unique map such that sX ◦ γX∗ = IdR[0] . The direct sum decomposition is therefore canonical and functorial. The commutativity of the square follows from Proposition 9.2.  ˜ Corollary 9.6. There is a canonical map RΓW (X , Z) → RΓ(X W , R). Proof. For a scheme of characteristic p this map is given by Theorem 7.2. For flat X , it is given by RΓW (X , Z) → RΓW (X , Z)R ≃ RΓ(X et , R) ⊕ RHom(RΓ(X , Q(d))≥0 , R[−δ])[1] ˜ → RΓ(X et , R) → RΓ(X W , R).  Proposition 9.7. There is a (non-canonical) direct sum decomposition (17)

RΓW,c (X , Z)R ≃ RΓc (Xet , R) ⊕ RHom(RΓ(X , Q(d))≥0 , R[−δ])[1]

inducing a canonical isomorphism detR RΓW,c (X , Z)R ≃ detR RΓc (Xet , R) ⊗ det−1 R RHom(RΓ(X , Q(d))≥0 , R[−δ]) Proof. Again we write DR := RHom(RΓ(X , Q(d))≥0 , R[−δ]) and RΓ(Xet , R) ≃ R[0] for brevity (we can assume that X is connected). Consider the morphism of exact triangles RΓW c (X , Z)R

/ RΓ (X , Z) W R

i∗∞ ⊗R

≃

∃u



RΓc (X et , R) ⊕ DR [1]

/ RΓ(X∞,W , Z)R /

≃



/ RΓ(X , R) ⊕ D [1] et R

 / RΓ(X∞ , R) /

Here all the maps but the isomorphism u are canonical. The non-canonical direct sum decomposition (17) follows. A choice of such an isomorphism u induces ∼

detR (u) : detR RΓW,c (X , Z)R −→ detR (RΓc (Xet , R) ⊕ DR [1])

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BAPTISTE MORIN

But detR (u) coincides by [17] with the following (canonical) isomorphism detR RΓW,c (X , Z)R ≃ detR RΓW (X , Z)R ⊗ det−1 R RΓ(X∞,W , Z)R ≃ detR (RΓ(X et , R) ⊕ DR [1]) ⊗ det−1 R RΓ(X et , R) ≃ detR (RΓc (Xet , R) ⊕ DR [1]) hence does not depend on the choice of u.



10. The conjecture B(X , d) Let X be a proper flat regular connected scheme of dimension d with generic fibre X = XQ . The "integral part in the motivic cohomology" H 2d−1−i (X/Z , Q(d)) is defined as the image of the morphism H 2d−1−i (X , Q(d)) → H 2d−1−i (X, Q(d)). p Let HD (X/R , R(q)) denote the real Deligne cohomology and let 2d−1−i 2d−1−i (X/R , R(d)) (X/Z , Q(d))R → HD ρi∞ : HM

be the Beilinson regulator. Conjecture 10.1. (Beilinson) The map ρi∞ is an isomorphism for i ≥ 1 and there is an exact sequence ρ0

∞ 2d−1 2d−1 (X/R , R(d)) → CH 0 (X)∗R → 0 → HD (X/Z , Q(d))R −− 0 → HM

for i = 0. Conjecture 10.2. (Flach) The natural map H 2d−1−i (X , Q(d))R → H 2d−1−i (X, Q(d))R is injective for i ≥ 0. We consider the composite map 2d−1−i (X/R , R(d)). H 2d−1−i (X , Q(d))R → H 2d−1−i (X, Q(d))R → HD

The conjunction of the two previous conjectures is equivalent to the following Conjecture 10.3. B(X , d) The map 2d−1−i (X/R , R(d)) H 2d−1−i (X , Q(d))R → HD

is an isomorphism for i ≥ 1 and there is an exact sequence 2d−1 (X/R , R(d)) → CH 0 (X)∗R → 0 0 → H 2d−1 (X , Q(d))R − → HD

for i = 0. If the scheme X lies over a finite field then Conjecture B(X , d) is always satisfied by assumption.

ZETA FUNCTIONS AT s = 0

29

11. The regulator map 11.1. Flat arithmetic schemes. Let X be a proper flat regular connected arithmetic scheme of dimension d satisfying L(X et , d). Let X = XQ be the generic fiber. By the work of Goncharov [13], there is a canonical morphism of complexes Γ(X, Q(d)) → CD (X/R , R(d)) inducing Beilinson’s regulator on cohomology [4], where the complex CD (X/R , R(d)) computes the (truncated) real Deligne cohomology. This complex is truncated so that H j (CD (X/R , R(d))) = 0 for j > 2d − 1. Let j : XZar → XZar the natural embedding. Consider the map RΓ(X , Q(d)) → RΓ(X , j∗ Q(d)) → RΓ(X , Rj∗ Q(d)) ≃ RΓ(X, Q(d)) ≃ Γ(X, Q(d)) In fact we have a quasi-isomorphism j∗ Q(d) ≃ Rj∗ Q(d) since the Zariskihypercohomology of Q(d) coincides with the cohomology (of the complex of abelian groups given its global sections) over a field. We consider the composite map ρ∞ : RΓ(X , Q(d)) → Γ(X, Q(d)) → CD (X/R , R(d)) Theorem 11.1. Assume that Conjecture B(X , d) holds. A choice of a direct sum decomposition (17) induces, in a canonical way, an isomorphism in D: ∼ e RΓW,c (X , Z) ⊗ R −→ RΓW,c (X , R)

Proof. Duality for Deligne cohomology (see [3] Cor. 2.28) 2d−1−i i (X/R , R(d − p)) → R HD (X/R , R(p)) × HD

yields an isomorphism in D CD (X/R , R) → RHom(CD (X/R , R(d)), R[−2d + 1]) Composing with the quasi-isomorphism RΓ(X∞ , R) → CD (X/R , R) we obtain RΓ(X∞ , R) → RHom(CD (X/R , R(d)), R[−2d + 1]) One has a morphism of complexes CD (X/R , R(d)) → CH 0 (X)∗R [−2d + 1] and we define C˜D (X/R , R(d)) to be its Kernel. Applying the functor RHom(−, R[−2d+ 1]), we obtain an exact triangle R[0] → RHom(CD (X/R , R(d)), R[−2d+1]) → RHom(C˜D (X/R , R(d)), R[−2d+1]) → R[1] We have a morphism of exact triangles R[0]

/ RΓ(X∞ , R)





/ RΓc (Xet , R)[1] ∃!

R[0]

/ RHom(CD (X/R , R(d)), R[−2d + 1])

 / RHom(C ˜D (X/R , R(d)), R[−2d + 1])

30

BAPTISTE MORIN

where the vertical map on the right hand side is uniquely determined. This yields a canonical quasi-isomorphism ∼ (18) RΓc (Xet , R)[−1] −→ RHom(C˜D (X/R , R(d)), R[−2d − 1]) It follows from Conjecture B(X , d) that the morphism of complexes ρ∞ : RΓ(X , Q(d)) → Γ(X, Q(d)) → CD (X/R , R(d)) induces a quasi-isomorphism ∼

eD (X/R , R(d)). ρ˜∞,R : RΓ(X , Q(d))R −→ C

Applying the functor RHom(−, R[−2d − 1]) and composing with the map (18), we obtain an isomorphism in D: (19) ∼ ∼ RΓc (Xet , R)[−1] → RHom(C˜D (X/R , R(d)), R[−2d−1]) → RHom(RΓ(X , Q(d)), R[−2d−1]) The inverse isomorphism in D ∼

RHom(RΓ(X , Q(d)), R[−2d − 1]) → RΓc (Xet , R)[−1] and the natural quasi-isomorphism ∼

RΓc (Xet , Z)R → RΓc (Xet , R) provide us with the desired map in D: ∼

RΓW c (X , Z)R → RΓc (Xet , Z)R ⊕ RHom(RΓ(X , Q(d)), R[−2d − 1]) ∼ ∼ ˜ → RΓc (Xet , R) ⊕ RΓc (Xet , R)[−1] → RΓW c(X , R)

 Corollary 11.2. If Conjecture B(X , d) holds then there is a canonical isomorphism ∼ ˜ detR RΓW,c (X , Z)R −→ detR RΓW,c (X , R) Proof. This follows from Proposition 9.7 and Theorem 11.1.



11.2. Schemes over finite fields. Let X be a proper smooth connected scheme of dimension d over a finite field satisfying L(Xet , d). Here we have RΓW c (X , Z) = RΓW (X , Z) and a canonical isomorphism RΓW (X , Z)R ≃ RΓ(Xet , R) ⊕ RHom(RΓ(X , Q(d)), R[−2d − 1]) It follows from L(Xet , d) that the natural map RΓ(Xet , R)[−1] → RHom(RΓ(X , Q(d)), R[−2d − 1]) is a quasi-isomorphism (see the proof of Theorem 7.2). This yields a canonical map ∼

RΓW (X , Z)R → RΓ(Xet , R) ⊕ RHom(RΓ(X , Q(d)), R[−2d − 1]) ∼ ∼ ˜ → RΓ(Xet , R) ⊕ RΓ(Xet , R)[−1] → RΓW (X , R)

We obtain the morphism (20)

˜ RΓW (X , Z) → RΓW (X , R)

ZETA FUNCTIONS AT s = 0

31

inducing the canonical isomorphism ∼ ˜ detR RΓW (X , Z)R −→ detR RΓW (X , R)

(21)

By construction, (20) coincides with ∼ ˜ RΓW (X , Z) ← RΓ(XW , Z) → RΓ(XW , R)

˜ where the right hand side map is induced by the morphism of sheaves Z → R on the Weil-étale topos XW . 12. Zeta functions at s = 0 The following theorem summarizes some results obtained previously. Theorem 12.1. Let X be a proper regular arithmetic scheme. i (X , R) ˜ are finite dimena) The compact support cohomology groups HW,c sional vector spaces over R, vanish for almost all i and satisfy X i ˜ = 0. (−1)i dimR HW,c (X , R) i∈Z

˜ yields an acyclic b) Cup product with the fundamental class θ ∈ H 1 (BR , R) complex ∪θ ∪θ ∪θ i+1 i ˜ − ˜ − (X , R) → ··· · · · −→ HW,c (X , R) → HW,c i (X , Z) c) If L(X et , d)≥0 holds, then the compact support cohomology groups HW,c are finitely generated over Z and they vanish for almost all i. d) If B(X , d) holds then there are isomorphisms ∼ i i ˜ HW,c (X , Z) ⊗Z R − → HW,c (X , R).

We refer to [17] for generalities on the determinant functor. By Corollary 11.2 we have canonical isomorphisms (detZ RΓW,c(X , Z))R ≃ detR (RΓW,c (X , Z)R ) ˜ ≃ detR RΓW,c(X , R) Moreover, the exact sequence b) above provides us with O ∼ ∼ ∼ i ˜ (−1)i −→ ˜ −→ λ : R −→ detR HW,c (X , R) detR RΓW,c (X , R) (detZ RΓW,c (X , Z))R i∈Z

The isomorphism λ can also be defined as follows: ∼

∼

λ : R −→ detR RΓc (Xet , R) ⊗ det−1 R RΓc (Xet , R) −→ ∼ ∼ ˜ −→ detR (RΓc (Xet , R)⊕RΓc (Xet , R)[−1]) −→ detR RΓW,c (X , R) (detZ RΓW,c (X , Z))R We also consider the (well defined) integer X i rankZ RΓW,c(X , Z) := (−1)i · i · rankZ HW,c (X , Z) i∈Z

and we denote by

ζ ∗ (X , 0)

the leading Taylor-coefficient of ζ(X , s) at s = 0.

Conjecture 12.2. Let X be a proper regular connected arithmetic scheme.

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BAPTISTE MORIN

e) Conjecture L(X et , d)≥0 holds, the function ζ(X , s) has a meromorphic continuation to s = 0 and one has ords=0 ζ(X , s) = rankZ RΓW,c (X , Z) f) Conjecture B(X , d) holds and one has Z · λ(ζ ∗ (X , 0)) = detZ RΓW,c (X , Z) where

∼

λ : R −→ detZ RΓW,c (X , Z) ⊗ R is the isomorphism defined by cup product with the fundamental class. 13. Relation to Soulé’s Conjecture We fix a regular connected arithmetic scheme X of dimension d. We assume that X is flat and projective over Spec(Z). The following conjecture is a reformulation (see [15] Remark 43) of Soulé’s Conjecture [26] in terms of motivic cohomology (thanks to [19] Theorem 14.7 (5)). It presupposes the meromorphic continuation of ζ(X , s) at s = 0 and the fact that the Q-vector space H i (X , Q(d)) is finite dimensional and zero for almost all i. Conjecture 13.1. (Soulé) One has X (−1)i+1 dimQ H 2d−i (X , Q(d)) ords=0 ζ(X , s) = i

Lemma 13.2. If X satisfies L(X et , d) and B(X , d) then X rankZ RΓW,c (X , Z) = (−1)i dimQ H 2d+1−i (X , Q(d)) i∈Z

Proof. Assuming L(X et , d) and B(X , d) one has X i rankZ RΓW,c (X , Z) = (−1)i · i · dimQ HW,c (X , Z)Q i∈Z

X = (−1)i · i · (dimQ Hci (Xet , Q) + dimQ H 2d+2−i−1 (X , Q(d))∗ ) i∈Z

X = (−1)i · i · (dimQ H 2d−i (X , Q(d))∗ + dimQ H 2d+1−i (X , Q(d))∗ ) i∈Z

X = (−1)i dimQ H 2d+1−i (X , Q(d)) i∈Z

The second equality follows from Proposition 9.7. The third equality follows from Conjecture B(X , d) and from duality for Deligne cohomology, see the proof of Theorem 11.1. The fourth equality also follows from B(X , d) since it implies that H 2d−i (X , Q(d)) is finite dimensional and zero for almost all i.  Theorem 13.3. Assume that X satisfies L(X et , d) and B(X , d). Then Conjecture 12.2 e) is equivalent to Conjecture 13.1. Proof. This follows from the previous Lemma.



ZETA FUNCTIONS AT s = 0

33

14. Relation to the Tamagawa Number Conjecture Let OF be a number ring and let X → Spec(OF ) be a flat, proper and smooth morphism. Assume that X is connected of dimension d and satisfies Spec(OF ) be a closed subset with open complement U . L(X et , d). Let Z If Xp := X ⊗OF k(p) satisfies L(Xp,et , d − 1) for any p ∈ Z, then we define RΓW,c(XU , Z) such that the triangle M RΓW (Xp , Z) (22) RΓW,c (XU , Z) → RΓW (X , Z) → p∈Z∪Z∞

is exact, where Z∞ denotes the set of archimedean primes of F . We don’t show nor use that RΓW,c (XU , Z) only depends on XU . In fact RΓW,c (XU , Z) is only defined up to a non-canonical isomorphism but detZ RΓW,c(XU , Z) is canonically defined and we have a canonical isomorphism detZ RΓW,c (XU , Z) ≃ detZ RΓW,c (X , Z) ⊗ det−1 Z RΓW (XZ , Z)

(23) We define

˜ := RΓ(X W , j! R) ˜ RΓW,c (XU , R) where j : XU,W → X W is the natural open embedding of topoi. The triangle M ˜ ˜ → ˜ → RΓW (X , R) RΓW (Xp , R) RΓW,c (XU , R) p∈Z∪Z∞

is exact, where the ` map on the right hand side is induced by the closed embedding of topoi p∈Z∪Z∞ Xp,W → X W which is the closed complement of the open embedding j : XU,W → X W . We obtain a canonical isomorphism (24)

˜ ˜ ≃ detR RΓW,c (X , R) ˜ ⊗ det−1 RΓW (XZ , R) detR RΓW,c (XU , R) R

By Corollary 11.2, (21), (23) and (24) we have a canonical isomorphism ∼ ˜ detR RΓW,c(XU , Z)R −→ detR RΓW,c (XU , R)

Theorem 14.1. (Flach-Morin) Let X be a proper flat regular connected arithmetic scheme of dimension d satisfying L(X et , d) and B(X , d). (1) If the Hasse-Weil L-functions L(hi (XQ ), s) of all motives hi (XQ ) satisfy the expected meromorphic continuation and functional equation, then Conjecture 12.2 e) holds for X . (2) Assume that X is moreover smooth over a number ring OF , that X satisfies Conjectures 1,3,5 of [7] and that L(hi (X), s) satisfies the expected meromorphic continuation for all i. Assume also that Xp = X ⊗OF k(p) satisfies Conjecture L(Xp,et , d − 1) for any finite prime p of F . Then the Tamagawa number Conjecture ([7] Conjecture 4) for the motive h(XQ ) =

i=2d−2 M

hi (XQ )[−i]

i=0

is equivalent to statement f ) of Conjecture 12.2 for X .

34

BAPTISTE MORIN

Proof. In view of Theorem 11.1 the first statement is a reformulation of the main result of [7]. By [7] Proposition 9.2, it is enough to check that our definition of the Weil-étale cohomology groups satisfies assumptions a)-j) except perhaps f) of [7] in order to show the second statement. Assumptions a), c), d) and e) are given by Theorem 12.1. b) follows from [7] Conjecture 1, Lemma 13.2 and Conjecture 10.2 which is part of B(X , d). f) We need Conjecture 12.2 f) to be satisfied by Xp = X ⊗OF k(p) for any finite prime of F . By Theorem 15.1, this follows from Conjecture L(Xp,et , d − 1). g) We need this property to hold only in the following special case. By (22) and Theorem 7.2, for any open subset U ⊆ Spec(OF ) with closed complement Z we an have exact triangle RΓW,c (XU , Z) → RΓW,c (X , Z) → RΓ(XZ,W , Z) inducing the canonical isomorphism (23), where XZ,W is the Weil-étale topos of XZ . h) By Corollary 6.2, the natural map RΓ(X et , Z) → RΓW (X , Z) induces i (X , Z) ⊗ Z ≃ H i (X , Z ) for any i ∈ Z and any an isomorphism HW et l l prime number l. We set Vli = H i (XQ,et , Ql ). By [7] Proposition 9.1 we obtain an isomorphism for i ≥ 0 ∼

∼

i+2 rli : Hf2 (Q, Vli ) → H i+2 (X et , Ql ) → HW (X , Z)Ql

˜ Here we i) follows from the definitions of RΓW c (X , Z) and RΓW c (X , R). obtain a canonical map ∼

i i+2 (X , Z)R r∞ : H i (X∞ , R) → HW i is induced by which is an isomorphism for i > 0. Indeed, the map r∞ the following canonical map:

RΓ(X∞ , R)[−2] → RΓc (Xet , R)[−1] → RHom(RΓ(X , Q(d)), R[−2d − 1]) → RΓW (X , Z)R where the second map is (19) and the third map is given by Proposition 9.5. j) By Lemma 5.2 and Theorem 11.1 we have isomorphisms λi : H i+2 (X W , Z)Q ∼ = H 2d−1−i (X/Z , Q(d))∗ for i ≥ 0 such that 9.4).

λiR

◦

i r∞

=

M i (ρ∞ )∗

and λiQl ◦ rli = ρil (see [7] section 

15. Proven cases 15.1. Varieties over finite fields. Let Fq be a finite field and let A(Fq ) be the class of smooth projective varieties over Fq defined in Section 3.2. Theorem 15.1. Let X be a variety over the finite field Fq . If X lies in A(Fq ) then Conjecture 12.2 holds for X .

ZETA FUNCTIONS AT s = 0

35

Proof. The variety X lies in A(Fq ) hence L(Xet , d) ⇔ L(XW , d) holds (see Proposition 3.10). In view of Theorem 7.2 the result follows from [21] Theorem 7.4.  15.2. Number rings. Theorem 15.2. If X = Spec(OF ) is the spectrum of a number ring, then Conjecture 12.2 holds for X . Proof. We set X = Spec(OF ). By Proposition 5.3 one has i (X , Z) = 0 for i < 0 and i > 3. HW 0 (X , Z) = Z, H 1 (X , Z) = 0, an exact sequence By Lemma 5.2 one has HW W

(25)

2 (X , Z) → Hom(H 1 (X et , Z(1)), Z) → 0 0 → H 2 (X et , Z)codiv → HW

and an isomorphism 3 3 (X et , Z)codiv = ((OF× )D )codiv = µD HW (X , Z) ≃ HW F.

since H 0 (X et , Z(1)) = 0. The sequence (25) reads as follows 2 0 → Cl(F )D → HW (X , Z) → Hom(OF× , Z) → 0

where Cl(F ) is the class group of F and OF× is the units group. The map ∼ i i ˜ HW,c (X , Z) ⊗ R → HW,c (X , R)

is trivial for i 6= 1, 2, the obvious isomorphism for i = 1 and the inverse of the transpose of the classical regulator map Y 2 HW,c (X , Z) ⊗ R ≃ Hom(OF× , R) → ( R)/R X∞ ∗ (X , R), ˜ ∪θ) is reduced to the identity map for i = 2. The acyclic complex (HW,c Y Y Id 2 1 ˜ ˜ = ( R)/R −→ (X , R) ( R)/R = HW,c HW,c (X , R) X∞

X∞

The result now follows from the analytic class number formula ords=0 ζ(X , s) = ♯X∞ − 1 and ζF∗ (0) = −hR/w. where h (respectively w) is the order of Cl(F ) (respectively of µF ) and R is the regulator of the number field F .  15.3. Projective spaces over number rings. Let B be a Dedekind ring of characteristic 0 with perfect residue fields, and let p : X → B be a connected smooth B-scheme of finite type and of absolute dimension d = dim(X ). Recall that Xet denotes the category of sheaves of sets on the small étale site of X and XZar denotes the category of sheaves of sets on the category of étale X -schemes

36

BAPTISTE MORIN

endowed with the Zariski topology. Let us consider the commutative square of topoi f

Xet

/ Bet ǫ

ǫ



XZar

p



/ BZar

where ǫ is the canonical embedding. Lemma 15.3. Let X be connected smooth over the Dedekind ring B of characteristic 0 with perfect residue fields. Then the map ǫ∗ Rp∗ Z(n) → Rf∗ (ǫ∗ Z(n)) is a quasi-isomorphism for n ≥ dim(X ). Proof. It is enough to check that the induced map on stalks are quasi-isomorphisms for any geometric point β → B over a closed point of B. On the one hand one has (26) (27)

(ǫ∗ Rp∗ Z(n))β

= (Rp∗ Z(n))β ≃ (p∗ Z(n))β ′ = lim −→ p∗ Z(n)(B ) ′ = lim −→ Z(n)(X ×B B )

(28) (29)

≃ Z(n)(X ×B Bβsh )

(30)

′ Here B ′ runs over the étale neighborhoods of β → B, and Bβsh = lim ←− B denotes the strict henselization. We denote by (−)β the stalk of both a Zariski and étale sheaf, i.e. (−)β denotes the inverse image of the morphisms βet → Bet and βZar → BZar . (26) is true since taking stalks commutes with sheafification as it follows from the commutative square of topoi

Set = βet

/ Bet

=



Set = βZar

ǫ



/ BZar

(27) is a quasi-isomorphism because p∗ Z(n) → Rp∗ Z(n) is a quasi-isomorphism (see [8] Corollary 3.3 (b)). Then (30) is true since the cycle complex commutes with inverse limits of schemes with affine transition maps (at least if the schemes are quasi-compact). One the other hand one has (31) (32)

′ (Rf∗ Z(n))β = lim −→ Rf∗ Z(n)(B ) ′ ′ = lim −→ Rf∗ Z(n)(B )

(33)

= Rf∗sh Z(n)(Bβsh )

(34)

= RΓet (X ×B Bβsh , Z(n))

(35)

≃ Z(n)(X ×B Bβsh )

ZETA FUNCTIONS AT s = 0

37

Here we denote by f ′ and f sh the morphisms of étale topoi induced by X ′ = X ×B B ′ → B ′ and X sh = X ×B Bβsh → Bβsh respectively. Then (32) is clear since the base change from B to B ′ is exact and preserves injectives. Take an injective resolution Z(n) ≃ I • . Then Z(n) |X ′ ≃ I • |X ′ is still an injective resolution, and Z(n) |X sh ≃ I • |X sh is an acyclic resolution, since étale cohomology commutes with inverse limit of schemes (quasi-compact and quasi-separated with affine sh , −) = transition maps), and (33) follows. The identity (34) is given by RΓ (Bβ, et sh , −) since Γ (B sh , −) is exact. Finally, the quasi-isomorphism (35) is Γ (Bβ, β, et et given by [11] Theorem 7.1.  Lemma 15.4. Let p : Pm B,Zar → BZar be the canonical map. Then one has Rp∗ Z(n) ≃

m M

Z(n − j)[−2j]

j=0

an isomorphism in the derived category D(Ab(BZar )). Proof. Consider an open-closed decomposition m m−1 :i j : Am B → PB ← PB

We have an exact triangle m−1

i∗ Z(n − 1)PB

m

m

[−2] → Z(n)PB → j∗ Z(n)AB

of complexes of Zariski sheaves on Pm B (see [8] Corollary 3.3). Applying Rp∗ we get m−1 m m p∗ i∗ Z(n − 1)PB [−2] → p∗ Z(n)PB → p∗ j∗ Z(n)AB where we use p∗ Z(n) ≃ Rp∗ Z(n), p∗ i∗ Z(n−1) ≃ R(p∗ i∗ )Z(n−1) ≃ (Rp∗ )i∗ Z(n− 1), and j∗ Z(n) ≃ Rj∗ Z(n) (see [8] Corollary 3.3 and the remark after it). But the map on the right hand side has the following section (see [8] Corollary 3.5) m

m

m

p∗ j∗ Z(n)AB ≃ R(p∗ j∗ )Z(n)AB ≃ Z(n)B → p∗ Z(n)PB Writing pm : Pm B → B, we obtain m

m−1

PB ≃ Z(n)B ⊕ pm−1 Z(n − 1)PB [−2] pm ∗ ∗ Z(n) L m B Pm By induction we get pm ∗ Z(n) B ≃ j=0 Z(n − j) [−2j].



Lemma 15.5. Let F be a number field and let n ≥ 2. For any i ≤ n + 1, one has H i (Spec(OF )et , Z(n)) ≃ H i (Spec(OF ), Z(n)) and this group is finitely generated. For any i > n + 1, one has H i (Spec(OF )et , Z(n)) ≃ H i (Fet , Z(n)) and this group is finite killed by 2. Proof. The first statement is given by the Beilinson-Lichtenbaum conjecture (see [8]). By [11] Corollary 7.2, for n ≥ 2 we have an exact sequence M → H i−2 (Fp,et , Z(n − 1)) → H i (Spec(OF )et , Z(n)) → H i (Fet , Z(n)) → p

38

BAPTISTE MORIN

For i > n + 1 ≥ 3 we have H i−2 (Fp,et , Z(n − 1)) = 0 hence an isomorphism ∼

H i (Spec(OF )et , Z(n)) −→ H i (Fet , Z(n)) But H i (Fet , Q(n)) = 0 for i > n hence we obtain ∼

∼

H i (Spec(OF )et , Z(n)) −→ H i (Fet , Z(n)) −→ H i−1 (Fet , Q/Z(n)) for i > n + 1. Moreover, for i − 1 > n ≥ 2 we have ∼

∼

H i−1 (Fet , Q/Z(n)) −→ H i−1 (Fet , µ⊗n ) −→

X

H i−1 (GR , µ⊗n )

v|R

where µ := lim −→ µm denotes the sheaf of all roots of unity, and v runs over the real places of F . Indeed, the first isomorphism follows from Z/mZ(n) ≃ µ⊗n m , and the second isomorphism is well known (see [23]).  Lemma 15.6. For any n ≥ 2 and any i ∈ Z, the map H i (Spec(OF ), Q(n)) → H i (Spec(F ), Q(n)) is an isomorphism. Proof. We have a localization sequence M ... → H i−2 (Spec(k(p)), Q(n−1)) → H i (Spec(OF ), Q(n)) → H i (Spec(F ), Q(n)) → ... p

where the sum is taken over the finite primes p of F (see [8] Corollary 3.4 and correct the obvious misprint). But the motivic cohomology groups H j (Spec(k(p)), Q(n − 1)) = 0 for all j and n ≥ 2 are all trivial.



Theorem 15.7. Let OF be the ring of integers in a number field F and let X = PnOF with n ≥ 0. (1) Conjecture L(X et , n + 1) and B(X , n + 1) hold. (2) Conjecture 12.2 e) holds for X = PnOF . (3) If F/Q is abelian then Conjecture 12.2 f ) hold for X = PnOF . Proof. (1) This follows from Lemmas 15.3, 15.5 and 15.6. (2) This follows from Theorem 14.1 and from the fact that the L-function of the motive hi (PnF ) satisfies the meromorphic continuation and the functional equation for any i. Indeed, hi (PnF ) = h0 (F )(−i/2) for i even with 0 ≤ i ≤ 2n and hi (PnF ) = 0 otherwise. (3) In view of Theorem 14.1, this will follow from [7] Conjecture 1, 2, 3, 5, for X = PnOF and from the Tamagawa number conjecture for the motive h0 (F )(j) with j ≤ 0. One also needs the validity of Conjecture L(Pnk(p),et , n) for any finite prime p of F . But Conjecture L(Pnk(p),et , n) is true by the projective bundle formula. The Tamagawa number conjecture for the motive h0 (F )(j) with j ≤ 0 is known. Conjectures 1, 2 of [7] for PnOF boil down to Borel’s theorem. Conjecture 3 of [7] for PnOF is a reformulation of Conjecture Motl for the motives h2n−i (PnF )(n + 1) and

ZETA FUNCTIONS AT s = 0

39

hi (PnF ), which are either 0 or of the form h0 (F )(j) for suitable j ∈ Z. Conjecture 5 of [7] for PnOF is reduced to the following statement: For any prime number p and j ≤ 0, the complex RΓf (Qp , Ql (j)) is semi-simple at 0.  15.4. Some open sub-schemes of projective spaces. Let F/Q be an abelian ` number field, let X = PnOF and let U = PnOF − Y where Y = i=s i=1 Yi is the disn joint union in POF of varieties Yi over finite fields Fqi lying in A(Fqi ). We define RΓW,c(U , Z) such that the triangle RΓW,c (U , Z) → RΓW (X , Z) → RΓW (Y, Z) ⊕ RΓ(X∞,W , Z) → RΓW,c (U , Z)[1] is exact. Again RΓW,c (U , Z) is only defined up to a non-canonical isomorphism but detZ RΓW,c (U , Z) is well defined and given with a canonical isomorphism detZ RΓW,c(U , Z) ≃ detZ RΓW,c (X , Z) ⊗ det−1 Z RΓW (Y, Z) Moreover, Corollary 11.2 and (21) yield a canonical isomorphism ∼

λU : R −→ detZ RΓW,c (U , Z) ⊗ R Corollary 15.8. The following holds: ords=0 ζ(U , s) = rankZ RΓW,c (U , Z) Z · λU (ζ ∗ (U , 0)) = detZ RΓW,c (U , Z) In particular rankZ RΓW,c(U , Z) and detZ RΓW,c (U , Z) only depend on U . Proof. In view of Theorem 15.7 and Theorem 15.1, the result follows from the formulas ζ(X , s) = ζ(U , s) · ζ(Y, s) rankZ RΓW,c (X , Z) = rankZ RΓW,c (U , Z) + rankZ RΓW (Y, Z) detZ RΓW,c (X , Z) ≃ detZ RΓW,c (U , Z) ⊗Z detZ RΓW (Y, Z) where the last isomorphism is compatible with (−) ⊗ R and λ.



15.5. Cohomology of projective spaces in terms of K-theory. Let F be a number field with r1 real primes and r2 complex primes. We set X = PnOF . Neglecting the 2-torsion, the Weil-étale cohomology of X is given by the following

40

BAPTISTE MORIN

identifications and exact sequences: i HW (X , Z)

= Z for i = 0, = 0 for i = 1, D 2 0 → Cl(F ) → HW (X , Z) → Hom(OF× , Z) → 0 =

µD F for i = 3,

4 0 → K2 (OF )D → HW (X , Z) → Hom(K3 (OF ), Z) → 0

= (K3 (OF )tors )D for i = 5, ... D 2n+2 0 → K2n (OF ) → HW (X , Z) → Hom(K2n+1 (OF ), Z) → 0 = =

(K2n+1 (OF )tors )D for i = 2n + 3, 0 for i > 2n + 3.

The complex ∪θ

∪θ

∪θ

... −→ Hci (X , Z) ⊗ R −→ Hci+1 (X , Z) ⊗ R −→ ... is canonically isomorphic to ∼

0

∼

0 −→ Rr1 +r2 /R −→ Hom(OF× , R) −→ Rr2 −→ Hom(K3 (OF ), R) 0

∼

0

∼

0

−→ Rr1 +r2 −→ Hom(K5 (OF ), R) −→ Rr2 −→ Hom(K7 (OF ), R) −→ ... in which the isomorphisms are the dual of the regulator maps. References [1] M. Bienenfeld, An étale cohomology duality theorem for number fields with a real embedding. Trans. Amer. Math. Soc. 303 (1987), no. 1, 71−96. [2] D. Benois; T. Nguyen Quang Do, Les nombres de Tamagawa locaux et la conjecture de Bloch et Kato pour les motifs Q(m) sur un corps abélien. Ann. Sci. École Norm. Sup. (4) 35 (2002), nno. 5, 641−672. [3] J. I. Burgos Gil, Semipurity of tempered Deligne cohomology. Collect. Math. 59 (2008), no. 1, 79−102. [4] J. I. Burgos Gil; E. Feliu; Y. Takeda, On Goncharov’s Regulator and Higher Arithmetic Chow Groups. [5] D. Burns, Perfecting the nearly perfect. Pure Appl. Math. Q. 4 (2008), no. 4, part 1, 1041−1058. [6] M. Flach, Cohomology of topological groups with applications to the Weil group. Compositio Math. 144 (2008), no. 3, 633−656. [7] M. Flach; B. Morin, On the Weil-étale topos of regular arithmetic schemes. Preprint (2010). [8] T. Geisser, Motivic cohomology over Dedekind rings. Math. Z. 248 (2004), no. 4, 773−794. [9] T. Geisser, Weil-étale cohomology over finite fields. Math. Ann. 330 (2004), no. 4, 665−692. [10] T. Geisser, Motivic cohomology, K-theory and topological cyclic homology. Handbook of K-theory. Vol. 1, 193−234. [11] T. Geisser, Duality via cycle complexes. Ann. of Math. (2) 172 (2010), no. 2, 1095−1126 [12] T. Geisser, Arithmetic cohomology over finite fields and special values of ζ-functions. Duke Math. J. 133 (2006), no. 1, 27−57. [13] A. B. Goncharov, Polylogarithms, regulators, and Arakelov motivic complexes. J. Amer. Math. Soc. 18 (2005), no. 1, 1−60. [14] A. Grothendieck, M. Artin and J.L. Verdier, Théorie des Topos et cohomologie étale des schémas (SGA4). Lectures Notes in Math. 269, 270, 305, Springer, 1972.

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[15] B. Kahn, Algebraic K-theory, algebraic cycles and arithmetic geometry. Handbook of K-theory. Vol. 1, 2, 351−428, Springer, Berlin, 2005. [16] B. Kahn, Équivalences rationnelle et numérique sur certaines variétés de type abélien sur un corps fini. Ann. Sci. École Norm. Sup. (4) 36 (2003), no. 6, 977−1002 (2004). [17] F. Knudsen, D. Mumford, The projectivity of the moduli space of stable curves I: Preliminaries on ‘det’ and ‘Div’. Math. Scand. 39 (1976), 19−55 [18] M. Kolster; J.W. Sands, Annihilation of motivic cohomology groups in cyclic 2-extensions. Ann. Sci. Math. Québec 32 (2008), no. 2, 175−187. [19] M. Levine, K-theory and motivic cohomology of schemes. [20] M. Levine, Techniques of localization in the theory of algebraic cycles. J. Algebraic Geom. 10 (2001), no. 2, 299−363. [21] S. Lichtenbaum, The Weil-étale topology on schemes over finite fields. Compositio Math. 141 (2005), no. 3, 689−702. [22] S. Lichtenbaum, The Weil-étale topology for number rings. Ann. of Math. (2) 170 (2009), no. 2, 657−683. [23] J. Milne, Arithmetic duality theorems. [24] B. Morin, On the Weil-étale cohomology of number fields. To appear in Trans. Amer. Math. Soc. [25] B. Morin, The Weil-étale fundamental group of a number field II. Selecta Math. (N.S.) 17 (2011), no. 1, 67−137. [26] C. Soulé, K-théorie et zéros aux points entiers de fonctions zêta. (French) Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983), 437−445 [27] M. Spiess, Artin-Verdier duality for arithmetic surfaces. Math. Ann. 305 (1996), no. 4, 705−792.

University of Muenster Mathematisches Institut Einsteinstr. 62 48149 Muenster Germany [email protected]