On the Weil-étale Cohomology of S-Integers Citation Chiu, Yi-Chih (2012) On the Weil-étale Cohomology of S-Integers. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/W4CA-6489. https://resolver.caltech.edu/CaltechTHESIS:12212011-095330433 Abstract We generalize the Lichtenbaum's prototype of Weil-étale cohomology to S-integers and study its relation to the Tate sequences. In the final part, we present a more natural way to define Weil-étale cohomology for one-dimensional arithmetic schemes motivated by a dual quasi-isomorphism between Weil-étale cohomology and étale cohomology. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: Weil-étale cohomology, S-integers, Tate sequences Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Mathematics Thesis Availability: Public (worldwide access) Research Advisor(s): Flach, Matthias Thesis Committee: Flach, Matthias (chair) Graber, Thomas B. Jorza, Andrei Mantovan, Elena Defense Date: 6 October 2011 Record Number: CaltechTHESIS:12212011-095330433 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:12212011-095330433 DOI: 10.7907/W4CA-6489 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 6756 Collection: CaltechTHESIS Deposited By: Yi-Chih Chiu Deposited On: 17 Feb 2012 21:36 Last Modified: 03 Oct 2019 23:53 Thesis Files Preview PDF - Final Version See Usage Policy. 426kB Repository Staff Only: item control page

On the Weil-étale Cohomology of S-integers Thesis by Yi-Chih Chiu In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 2012 (Defended October 5, 2011) ii c 2012 Yi-Chih Chiu All Rights Reserved iii Acknowledgments I would sincerely like to thank my supervisor Matthias Flach for his patience and guidance throughout my graduate studies. It has been a pleasure to work with him and his expertise and advice have been invaluable. Without his helpful instructions and constant guidance, this work would not have been possible. I am grateful to Tom Graber, Andrei Jorza, and Elena Mantovan for agreeing to serve on my dissertation committee. I would also like to thank the Department of Mathematics at Caltech for its support during my graduate study program and for providing a great environment for research. Finally, and most of all, I want to thank my family. My parents have always been a tireless source of love and support; I dedicate this thesis to them. iv Abstract In this thesis, we first briefly introduce the history of the Weil-étale cohomology theory of arithmetic schemes and review some important results established by Lichtenbaum, Flach and Morin. Next we generalize the Weil-étale cohomology to S-integers and compute the cohomology for constant sheaves Z or R. We also define a Weil-étale cohomology with compact support Hc (YW , −) for Y = Spec OF,S where F is a number field, and computed them. We verify that these cohomology groups satisfy the axioms state by Lichtenbaum. As an application, we derive a canonical representation of Tate sequence from RΓc (YW , Z). Motivated by this result, in the final part, we define an étale complex R Gm , such that the complexes RHomZ (RΓ(Uét , R Gm ), Z)[−2] and τ ≤3 RΓc (UW , Z) are canonically quasi-isomorphic for arbitrary étale U over Spec OF . This quasi-isomorphism provides a possible approach to define the Weil-étale cohomology for higher dimensional arithmetic schemes, as the Weil groups are not involved in the definition of R Gm . v Contents Acknowledgments iii Abstract iv Introduction 1 Notation 6 1 Preliminary 8 1.1 The Weil Groups for Number Fields and Local Fields . . . . . . . . . 8 1.1.1 Weil Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.1.2 Weil Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2 The Classifying Topos and Cohomology of a Topological Group . . . 10 1.3 Artin-Verdier Topology . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.3.1 The Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.3.2 The Norm Maps . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.3.3 The Étale Cohomology with Compact Support . . . . . . . . . 18 2 The Weil-Étale Cohomology of S-integers 20 2.1 The Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.2 The Weil-Étale Sites and the Artin-Verdier Étale Sites . . . . . . . . 27 3 Cohomology with Compact Support for the Ring of S-integers 30 3.1 The Computation of the Cohomology with Compact Support . . . . . 30 3.2 Verification of the Axioms of Weil-Étale Cohomology . . . . . . . . . 37 vi 3.3 3.2.1 Axiom (a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.2.2 Axiom (b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.2.3 Axiom (d) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.2.4 Axiom (c) and (e) . . . . . . . . . . . . . . . . . . . . . . . . . 39 A Canonical Representation of Tate Sequences . . . . . . . . . . . . . 43 4 The Construction of R Gm 49 4.1 The Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.2 The Duality Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Bibliography 68 1 Introduction Stephen Lichtenbaum conjectured that there exists a new cohomology theory, called Weil-étale cohomology, for arbitrary arithmetic schemes X, such that its cohomologies are connected to the zeta function of X. In [9], he defined a prototype of such a cohomology theory for number rings. The main idea is replacing the role of Galois groups in the étale cohomology by Weil groups. There is still no exact definition for the Weil-étale cohomology. Let φ : X ,→ X be a fixed Nagata and Artin-Verdier style compactification, and here we list the axioms suggested by Lichtenbaum: a) The Weil-étale cohomology groups with compact support Hcq (X, Z) := H q (X, φ! Z) are finitely generated abelian groups that equal to 0 almost everywhere, and independent of the choice of compactification of X. e and H q (X, Z) e denotes the sheaf of real valued functions on X, then H q (X, R) b) If R c c are independent of the choice of φ. Moreover, the natural map from Hcq (X, Z)⊗Z e is an isomorphism. R to Hcq (X, R) c) There exists an element ψ ∈ H 1 (X, R) such that the complex `ψ / H i (X, R) e `ψ c / H i+1 (X, R) e `ψ / c is exact. In particular, X e = 0. (−1)i dim Hci (X, R) i≥0 d) ords=0 ζX (s) = X i≥0 (−1)i i rankZ Hci (X, Z). 2 e) The Euler characteristic χc (X)(Def. 3.1) of the complex Hcq (X, Z) is well-defined, and the leading coefficient of the Taylor’s expansion of ζX (z) at z = 0 is ±χc (X). The cohomology theory H ∗ (X W , −) for X = Spec OF , where F is a number field, defined in [9], is an inductive limit of cohomology for a projective system of sites. And the above axioms hold only under the assumption H i (X W , Z) = 0 for all j > 3. However, Matthias Flach [5] proved that the cohomology group H i (X W , Z) is an infinitely generated abelian group for even i ≥ 4, and is vanishing for odd i ≥ 5. In a recent paper [6], by modifying the prototype defined by Lichtenbaum, Flach and Baptiste Morin constructed a topos that recovers the Lichtenbaum Weil-étale cohomology groups for number fields. By this result, one may avoid the limit process for the case of number fields. However, we still need to truncate the cohomologies of higher degrees. Recall that Lichtenbaum only defined the Weil-étale cohomology for Spec OF , but one can generalize it to its arbitrary open subschemes Y = Spec OF,S in an obvious way. In Chapter 2, we compute the Weil-étale cohomology of Y with Z-coefficients and give a brief review of Morin’s results concerning Weil-étale cohomology. From Chapter 3, we assume that the ground field F is totally imaginary. Under this assumption, in section 3.1, we show that the Weil-étale cohomology groups with compact support e for S-integers with Z or R-coefficients are as follows, Hcp (YW , Z) =    0      Q   Z/Z p = 0, p = 1, S    Pic(Y )D × Hom(US , Z) p = 2,      µ D p = 3, F and e = Hcp (YW , R)  Q   S R/R p = 1, 2,  0 otherwise. 3 The computation is based on Theorem 3.1 that there exists an exact triangle RΓc (Yét , Z) → τ ≤3 RΓc (YW , Z) → Hom(US , Q)[−2] → . In section 3.2, we show that all the axioms (a)-(d) hold for S-integers if we truncate the cohomologies of higher degrees. The proof relies on that in Lichtenbaum’s original paper and a closer look of the effect of the cup product by ψ. Recall that for any Galois extension F/L of number fields with Galois group G, and a G-stable subset S of valuation of F containing all the infinite places and those ramified in F/Q and so that Pic(OF,S ) = 0, then, by using class field theory, one can define an exact sequence ([15] Théorèm 5.1) 0 → US → A → B → XS → 0, where US is the group of S-units, XS := {(xv ) ∈ ` v∈S Z| P xv = 0} and A and B are finitely generated cohomologically trivial Z[G]-modules. We call the complex A → B the Tate sequence associated to S. The Tate sequence is not unique, but one can choose the complex ΨS : A → B so that it represents a certain canonical e S the complex A → B → XS ⊗Z Q and class of Ext2G (XS , US ). We denote by Ψ (−)D := RHomZ (−, Q/Z) = HomZ (−, Q/Z). Burns and Flach([2] Prop. 3.1) showed that there is a map of complexes, e S → RΓc (Yét , Z)D [−3], Ψ b cS := US ⊗Z Z inducing an isomorphism on H i for i 6= 0 and the inclusion US ,→ U on H 0 . In section 3.3, we are able to show that τ ≤3 RΓc (Y W , Z) gives a canonical description of the complex ΨS in the sense that there is a quasi-isomorphism, 
 qis ΨS −→ RHom τ ≤3 RΓc Y W , Z , Z [−2]. (1) In the final section, we construct a canonical complex R Gm of Y ét -sheaves as 4 f, Gm ), where X f is the Y ét -sheaf associated to the presheaf an element of ExtY2 ét (X Uét 7→ XU −U . We show that   H i (Uét , R Gm ) = H i RHomZ (τ ≤3 RΓc (UW , Z) , Z)[−2] , ∀ i ≥ 0, for any étale U → Spec OF . Therefore, it may be possible to define certain Weil-étale cohomology for arbitrary arithmetic scheme by generalizing R Gm . In the final section, we prove that, in fact, there is a canonical quasi-isomorphism: qis RΓ(Uét , R Gm ) −→ RHomZ (τ ≤3 RΓc (UW , Z) , Z)[−2] in D(Z[G]), for any étale U → Spec OF , when S is stable under the action of G. When U small enough so that the Tate sequence exists, this together with (1) show that RΓ(Uét , R Gm ) ≃ ΨS . When G is the trivial group, this implies the quasi-isomorphism, RHom(τ ≤3 RΓc (UW , Z), Z)[−2] ≃ RΓ(Uét , R Gm ), in D(Z), for any étale U → Spec OF . Therefore, one sees that the the complex RΓ(Uét , R Gm ) generalizes the Tate sequences and its Z-dual recovers the truncated Weil-étale cohomology groups of Sintegers. Also, this suggests the following duality property: Conjecture 1. There is a (perfect) paring RHomUét (F , R Gm ) × τ ≤3 RΓc (UW , F ) → Z[−2], for a certain class of étale sheaf F on U . The reason that it is hard to generalize Lichtenbaum’s prototype to higher dimen- 5 sional arithmetic schemes X is that there are no Weil groups for higher dimensional fields. However, Theorem 4.3 shows us a probability to generalize Lichtenbaum’s prototype, because we do not use Weil groups when defining the complex R Gm . One direct conjecture is that Conjecture 2. For an arithmetic scheme X of dimension n, one may define a complex RZ(n) in Ext2X (Z(n), F • (n)), so that the Z-dual of RΓ(X et , RZ(n)) defines certain Weil-étale cohomology theory, where F • (n) is a complex of étale sheaves that depends on n. 6 Notation Here we list the notations that are used frequently throughout this thesis. • F is a number field if not specified, GF is its absolute Galois group. G is reserved to be the Galois group of any Galois extension F/L. • OF : the ring of integers of F . • Spec OF : the Artin-Verdier compactification of Spec OF , which, as a set, is the same as the set of the equivalence classes of valuation of F (the trivial valuation is included). We say a subset of Spec OF is open if it has finite complement. • For any open subset U of Spec OF , we denote by – U := Spec OF , – K(U ) : the function field of U , – U∞ (resp. Uf ): the set of points of U corresponding to infinite (resp. finite) places of K(U ). • For any field K, MK is the collection of equivalence classes of valuations of K. • v0 : the trivial valuation of F . • S: a finite subset of MK . We always assume that it contains all the infinite places then S = Spec OF − Spec OF,S as sets. • S∞ (resp. Sf ): the subset of all the infinite (resp. finite) places in S. • US : the group of units of OF,S . 7 • (−)∨ := HomZ (−, Z), (−)D := RHomZ (−, Q/Z) = HomZ (−, Q/Z) and (−)D := Homcont (−, R/Z) the Pontryagin dual. • XS : = {(xv ) ∈ ` ∨ v∈S Z|Σv xv = 0}. Hence XS = ` ∨ ∨ S Z / Z and (XS ) = XS . • CF : the usual idèle class group CF (resp. F × ) when the field F is global (resp. local). • For any site C, Sh(C) is the category of abelian sheaves on C. 8 Chapter 1 Preliminary 1.1 The Weil Groups for Number Fields and Local Fields 1.1.1 Weil Groups Idèle class groups and Galois groups are the central objects in class field theory. In 1951, Weil introduced the Weil groups, which carry the information of both. There are many equivalent ways to define the Weil groups. (cf. [14] sec.1). A brief definition of the Weil group for F (global or local) is as follows : let K/F be any finite Galois extension, and the relative Weil group WK/F be the extension 1 → CK → WK/F → Gal(K/F ) → 1 representing the canonical generator αK/F of the group H 2 (Gal(K/F ), CK ) ∼ = 1 Z / Z. [K : F ] By abuse of language, the absolute Weil group for F is defined as WF := lim WK/F , ←− K/F where the index runs over all finite Galois extension K/F . This induces a contin- 9 uous morphism αF : WF → GF with a dense image, and we know that there is an ∼ isomorphism of topological groups rF : CF − → WFab ([1] p.238-239). From now on we restrict F to be a number field. Let K/F be a finite Galois extension, and S be a finite set of places of K containing all the infinite ones and those ramified in K/F , we define WK/F,S to be the extension CK,S := CK /UK,S by Gal(K/F ) that represents the canonical class αK/F,S of the group H 2 (Gal(K/F ), CK,S ) ∼ = H 2 (Gal(K/F ), CK ), × where UK,S is the idèles of the form (av ) in which av = 1 (resp. av ∈ OK ) for v all v ∈ S (resp. v ∈ / S). It is easy to see that WK/F,S ∼ = WK/F /UK,S and then ab ∼ WK/F,S = CK,S . Together with the map log | · | : CK,S → R, one defines a map lL,S : WK/F,S → R. Further, we know that WF = limK/F,S WK/F,S . ([9] Lemma 3.1), ←− and there is a canonical map lv0 : WF → R. 1.1.2 Weil Maps For any non-trivial place v of F , we choose a place v of F lying over v once for all, and denote by Dv the associated decomposition group and Iv the inertia group. This induces an embedding ov : Dv = GFv → GF . We set Gk(v) := Dv /Iv . Note that then Gk(v) = 1 for any infinite place v. By the construction of Weil groups, there exists a so-called Weil map Θv : WFv → WF , which is also an embedding such that the following diagram commutes. WFv  Θv αFv GFv ov / WF  αF / GF . p ∼ |.| For any non-trivial place v, the composition lv : WFv → − WFabv − → Fv× − → R has kernel p−1 (OF×v ). Thus, Wk(v) := WFv /p−1 (OF×v ) ∼ = Z (resp. ∼ = R) when v - ∞ (resp. 10 v|∞.). We denote the quotient WFv → Wk(v) by πv . Summarily, the diagram WFv  qv Θv WF lv0 / Wk(v) (1.1) lv  / R, is commutative. Note that, for any finite place v, lv sends the generator σv of Wk(v) to log N (v). Consider the commutative diagram WFv    WKw /Fv / WF  / WK/F,S = WK/F /UK,S . It is not hard to show that the image of WFv in WK/F,S is isomorphic to WKw /Fv ( × ) when v ∈ S (resp. v ∈ / S). This implies the canonical map resp. WKw /Fv /OK w g g πv : WFv → Wk(v) factors through W Fv , where WFv is the image of WFv in WK/F,S . g g We denote qv : W Fv → Wk(v) and θv : WFv → WK/F,S . 1.2 The Classifying Topos and Cohomology of a Topological Group Let G be a topological group. We denote by B sm G the small classifying topos of G the category of discrete sets on which G acts continuously, and by BT op G the classifying site of G, the category of G-topological spaces endowed with the local section topology Jls (see [9] sec. 1), and BG := Sh(BG, Jls ) the topos of sheaves on this site. Let eG be the final object of BG , we define the global section functor ΓG as HomBG (eG , −). For any abelian object A of BG , we define the cohomology of G with coefficient A by H i (G, A) := Ri (ΓG )(A). 11 Let A be any continuous G-module, and A : X → HomBG (X, A), then H i (G, A) coincides with the usual group cohomology group H i (G, A) when G is a profinite group ([9] corollary 2.4). Theorem 1.1 ( [9] Thm. 3.6 & [5] Thm. 10.1). H q (WF , Z) =    Z       C 1,D F q = 0, q = 2,    0 odd q,      is of infinite rank even q ≥ 4, where CF1 is the idèle class group with norm one and D := Homconti (−, R/Z) the Pontryagin dual. 1.3 Artin-Verdier Topology 1.3.1 The Definition For any subset U of X, one can define a so-called Artin-Verdier étale site EtU , which generalizes the usual étale topology for schemes. The objects in EtU are pairs V = (Vf , V∞ ) where Vf (resp. V∞ ) is a subset of the finite (resp. infinite) places of K(Vf ) and Vf (resp. V∞ ) is étale over Uf (resp.U∞ ). We say that V is étale over U if V is an object of EtU . A morphism from V 0 = (Vf0 , V∞0 ) to V is an Uf -étale morphism f : V 0 → V that induces a map sending V∞0 to V∞ . The fibre product is defined componentwise in the obvious way, and a covering is a surjective family. We denote by Sh(Uét ) the category of abelian sheaves on the site EtU . Let j : V ,→ U be nonempty open subsets of X and j −1 W = W ×U V is a map of topology. We have an embedding j∗ : Sh(Vét ) → Sh(Uét ). For each x ∈ U , there is 12 an embedding of closed point ix : Sh(xét ) → Sh(Uét ). In this case Z := U − V has ` only finitely many points, and it is easy to see that Sh(Zét ) = v∈Z Sh(vét ) where sm Sh(vét ) = BG . We set Fη = α∗ F , where α : Spec K(U ) → U . k(v) Proposition 1.1. The category Sh(Uét ) of abelian sheaves is equivalent to the category that consists of (F , {Ax }U −V , {φx : Ax → FηIx }U −V ) where Ax is a continuous Gk(x) -module and φx is an Gk(x) -map. ( The morphisms between objects (F , {Ax }, {φx }) and (F 0 , {A0x }, {φ0x }) are collection of morphisms {f : F → F 0 , {fx : Ax → A0x }} such that (i∗x f )(φx ) = φ0x (fx ) for all x ∈ U − V .) Proof. This is just a generalization of [18] section 1.3.3. By the same argument we are led to prove the following condition holds. a. ix,∗ and j∗ are fully faithful, b. j ∗ F = 0 if and only if F is of the form Πx∈U −V Mx where Mx ∈ Sh(x). Condition (a) follows from the fact ix,∗ M (W ) = Πx×U W M and j ∗ j∗ = id. To see condition (b) is true, first note that the sufficient is obvious. For the necessary, let (W, y) → (U, x) be any étale morphism and W → U together with (Ui → U ) be a covering of U , then (W, y) → (U, x) together with (Ui → U ) form a covering of (U, x). By the sheaves condition and j ∗ F = 0, we see that F ((W, y)) ≃ F ((U, x)), which shows that i∗x F ≃ Mx where Mx = F ((U, x)). Therefore F is of the desired form. And the result follows from the above and [3] III, Prop. 2.3. Remark. In the proof of the above property, we actually proved that F ∈ Sh(Uét ) is vanishing if and only if the stalks Fx = 0 for all x ∈ U . Moreover, to verify the exactness of complexes of étale sheaves it is enough to check it on stalks. As the open-closed decomposition of étale sheaf for schemes, there are functors j! Sh(Uét ) o j∗ j∗ ix,∗ / / Sh(Vét ), Sh(xét ) o / i∗x ix,! k∗ / Sh(Vét ), Sh((V∞ )ét ) o k∗ / Sh(Vét ) 13 such that each functor is left adjoint to the one below it. Explicitly, let F = (G , {Ax }, {φx }) ∈ Sh(Uét ) then j! G = (G , {0}, {0}), j ∗F = G , j∗ G = (G , {GηIx }, {id}), k ∗ F = {Ax }, k∗ {Ax } = (0, {Ax }, {0}), k ! F = {ker φx }. Some immediate properties: Proposition 1.2. a. j! , j ∗ , k ∗ and k∗ are exact, b. j ∗ , j∗ , k∗ and k ! preserve injectives, c. There is a short exact sequence 0 → j! G → j∗ G → k∗ k ∗ j∗ G → 0. Proof. Part (a) follows from the formula above and part (b) follows directly from (a) because any functors listed in (b) has an exact right adjoint. The exact sequence in part (c) is the same as the exact sequence 0 → (G , 0, 0) → (G , {GηIx }, {id}) → (0, {GηIx }, 0) → 0. Remark. By the construction of mapping cylinder, the section function Γ(−, F ) is the sheaf sending an étale W over U to the abelian group j∗ j ∗ F (W) ×k∗ k∗ j∗ j ∗ F (W) k∗ k ∗ F (W). 14 We denote by Gm ∈ Sh(Uét ) to be the j∗ Gm , where j : U0 → U . Because Sh((U0 )ét ) and Sh((U∞ )ét ) have enough injectives and j∗ and k∗ are both preserve injective, so Sh(Uét ) has enough injectives as well. This enables us to define right derived functors. For any F ∈ Sh(Uét ), we define H p (Uét , F ) = Rp Γ(−, F ) and Hxp (Uét , F ) = Rp kx! F . Then applying for Ext∗Uét (−, F ) to the short exact sequence, 0 → j! Z → j∗ Z → k∗ k ∗ j∗ Z → 0, we get a long exact sequence, → H p−1 (Vét , F |V ) → a Hxp (Uét , F ) → H p (Uét , F ) → H p (Vét , F |V ) → . (1.2) (U −V )ét Lemma 1.1.    ker φx p = 0,   Hxp (Vét , F ) = coker φx p = 1,     H p−1 (I , F ) p ≥ 2. x η Corollary 1.1. Let U and V be the same as before, and F ∈ Sh(Uét ). If V − U consists of only complex places, then H p (Vét , j∗ F ) ≃ H p (Uét , F ). Corollary 1.2. Assume that F is totally imaginary. Let U be any open subset of X = Spec OF and S its complement, then  ×   OF,S       P ic(OF,S )   H p (Uét , Gm ) = {elements of Br F with no invarants in U }      Q/Z      0 p = 0, p = 1, p = 2, p = 3 and U ⊃ X f , otherwise. Proof. By the previous Lemma, we have H p (Uét , Gm ) = H p ((U ∩ X)ét , Gm ). 15 Corollary 1.3. Under the same assumption of the previous corollary. Let v be a place of F , then a. Hv3 (X ét , Gm ) = 0 if v is complex, b. The morphism Hv3 (X ét , Gm ) → H 3 (X ét , Gm ) is an isomorphism if v is a finite place. Proof. Part (a) follows from Lemma 1.1 directly. To see part (b), we set U = X − v and V = X in the long exact sequence (1.2) and note that, by the previous corollary, H 2 (Uét , Gm ) = H 3 (Uét , Gm ) = 0. Thus, the morphism Hv3 (X ét , Gm ) → H 3 (X ét , Gm ) has to be an isomorphism. 1.3.2 The Norm Maps Let Y = Spec OK , where K is a finite extension of F , and π0 : Y0 → X0 the natural projection. There is also a natural projection π∞ : Y∞ → X∞ sending a valuation to the unique one it lays above. We can combine these two projections into π : Y → X, which indeed defines a morphism between sites. More general, let U = (Uf ; U∞ ) be any open subscheme of X and V = π −1 (U ) an open subscheme of Y , then we also denote by π the fibre product of morphism π × (V ,→ X) : V → U . From now on, j denotes either Uf → U or Vf → V and i denotes either U∞ → U or V∞ → V . a. j ∗ π∗ = π∗ j ∗ , Proposition 1.3. b. i∗x π∗ k∗ k ∗ F = Q y/x Fy . c. π ∗ and π∗ are exact. Proof. For any étale Wf → Uf , j ∗ π∗ F (Wf ) = π∗ F (Wf ) = F (Wf ×U V ) = F (Wf ×Uf Vf ) = π∗ j ∗ F (Wf ), which proves the part (a). 16 For proving part (b), note that k∗ k ∗ F = (0, {Fy }y , 0), and then i∗x π∗ k∗ k ∗ F = − lim → π∗ k∗ k ∗ F ((W, w)) (W,w)→(Uf ,x) =− lim → (W,w)→(Uf ,x) =− lim → (W,w)→(Uf ,x) (0, {Fy }y , 0)(π −1 ((W, w))) Y Fy y/x = Y Fy . y/x As π is a morphism between sites, π ∗ is exact. To see π∗ is exact, suppose F ∈ Sh(Vét ) has decomposition (G , {Gv }, {φv }), then π∗ F is the sheaf sending W to i∗x π∗ F = − lim → (W,w) π∗ j∗ j ∗ G ((W, w)) ×π∗ k∗ k∗ j∗ j ∗ G ((W,w)) π∗ k∗ k ∗ G ((W, w)), where the limit runs over all étale (W, w) → U than sends w to x. Since taking stalks commute with fibre products, so the above direct limit equals to i∗x π∗ j∗ j ∗ G ×i∗x π∗ k∗ k∗ j∗ j ∗ G i∗x π∗ k∗ k ∗ G . By part (b), we know that i∗x π∗ k∗ k ∗ G = Q ∗ ∗ ∗ y/x Gy and ix π∗ k∗ k j∗ j G = Q Iy y/x Gη . Also observe that Ix Ix i∗x π∗ j∗ j ∗ G = i∗x j∗ π∗ j ∗ G = f ∗ π∗ G (Spec F ) = π∗ f 0∗ G (Spec F ) = f 0∗ G ( Y y/x where f : Spec F Ix → U and f 0 and π 0 are defined by the fibre product Q y/x Spec K  Iy f0 /V π0 Spec F π Ix f  /U Iy Spec K ) = Y y/x GηIy , 17 Therefore, i∗x π∗ F = Y GηIy ×Q Y Iy y/x Gη y/x Gy = y/x Y Gy = y/x Y Fy , y/x and so the functor i∗x π∗ is exact for all x ∈ U∞ . Together with the fact, j ∗ π∗ = π∗ j ∗ is exact, we see that π∗ have to be exact. One can generalize the classical norm map K × → F × to the level of étale sheaf, ie. N : π∗ Gm,Vf → Gm,Uf . Applying j∗ to N , where j : Uf → U , one obtains a morphism π∗ Gm,V → Gm,U . We also denote this morphism by N . For any F ∈ Sh(Vét ), we define a norm map N as the following composition, adj norm N (F ) : RHomSh(Vét ) (F , Gm,V ) −→ RHomSh(Uét ) (π∗ F , π∗ Gm,V ) −−−→ RHomSh(Uét ) (π∗ F , Gm,U ), recalling that Ri π∗ = 0 for i > 0 as π∗ is exact. Here [A, B] denotes RHomD(Sh(Vét )) (A, B) for A, B ∈ Sh(Vét ). Theorem 1.2 (Norm Theorem). Let π : V → U be the same as before, and assume that F is totally imaginary, then N (j∗ F ) is a quasi-isomorphism for all Zconstructible sheaf F on V0 . Proof. The norm functor transfers the short exact sequence, 0 → j! F → j∗ F → k∗ k ∗ j∗ F → 0, to the morphism of exact triangles RHomSh((Vf )ét ) (F , Gm,Vf )  N1 RHomSh((Uf )ét ) (π∗ F , Gm,Uf ) / RHomSh(Vét ) (j∗ F , Gm,V ) N2  / RHomSh(Uét ) (π∗ j∗ F , Gm,U ) / RHomSh(V ) (k∗ k ∗ j∗ F , Gm,V )  N3 / RHomSh(Uét ) (π∗ k∗ k ∗ j∗ F , Gm,U ) (Note j! commutes with π∗ .) By [10] III.3.9, N1 is a quasi-isomorphism and [4] Lemma 3.8 ensures that both of the domain and the codomain of N3 are vanishing. Therefore, N2 is a quasiisomorphism. / / 18 1.3.3 The Étale Cohomology with Compact Support There are many kinds of étale cohomologies with compact support. Let φ : U = Spec OL,S → Spec OL . We denote by Hci (Uét , F ) := H i ((Spec OL )ét , φ! F ) the cohomology with compact support, which is equivalent to be defined by the exact triangle RΓc (Uét , F ) → RΓ(Uét , F ) → Y RΓ(Lw , F ) → . w∈S e i (Uét , F ), by One can define a modified cohomology with compact support, H c replacing the complex RΓ(Lw , F ) in the above exact triangle by RΓT ate (Lw , F ) for each infinite places w. Since Tate cohomology is isomorphic to the original cohomology for degree ≥ 1, these two cohomologies with compact support are the same for degree ≥ 2. Moreover, the difference between these two kinds of cohomologies is measured by the exact triangle, f c (Uét , F ) → RΓc (Uét , F ) → RΓ M RΓ∆ (Lw , F ) →, w∈S∞ where RΓ∆ (Lw , F ) := cone(RΓ(Lw , F ) → RΓT ate (Lw , F ))[−1]. Assume that L is totally imaginary and set F = Z, we have the following exact triangle, f c (Uét , Z) → RΓc (Uét , Z) → RΓ M Z [0] →, w|∞ Artin-Verdier Duality Consider the pairing f c (Uét , F ) × RHomU (F , Gm ) → RΓc (Uét , Gm ) → Q/Z[−3]. RΓ ét It induces a morphism g (F ) : RHomU (F , Gm ) → RΓ f c (Uét , F )D [−3]. AV ét (1.3) 19 g (Z)) is The Artin-Verdier duality theorem (cf. [10] Theorem 3.1) shows that H i (AV g (Z)) : US ,→ U bS , which can be summarized in an isomorphism for i ≥ 1 and H 0 (AV an exact triangle f c (Uét , Z)D [−3] → U bS /US [0] → . RΓ(Uét , Gm ) → RΓ (1.4) Together with the Q/Z-dual of (1.3), we obtain the following exact triangle bS /US [0] ⊕ (Πv|∞ Q/Z)[−2] → . RΓ(Uét , Gm ) → RΓc (Uét , Z)D [−3] → U (1.5) Exact triangles (1.3), (1.4), and (1.5) will be used in the last part of chapter 4. 20 Chapter 2 The Weil-Étale Cohomology of S-integers 2.1 The Definition We fix a number field F and set Y = Spec OF . Let K/F be a finite Galois extension and S be a finite set of non-trivial valuations of F , containing all those which ramify in K/F . The site TK/F,S was defined in [9] as follows: The objects of TK/F,S are the collections ((Xv ), (fv ))v∈Y , where Xv is an Wk(v) -space, and fv : Xv → Xv0 is map of WFv -spaces. ( We regard Xv as an WFv -space via πv , and Xv0 as an WFv -space via Θv ). Further, we require that the action of WF on Xv0 factors through WL/F,S . Let X = ((Xv ), (fv )) and X 0 = ((Xv0 ), (fv0 )) be objects of TK/F,S , then Hom(X , X 0 ) is the collections of WFv -maps gv : Xv → Xv0 such that gv0 fv = fv0 0 gv for all v. And the fibre product of two morphisms with the same codomain is defined componentwise. The covering Cov(TK/F,S ) consists of the family of morphisms {((Xi,v ), (fi,v )) → ((Xv ), (fv ))}i such that {Xi,v → Xv }i is a local section covering of Xv for all v. We denote T^ K/F,S the topos Sh(TK/F,S , Jls ). Clearly, ∗K/F,S = ((pt), (id)) is the final object of TK/F,S . For any abelian sheaf FK,S on it, we define the cohomology H p (TK/F,S , FK,S ) by H p (TK/F,S , ∗, FK,S ). 21 For each non-trivial place v, the map i−1 K,S,v : ((Xv )) 7→ Xv induces an embedding of topoi iK,S,v : BWk(v) → T^ K/F,S . When v = v0 , we denoted by jK,S the embedding iK,S,v : BWK/F,S → T^ K/F,S . For any finite Galois extension K 0 of K, and S 0 ⊃ S, the canonical morphism ^ p : WK 0 /F,S 0 → WK/F,S defines a transition map tK 0 /K,S 0 /S : T^ K 0 /F,S 0 → TK/F,S ( by regarding any WK 0 /F,S 0 -space as an WK/F,S -space via p). We will throw out K, S from the index if there is no risk of confusing. a) i∗v , iv,∗ and j ∗ are exact, and iv,∗ i∗v = Proposition 2.1 ([12] Prop. 3.5, 3.14). id. b) We have the following commutative diagram j 0 0 o BWK 0 /F,S0 K ,S / T^ K 0 /F,S 0 iK 0 ,S 0 ,v  BWK/F,S jK,S BWk(v) v vv vv t v v  {vv iK,S,v / T^ . K/F,S Proposition 2.2 (ibid., Prop. 3.15, 3.16). a) There is a morphism fK,S : T^ K/F,S → BR so that fK,S ◦ iv is isomorphic to Blv : BWk(v) → BR , for any closed point v of Y. b) The following diagram is commutative / T^ K/F,S JJ JJ JJ JJ J$  T^ K 0 /F,S 0 BR for any F /K 0 /K/F and S ⊂ S 0 . Definition 2.1. A compatible system of abelian sheaves on the sites (TK/F,S )K,S is a collection {(FK,S ), (φK 0 /K,S 0 /S )}, where FK,S ∈ T^ K/F,S and φK 0 /K,S 0 /S is a mor- 22 phism t∗K 0 /K,S 0 /S FK,S → FK 0 ,S 0 , such that if K 00 /K 0 /K a series of finite Galois extensions and S 00 ⊃ S 0 ⊃ S, then φK 00 /K,S 00 /S = t∗K 00 /K 0 ,S 00 /S 0 (φK 00 /K 0 ,S 00 /S 0 ) ◦ φK 0 /K,S 0 /S . Definition 2.2. Let F = {(FK,S ), (φK 0 /K,S 0 /S )} be any compatible system of sheaves on the sites (TK/F,S )K,S , then we define the Weil-étale cohomology on Spec OF for F , H p (Y W , F ) to be lim −→ K,S H p (TK/F,S , FK,S ). Example 2.1. Let pK,S be the morphism BWF → BWK/F,S and A be an abelian object e := {(jK,S,∗ pK,S,∗ A), (φ)} of BWF associated to a topological abelian WF -group A. Then A forms a compatible system of sheaves, where φK 0 /K,S 0 /S : t∗ jK,S,∗ pK,∗ A = t∗ t∗ jK 0 ,S 0 ,∗ pK 0 ,∗ A → jK 0 ,S 0 ,∗ pK 0 ,∗ A e is the natural map induced by the adjunction. We define H p (Y W , A) to be H p (Y W , A). e is In particular, if A is a topological abelian group on which WF acts trivially, then A called the constant sheaf with A-value. We denote by A the constant sheaf defined by A if A has the discrete topology. g In [12], Morin gave a direct description of the topos T^ K/F,S . Recall that WKv g g denotes the image of WKv in WK/F,S , and θv : W Kv → WK/F,S and qv : WKv → Wk(v) are the induced continuous maps. Definition 2.3 (ibid., Def. 3.1). We define a category FK/F,S as follows. The objects of this category are of the form F = (Fv ; fv )v∈Y , where Fv is an object of BWk(v) for v 6= v0 ( resp. of BWK/F,S for v = v0 ) and fv : qv∗ (Fv ) → θv∗ (Fv0 ) is a morphism of BWg so that fv0 = idFv0 . A morphism φ from F = (Fv ; fv )v∈Y K v to F 0 = (Fv0 ; fv0 )v∈Y is a family of morphisms φv : Fv → Fv0 ∈ F l(BWKv )(and φv0 ∈ F l(BWK/F,S )) so that qv∗ (Fv )  qv∗ (φv ) / q ∗ (F 0 ) v fv qv∗ (Fv0 )  v θv∗ (φv0 ) fv0 / θ ∗ (F 0 ) v v0 23 is a commutative diagram of BWg . K v For K = F and S the set of all non-trivial valuations of F , one has WK/F,S = g WF , W Fv = WFv and we set FK/F,S = FW ;Y . Proposition 2.3 (ibid., Thm. 5.9). FK/F,S is a topos and is equivalent to T^ K/F,S as a topos. In the following, we identify FK/F,S and T^ K/F,S . Proposition 2.4 (a special case of [12] Lemma 4.5 & Prop. 4.6). Let A be an abelian object in BWF defined by A on which WF acts trivially, then a) Rp j∗ A := {(Rp (jK,S,∗ )(pK,S,∗ A), (φ)} is a compatible system of sheaves. b) There exists a spectral sequence H p (Y W , Rq j∗ A) =⇒ H p+q (WF , A). Proposition 2.5 ([9] Thm 5.10, [5] Theorem 11.1). H p (Y W , Z) =     Z        0       Pic1 (Y )D    µD F        is of infinite rank       0 p = 0, p = 1, p = 2, p = 3, even p ≥ 4, odd p ≥ 5, where Pic1 (Y ) is the kernel of the absolute value map from Pic(Y ) to R>0 and Pic(Y ) is the Arakelov Picard group of F , which is obtained by taking the idèle group of F and dividing by the principal idèles and the unit idèles( i.e. those idèle (uv ) s.t. |uv |v = 1 for all v.). 24 Proposition 2.6 ([9] Theorem 5.11). e = H p (Y W , R)   R p = 0, 1,  0 p > 1. More generally, for any open subscheme U = Spec OF,S of Y , U defines an open sub-topos FK/F,S 0 /U → FK/F,S 0 , the full sub-category of FK/F,S 0 whose objects are of the form F = (Fv ; fv )v∈U (i.e. Fv = ∅ for v ∈ Y − U ). One can show that FK/F,S 0 /U U U is equivalent to T^ 0 is defined similarly as TK/F,S 0 by throwing out 0 , where T K/F,S K/F,S from each object all the v-components for v ∈ Y − U . We have the usual embeddings S S jK/F,S 0 : BWK/F,S 0 → FK/F,S 0 /U and iv : BWk(v) → FK/F,S 0 /U . Definition 2.4. We define the cohomology H p (FK/F,S 0 , U, −) := H p (FK/F,S 0 /U , −), p 0 and H p (UW , Z) = − lim → H (FK/F,S , U, Z). Proposition 2.7. For any proper open subscheme U = Spec OF,S of Y , H p (UW , Z) =    Z       0 p = 0, p is odd, `    (CF1 / v∈U Uv )D      H p (W , Z) F p = 2, p ≥ 3. Proof. There exists the Leray spectral sequence induced by the inclusion of the generic point p q S p+q 0 lim −→K/F,S 0 H (FK/F,S /U , R jK/F,S 0 ,∗ Z) =⇒ H (WF , Z). S By the same argument of [9] Theorem 4.8, for q > 0, one has Rq jK/F,S 0 ,∗ Z = S S∗ q S S∗ q S q S∗ S v∈S 0 \S iv ∗ iv R jK/F,S 0 ,∗ Z. Moreover, the functor iv R jK/F,S 0 ,∗ = R (iv jK/F,S 0 ,∗ ) = ∗ ∗ S q ∗ ∗ q S Rq (iSv iS iS ∗ jK/F,S is the open embed0 ,∗ ) = R (iv jK/F,S 0 ,∗ ) = iv R jK/F,S 0 ,∗ , where i ` ding FK/F,S 0 /U → FK/F,S 0 . S p ∗ q Therefore, for any q > 0, H p (FK/F,S 0 /U , Rq jK/F,S 0 ,∗ Z) = Πv∈S 0 \S H (Wk(v) , iv R jK/F,S 0 ,∗ Z). 25 By the computation due to Flach [5], we have     Z    Q p q S D 0 lim v∈U Uv −→K/F,S 0 H (FK/F,S /U , R jK/F,S 0 ,∗ Z) =      0 q = 0, p = 0 and q = 2, p > 0 and q 6= 2. So the 2nd page of the spectral sequence looks like the following, 0 UvD Q v∈U 0 0 H 0 (UW , Z) H 1 (UW , Z) H 2 (UW , Z) H 3 (UW , Z). This gives us H 0 (UW , Z) = Z, H 1 (UW , Z) = H 1 (WF , Z) = 0 and the exact sequence, 0 / H 2 (U W , Z) / H 2 (W F , Z) / Q D v∈U Uv / H 3 (U / C1 / H 2 (U W , Z) / H 3 (W F , Z) = 0, CF1,D which is dual to 0 / H 3 (U D W , Z) / ` v∈U Uv F D W , Z) / 0. As the middle map is injective when U is a proper subset of Y , we see that H 2 (UW , Z) = ` (CF1 / v∈U Uv )D and H 3 (UW , Z) = 0. And for p ≥ 3, H p (UW , Z) = H p (WF , Z), which is vanishing for odd p ≥ 3. Definition 2.5. Let F = (FK/F,S ) be a compatible system on sheaves on the site 26 (TK/F,S ). Then i∗v F := (i∗K,S,v FK/F,S ) a compatible system of sheaves on the site BWk(v) and it gives rise to restriction maps H ∗ (TK/F,S , FK/F,S ) → H ∗ (k(v)W , i∗K,S,v FK/F,S ). Let U be any open subscheme of Y . Since U = Y , we define the Weil-étale cohomology with compact support for an open subscheme U with coefficient F as the cohomologies of the mapping cone   M RΓc (UW , F ) := cone RΓ(Y W , F ) → RΓ(k(v)W , i∗v F ) [−1], v∈Y \U where k(v)W := BWk(v) . The map RΓ(Y W , F ) → RΓ(k(v)W , i∗v F ) is derived by first applying global section functor to IF•K/F,S → iK,S,v,∗ Ii•∗K,S,v FK/F,S and then taking the inductive limits, where IF•K/F,S (resp. Ii•∗K,S,v FK/F,S ) is an injective resolution of FK/F,S (resp. i∗K,S,v FK/F,S ). Note that the complex RΓc (UW , F ) is uniquely determined up to homotopy equivalences. As H p (k(v)W , Z) = H p (Wk(v) , Z) has non-vanishing cohomology (all equals to Z) only when v|∞ and p = 0 or v - ∞ and p = 0, 1. And H p (k(v)W , R) equals to R (resp. 0) if p = 0, 1 (resp. p > 1). One can easily deduce the following, Proposition 2.8 ([9] Thm. 6.3). Let Y = Spec OF , then     0 p = 0,    ` Hcp (YW , Z) = ( v|∞ Z)/Z p = 1,      H p (Y W , Z) p ≥ 2, and Hcp (YW , R) =  `  (  0 v|∞ R)/R p = 1, 2, otherwise. 27 2.2 The Weil-Étale Sites and the Artin-Verdier Étale Sites All the material in this subsection can be found in [12]. We will only give brief definitions and state the results. Proposition 2.9 (ibid., Prop 7.1). There exists a morphism of left exact sites ζ ∗ : (Et(Y ); Jét ) → (TW ;Y ; Jls ) 7→ X = ((Xv ); (ϕv )), X sh sh where Xv := HomY (Y v ; X) and Y v = Spec(k(v)) (resp. = (∅; v)) when v - ∞ (resp. v|∞). Let K/F be a finite Galois extension. We denote by EtK/F the full sub-category of Et(Y ) consisting of étale Y -schemes X such that the action of GF on the finite set Xv0 := Hom(Spec(F ); X) factors through Gal(K/F ). This category is again endowed with the topology induced by the étale topology on Et(Y ) via the inclusion functor EtK/F → Et(Y ). These morphisms are compatible and induce a projective system ((Et^ K/F , Jét ))K . Proposition 2.10 (ibid., Prop. 7.7). The canonical morphism ^ Y ét → lim ←− (EtK/F , Jét ) is an equivalence. Proposition 2.11 (ibid., Prop. 7.5 & 7.8). a) ζ ∗ : (Et(Y ); Jét ) → (TW ;Y ; Jls ) is the inverse image of a morphism of topoi ζ : FW ;Y → Y ét . b) ζ induces a morphism of topoi ζK,S : FK/F,S → (Et^ K/F , Jét ). Moreover, the 28 diagram FK 0 /F,S 0  FK/F,S ζK 0 ,S 0 / ^ 0 /F , Jét ) (EtK ζK,S /  (Et^ K/F , Jét ) is commutative, for K 0 /K/F and S ⊂ S 0 . Proposition 2.12 (ibid., Prop. 4.69). For any closed point v ∈ Y , the diagram BWk(v)  αv sm / BG k(v) iv FW,Y ζ  uv / f Y ét . is a pull-back of topoi. Theorem 2.1 (ibid., Thm. 8.5 & Prop. 8.6). Let F = (FK,S ; ft ) be a compatible system of sheaves on the sites (TK/F,S )K,S . There exists a bounded below complex RF of abelian Y ét -sheaves and an isomorphism H p (Y ét , RF ) ≃ H p (Y W , F ), where the left-hand side is the étale hypercohomology of the complex RF . In particular, one has a spectral sequence relating Lichtenbaum’s Weil-étale cohomology to étale cohomology, H p (Y ét , Rq F ) =⇒ H p+q (Y W , F ), where Rq F := H q (RF ). The complex RF is well-defined up to quasi-isomorphism and functorial in F . Moreover, Rq F is the sheaf associated to the presheaf P q F : Y ét −→ Ab U 7→ H q (UW , F ). Remark. For the future use, we describe the construction of the complex RA. 29 • For each pair (K, S), we fix an injective resolution 0 → FK,S → IK,S in FK,S . • • and with morphism of complexes t∗ IK,S → IK 0 ,S 0 , for any transition map t. These • complexes are compatible. Then one can show that (ζK,S,∗ IK,S )S form a direct systems of complexes. We set • • := − lim IK → ζK,S,∗ IK,S . S • Then (IK )K defines a compatible system of complexes of sheaves on the sites (EtK/F )K . Finally, the complex ∗ • RF := − lim → uK IK , K where u : Y ét → (Et^ K/F , Jét ). We are interested in the complex τ ≤3 RΓ(Y W , Z), and Morin proved the following. Theorem 2.2 (ibid., Thm 8.5 & 9.5). Suppose F is totally imaginary then the following hold qis a) RΓ(Y ét , τ ≤3 RZ) −→ τ ≤3 RΓ(Y W , Z), qis b) RΓ(Y ét , R2 Z) −→ HomZ (UF , Q)[0]. Here is an immediate corollary of this theorem: Corollary 2.1. We have the following exact triangle RΓ(Y ét , Z) → τ ≤3 RΓ(Y W , Z) → HomZ (UF , Q)[−2] → . Proof. By Prop. 2.7, H 1 (UW , Z) = 0 for all open étale U over Y ét , so R1 Z = 0 by Theorem 1.2. Thus, we have an exact triangle Z[0] −→ τ ≤3 RZ −→ R2 Z[−2] −→ . Applying RΓ(Y ét , −) to the above exact triangle and by using the previous Theorem, we obtain the desired exact triangle. 30 Chapter 3 Cohomology with Compact Support for the Ring of S-integers In this whole chapter, we assume that F is a totally imaginary number field, and e Y = Spec OF,S . We shall compute the cohomology groups Hc∗ (YW , Z) and Hc∗ (YW , R) in section 3.1 and prove that the axioms of the Weil-étale cohomology theory hold for Y in section 3.2. 3.1 The Computation of the Cohomology with Compact Support Let A be an abelian group with trivial WF -action and A the constant sheaf defined by it. Recall that (cf. Def. 2.5) the Weil-étale cohomology groups with compact support Hc∗ (YW , A) are defined by the cohomologies of the mapping cone   M RΓc (YW , A) = cone RΓ(Y W , A) → RΓ(k(v)W , A) [−1], v∈S which is equivalent to the exact triangle RΓc (YW , A) → RΓ(Y W , A) → M RΓ(k(v)W , A) → . v∈S We are interested in the complex τ ≤3 RΓc (YW , Z), and want to prove first the 31 following. Theorem 3.1. There is an exact triangle RΓc (Yét , Z) / τ ≤3 RΓ (Y / Hom(US , Q)[−2] → . W , Z) c (3.1) Before proving Theorem 3.1, we show the following lemma. Lemma 3.1. For any constant sheaf A, the inclusion A[0] → RA induces a morphism RΓ(Y ét , A) −→ RΓ(Y W , A), and the following diagram commutes / RΓ(k(v)ét , A) RΓ(Y ét , A) α∗v   / RΓ(k(v)W , A), RΓ(Y W , A) where the morphisms of the rows are the canonical ones and αv∗ is induced by the sm . canonical morphism of topos αv : BWk(v) → BG k(v) Hence, taking the truncation functor τ ≤3 on every terms gives rise to the following commutative diagram. τ ≤3 RΓ(Y ét , A) / RΓ(k(v)ét , A)   α∗v / RΓ(k(v)W , A). τ ≤3 RΓ(Y W , A) Proof. Recall that by the knowledge of section 2.2, there exists a commutative diagram of topoi k(v)W  iK,S αv iv Tf Y   T^ K/F,S ζ ζK,S / k(v)ét i  / g EtF  uK / Et ^, K/F sm where αv : BWk(v) → BG . Here we drop v in the indexes. k(v) 32 Clearly, the following diagram commutes. / RΓ(Y RΓ(Y ét , A)  RΓ(Y W , A) = RΓ(Y ét , RA) ∗ ét , i∗ i A) = RΓ(k(v)ét , A) / RΓ(Y  ∗ ét , i∗ i RA) . To complete the proof, we need to show the morphism RΓ(Y W , A) → RΓ(k(v)W , A) factors through RΓ(Y W , A) → RΓ(Y ét , i∗ i∗ RA). Note that the morphism RΓ(Y W , A) → RΓ(k(v)W , A) is induced by ∗ ^ ^ lim −→ RΓ(TK/F,S , A) −→ lim −→ RΓ(TK/F,S , iK,S,∗ iK,S A). By Theorem 2.1, this is the same as the morphism of complexes of Y ét -sheaves ∗ • 0 ∗ • RA := − lim lim → uK lim −→ ζK,S,∗ IK,S −→ R A := − → uK lim −→ ζK,S,∗ iK,S,∗ JK,S , • • ) is any fixed injective resolution of A (resp. i∗K,S A) and note (resp. JK,S where IK,S that iK,S,∗ is exact and preserves injectives. By applying the natural functor id → i∗ i∗ to RA → R0 A, we get the following canonical commutative diagram RA pRA  i∗ i∗ RA / R0 A  pR 0 A / i i∗ R0 A. ∗ 33 Note that ∗ • R0 A = − lim →K uK lim −→S ζK,S,∗ iK,S,∗ JK,S ∗ • =− lim →K uK lim −→S uK,∗ i∗ αv,∗ JK,S ∗ • =− lim →K lim −→S uK uK,∗ i∗ αv,∗ JK,S • =− lim →K lim −→S i∗ αv,∗ JK,S ∗ (lim lim −→K lim −→S uK uK,∗ FK,S = − →K lim −→S FK,S (c.f. [11] lemma 7.5)) = i∗ Rαv,∗ A. Because applying with i∗ induces the adjoint isomorphism Hom(i∗ P, i∗ Q) = Hom(P, Q), and i∗ pR0 A = id, by definition, so pR0 A = id and the result follows. Remark. Whenever there is a square commutative diagram of complexes in an abelian category A, then by taking mapping cones, one can complete it as a semi-commutative 3 by 3 diagram of complexes in which rows and columns are exact triangles in the derived category D(A). (c.f. [17] Exercise 10.2.6). Proof of Theorem 3.1. By [11] Prop. 6.5 and 6.6, Rq αv,∗ Z =   Z, Q, 0 for q = 0, 1, and q ≥ 2 (v - ∞),  Z, 0 for q = 0, and q ≥ 1 (v|∞). Since Rαv,∗ Z has trivial cohomology in degree greater than 1, we have an exact triangle Z[0] → Rα∗ Z → R1 αv,∗ Z[−1] →. On the other hand, by Prop. 2.7, H 1 (UW , Z) = 0 for all open étale subset U of Y ét , so R1 Z = 0 by Theorem 1.2. And then we have an exact triangle Z[0] −→ τ ≤3 RZ −→ R2 Z[−2] −→. Together with Lemma 3.1 and Theorem 2.2, there exists a morphism of exact triangles / τ ≤3 RΓ(Y RΓ(Y ét , Z) L  v ∈Y / RΓ(k(v)ét , Z) / L / RΓ(Y W , Z)  v ∈Y / RΓ(k(v)W , Z) / 2 ét , R Z)[−2]  ∃γ 1 v ∈Y / RΓ(k(v)ét , R αv,∗ Z)[−1] L / where the rows are exact triangles, and the vertical morphisms in the first 2 columns /, 34 are the canonical ones. By the remark above, we get an exact triangle / τ ≤3 RΓ (Y RΓc (Yét , Z) c / RΓ g (Y W , Z) c /, 2 ét , R Z)[−1] where  gc (Yét , R2 Z) := cone RΓ(Y ét , R2 Z)[−2] RΓ γ / L  1 RΓ(k(v) , R α Z)[−1] . ét v,∗ v∈S Since RΓ(Y ét , R2 Z) = Hom(OF× , Q)[0] (Theorem 2.2) and R1 αv,∗ Z = 0 (resp. Q) when v is an infinite place (resp. a finite place),   M × 2 g RΓc (Yét , R Z) = cone Hom(OF , Q)[−2] → Q[−1] . v∈Sf gc (Yét , R2 Z) By looking at the long exact sequence of cohomology, one can identify RΓ with Hom(US , Q)[−1]. Theorem 3.2. Let Y = Spec OF,S , then Hcp (YW , Z) =    0      Q   Z/Z p = 0, p = 1, S    Pic(Y )D × Hom(US , Z) p = 2,      µD p = 3. F Proof. By the definition of the cohomology with compact support, we have the following long exact sequence 4 0 / H 0 (Y c W , Z) /Z → ⊕v∈S<∞ Z / H 2 (Y W , Z) / Pic1 (Y )D c / ⊕v∈S Z / H 1 (Y /0→ /0 / H 3 (Y / µD c c W , Z) W , Z) F Since 4 is the diagonal map, we see that Hc0 (YW , Z) = 0 and Hc1 (YW , Z) = Z|S| /Z. And Hc3 (YW , Z) = µD F , trivially. Finally, the exact triangle in Theorem 3.1 induces a / 0. 35 long exact sequence of cohomology groups 0 / H 2 (Y c ét , Z) / H 2 (Y c W , Z) / H 3 (Y / Hom(US , Q) Pic(Y )D c ét , Z) / µD F /0 Hom(US , Q/Z) Since Hom(US , Q) is torsion free, the injection on the left implies Hc2 (YW , Z)tor = Pic(Y )D . On the other hand, the long exact sequence in the beginning of the proof shows rankZ (Hc2 (YW , Z)) = #{v|v ∈ / Y } = rankZ (US ), as Pic1 (Y )D = Pic(OF )D × Hom(OF× , Z)([9] Prop. 6.4). As any free abelian subgroup of Hom(US , Q) of rank # US is isomorphic to Hom(US , Z), Consequently, Hc2 (YW , Z) = Pic(Y )D × Hom(US , Z). By a direct computation of the long exact sequence defining the cohomologies with compact support, we see that Proposition 3.1. e = Hcp (YW , R)  Q   R/R p = 1, 2,  0 otherwise. S In the end of this section, we compare RΓc ((Spec OF,S0 )W , A) and RΓc ((Spec OF,S1 )W , A) for future use, where Y ⊃ S1 ⊃ S0 . Proposition 3.2. Let Y ⊃ S1 ⊃ S0 ⊃ Y ∞ and F a compatible system of sheaves on the sites (TK/F,S ), then we have an exact triangle RΓc ((Spec OF,S1 )W , F ) → RΓc ((Spec OF,S0 )W , F ) → M RΓ(k(v)W , i∗v F ). v∈S1 \S0 Proof. We first have the following commutative diagram of complexes in which the 36 rows are exact triangles, RΓc (Spec OF,S1 , F )  RΓc (Spec OF,S0 , F ) / RΓ(Y , F ) / L / RΓ(Y , F ) / L ∗ v∈S1 RΓ(k(v)W , iv F )  ∗ v∈S0 RΓ(k(v)W , iv F ) / /. This diagram can be completed by the following 3 by 3 commutative diagram of complexes, / RΓ(Y , F ) / L RΓc (Spec OF,S0 , F ) / RΓ(Y , F ) / L   /0   RΓc (Spec OF,S1 , F )  L ∗ v∈S1 \S0 RΓ(k(v)W , iv F ) / / ∗ v∈S1 RΓ(k(v)W , iv F ) L  ∗ v∈S0 RΓ(k(v)W , iv F )  ∗ v∈S1 \S0 RΓ(k(v)W , iv F )[1] / /,  (3.2) where all the rows and columns are exact triangles. Then the result follows from the first column. Corollary 3.1. Let A be Z or R on which WF acts trivially, then there is an exact triangle τ ≤3 RΓc ((Spec OF,S1 )W , A) → τ ≤3 RΓc ((Spec OF,S0 )W , A) → M RΓ(k(v)W , A) → . v∈S1 \S0 Proof. This follows easily from the fact τ ≥2 RΓ(k(v)W , A) = 0 when A = Z, R. Corollary 3.2. For A = Z, the long exact sequence induced by the exact triangle in the above corollary splits into short exact sequences 0→ a H i (k(v)W , Z) → Hci+1 ((Spec OF,S1 )W , Z) → Hci+1 ((Spec OF,S0 )W , Z) → 0, S1 \S0 for all i ≥ 0. 37 Proof. It is suffice to prove the induced morphisms Hci ((Spec OF,S0 )W , Z) → ` i S1 \S0 H (k(v)W , Z) are zero maps for all i. As Hci ((Spec OF,S0 )W , Z) is concentrated in degree 1,2 and 3 and H i (k(v)W , Z) is concentrated in degree 0 and 1, we only need to consider the case i = 1. Since the 3rd row of diagram (3.2) comes from the splitting exact sequence of complexes M 0→ RΓ(k(v)W , Z) → M RΓ(k(v)W , Z) → v∈S1 v∈S1 \S0 M RΓ(k(v)W , Z) → 0. v∈S0 It is then clear that the morphisms M δ H i−1 (k(v)W , Z) → − v∈S0 M H i (k(v)W , Z) v∈S1 \S0 are zero maps for i ≥ 0. Therefore, the morphism of long exact sequences induced by the morphism of the 2nd and the 3rd rows in diagram (3.2) looks like / / L L v∈S0 H 0 (k(v)W , Z) v∈S1 \S0 H  f / H 1 ((Spec OF,S )W , Z) c 0 1 (k(v)W , Z) ∼ / L v∈S1 \S0 H  0 / H 1 (Y 0 h 1 (k(v)W , Z)  /0 As f is surjective and the diagram is commutative, h has to be the zero map. 3.2 Verification of the Axioms of Weil-Étale Cohomology As before, F is totally imaginary and Y = Spec OF,S . In this subsection, we verify that the axioms (a)-(d) stated in the introduction hold for the generalized Weil-étale cohomology theory. / W , Z)(= 0) / 38 3.2.1 Axiom (a) We have seen that Hcp (YW , Z) are finitely generated abelian groups for p = 0, 1, 2, 3. Unfortunately, Hcp (YW , Z) is still of infinite rank, for odd p ≥ 5 as Hcp (YW , Z) ∼ = Hcp (Spec OF , Z). 3.2.2 Axiom (b) We need to show that Hcp (YW , Z) ⊗Z R ≃ Hcp (YW , R). But, obviously, this is false for odd p ≥ 5, as Hcp (YW , R) = 0 then. But we can prove the following. Proposition 3.3. Hcp (YW , Z) ⊗Z R ≃ Hcp (YW , R) for p = 0, 1, 2, 3. Proof. By corollary 3.1 and taking tensor product over Z, we get a morphism between exact triangles τ ≤3 RΓc ((Spec OF,S )W , Z) ⊗Z R / τ ≤3 RΓc ((Spec OF )W , Z) ⊗Z R  τ ≤3 RΓc ((Spec OF,S )W , R)  / τ ≤3 RΓc ((Spec OF )W , R) / L v∈S\S∞ RΓ(k(v)W , Z) ⊗Z R qis / qis L  RΓ(k(v) W , R) v∈S\S∞ Clearly, for v - ∞, H ∗ (k(v)W , Z) ⊗Z R ≃ H ∗ (k(v)W , R), so the right vertical map is a quasi-isomorphism. Also, the middle one is a quasi-isomorphism ([9] Thm. 8.1). Thus, by the property of exact triangles, the left vertical morphism is also a quasi-isomorphism. 3.2.3 Axiom (d) Recall that the L-function ζY (z) for Y is the same as the S-zeta function ζS (s) := P −s a⊂OF,S N (a) . The analytic class number formula shows its Taylor expansion at z = 0 is ζS (z) = − / hS RS |S|−1 z + O(z |S| ), w where hS = # Pic Y , RS the S-regulator, and w = #µF . As Hcp (Y, Z) does not vanish for all even p ≥ 4, we are unable to show that P ords=0 ζY (s) = i≥0 (−1)i i rankZ Hci (X, Z). However, by the computation in previous /. 39 section, we see that 3 X (−1)i i rankZ Hci (X, Z) = −(|S| − 1) + 2(|S| − 1) i=0 = |S| − 1 = ordz=0 ζY (z). 3.2.4 Axiom (c) and (e) Recall that the Euler character is defined as the following Definition 3.1. Let A0 , . . . , An be finitely generated abelian groups, Vi := Ai ⊗Z R, and Ti : Vi → Vi+1 are R-linear maps s.t. 0 → V0 → V1 → · · · → Vn → 0 is exact. We define the Euler characteristic χc (A0 , · · · , An , T0 , · · · , Tn ) to be ± Y i |(Ai )tor |(−1) / Det(V0 , . . . , Vn ; b0 , · · · , bn ), i where bi = (bij )j is an arbitrary choice of Z-basis of Ai /(Ai )tor for all i and Det(V0 , . . . , Vn ; b0 , · · · , bn ) (−1)i is the image of ⊗i (∧j bij ) under the canonical map i ⊗i Det(−1) Vi → R, and b−1 ij is the dual of bij in Hom(Vi , R). Recall that for any R-vector space V , Det V := ∧rank(V ) V , Det−1 V := HomR (Det V, R) and Det0 V := R. Clearly, the Euler characteristic is independent to the choice of basis. 40 Example 3.1. Suppose 0 → A0 → A1 → · · · → An → 0 is an exact sequence of finitely generated abelian groups. It induces an exact sequence of R-vector spaces 0 → A0 ⊗Z R → A1 ⊗Z R → · · · → An ⊗Z R → 0. The determinant of this complex i equals to Πi |(Ai )tor |(−1) . Therefore, the Euler characteristic of this complex is ±1. Example 3.2. Let {Ai,j }i=0,..,m;j=0,..,n be a set of finitely generated non-trivial abelian groups and Bi,j := Ai,j ⊗Z R. Suppose that there exist exact sequences 0 → Ai,0 → Ai,1 → · · · → Ai,n → 0, ∀ i = 0, · · · , m. And suppose there exist R-linear maps Ti,j : Bi,j → Bi+1,j for all i = 0, . . . , m − 1 and j = 0, . . . , n, such that the following diagram commutes and all rows and columns are exact. 0 0 0  / B0,0  / B0,1 T0,0  0 / B1,1 T1,0   .. .  0 / ··· Tm−1,0  /0 T1,n .. . / ···  Tm−1,n  / Bm,n /0  0
(−1)i / B1,n Tm−1,1  /0 T0,n   / Bm,1 0  / B0,n T1,1 .. . / Bm,0 Then Πi χc (Ak,i )k , (Tk,i )k / ··· T0,1  / B1,0 0 0 = ±1. This can be proved by showing the diagram i+j ⊗i (⊗j Det(−1) B ) PPP ij PPP ∼ PPPP PPP ' ∼ R / ⊗ (⊗ Det(−1)i+j B ) j i ij nn n n nn nnn ∼ n n wnn is commutative and all arrows are the canonical isomorphisms.(cf.[8]). 41 Let F = ((FL,S ); (ft )) be a compatible system of abelian sheaves on the sites (TL/F,S ). And observe that H p (TL/F,S , FL/F,S ) = Extp^ (R, FL/F,S ) for any sheaf of TL/F,S p e R-modules F , so H p (Y W , F ) = − lim → Ext ^ (R, FL,S ). This allow us to define a cup TL/F,S product H p (Y W , F ) × H q (Y W , R) → H p+q (Y W , F ). Consider the compatible system ((iv,∗ i∗v FL,S ); (iv,∗ i∗v ft )), and take mapping cone, we can actually define a cup product Hcp (YW , F ) × H q (Y W , R) → Hcp+q (YW , F ). The Leray spectral sequence for jK/F,S yields 0 → H 1 (Y W , R) → H 1 (WF , R) → H 0 (Y W , R1 j∗ R) = 0 (see [9]). Since H 1 (WF , R) = Homcont (WF , R) ∼ = Homcont (R, R)([9]Lemma 3.4) , we may choose an element ψ of H 1 (Y W , R) which corresponds to the composition log |.| ab WF − → CF −−−→ R. Note that ψ corresponds to the identity map in Homcont (R, R), and ψ ` ψ = 0, so the sequence (Hci (Y, F ), ` ψ): `ψ · · · → Hci (YW , F ) −−→ Hci+1 (YW , F ) → · · · is a complex. In the following, we will prove that (Hci (YW , R), ` ψ) is exact. Together with e ∼ the isomorphisms Hci (YW , R) = Hci (YW , Z) ⊗Z R for i ≤ 3, the Euler characteristic
χc (Hcq (YW , Z))q≤3 , (` ψ) is well-defined and called the Euler characteristic χc (Y ) for Y . It was already known that χc (Spec OF ) = Rh/w.([9] Thm. 8.1). 42 Proposition 3.4. e ` ψ) is exact. (a) The complex (Hci (YW , R), (b) Moreover, its Euler character equals ± RSwhS , where RS is the S-regulator. e 6= 0 only for i = 1, 2. So we only need to show (a) Recall that Hci (YW , R) Proof. `ψ e −−→ H 2 (YW , R) e Hc1 (YW , R) c is an isomorphism. For this, consider the morphism between two exact sequences / 0 0 / ` ` S\S∞ H S\S∞ H 0 1 / H 1 (Y (k(v)W , R)  c `ψ / H 2 (Y (k(v)W , R) c / H 1 ((Spec O W , R) e  c `ψ  e c /0 F )W , R) / 0, `ψ / H 2 ((Spec O W , R) e F )W , R) e (3.3) in which the rows are obtained by tensoring with R over Z the exact sequences derived in corollary 3.2. Recall that the right vertical map is an isomorphism ([9]Thm. 8.1). We are now showing the left vertical map is also an isomorphism. Note that iv is an embedding and preserves injectives, so that we have the following commutative diagram H 0 (k(v)W , R) × e H 1 (Y , R) H 0 (k(v)W , R) × H 1 (k(v)W , R)  ` / H 1 (k(v) ` / H 1 (k(v) W , R) W , R). For any finite place in S, the diagram (1.1) ensures that the image of ψ in H 1 (k(v)W , R) = Hom(Wk(v) , R) is lv : σ 7→ log N (v), where σ is the generator of Wk(v) . Let 1v is the identity element in H 0 (k(v)W , R) = R. Then 1v ` ` ψ = 1v ` lv = lv . So, cup product with ψ identifies S\S∞ H 0 (k(v)W , R) and ` 1 S\S∞ H (k(v)W , R). Therefore, the middle vertical map of diagram (3.3) is also an isomorphism. 43 (b) The rows of (3.3) are induced by tensoring with R the exact sequences 0→ a H i (k(v)W , Z) → Hci+1 (YW , Z) → Hci+1 ((Spec OF )W , Z) → 0, i = 0, 1. S<∞   Let χ0 := χc (0, H 0 (k(v)W , Z), H 1 (k(v)W , Z)), (` ψ) . Applying Example 3.2 to (3.3), we see that χc (Y ) = ±χc (Spec OF )χ0 . Because H i (k(v)W , Z) is torsion free, and we has seen in (a) that ` ψ sends 1v to lv : σ 7→ log N (v), so χ0 = ±ΠS<∞ log N (v). Thus, χc (Y ) = ± Rh Y RS hS log N (v) = ± w S w <∞ (c.f. [15] Lemma 2.1 for the last equality). 3.3 A Canonical Representation of Tate Sequences As before Y = Spec OF,S and G = Gal(F/L) for a subfield L of F . Recall that Hcp (YW , Z) =    0      Q   Z/Z (= X ∨ ) S S    Hom(US , Z) ⊕ Pic(Y )D      µ D F p = 0, p = 1, p = 2, p = 3. Proposition 3.5. The cohomologies of RHomZ (τ ≤3 RΓc (YW , Z), Z)[−2] concentrate at degrees 0 and 1 and H 0 = US , H 1 = XS ⊕ Pic(Y ). 44 Proof. Consider the spectral sequence E2pq = ExtpZ (H −q (C • ), Z) ⇒ H p+q (RHom(C • , Z)), in which C • = τ ≤3 RΓc (YW , Z). It is clear that E2pq = 0 for all p < 0 or q > 0, i.e. the E2 has only non-trivial terms on the 4th quadrant 0 0 XS = XS∨∨ 0 US /tor = US∨∨ Ext1 ((Pic Y )D , Z) Ext1 (µD F , Z) 0 0 We have exact sequences 0 → Pic Y → H −1 → XS → 0 and 0 → µF → H −2 → US /tor → 0. Thus, H −1 = XS ⊕ Pic(Y ) and H −2 = (US /tor) ⊕ µF = US . This shows that RHom(τ ≤3 RΓc (YW , Z), Z)[−2] has only two non-trivial cohomologies H 0 = US and H 1 = XS ⊕ Pic Y . From now on we assume that S is large enough so that Pic Y = 0 and is stable under the action of G. We will prove RHom(τ ≤3 RΓc (YW , Z), Z)[−2] is indeed quasiisomorphic to ΨS , the canonical Tate sequence. (see the introduction section). Lemma 3.2. Let A and B ∈ D, the derived category of Z[G]-modules. a) If H i (B) are Z[G]-injective for all i, then HomD (A, B) ∼ = Y i HomG (H i (A), H i (B)). 45 b) If H i (A) are Z[G]-cohomologically trivial, then we have an exact sequence 0→ Y Ext1G (H i (A), H i−1 (B)) → HomD (A, B) → i Proof. Y HomG (H i (A), H i (B)) → 0. i a) By [16], there is a spectral sequence Y ExtpG (H i (A), H q+i (B)) ⇒ H p+q (RHomZ[G] (A, B)). i By condition, Extp (H i (A), H q+i (B)) is non-trivial only if p = 0, so the above spectral sequence simply shows HomD (A, B) = H 0 (RHomZ[G] (A, B)) = Y HomG (H i (A), H i (B)). i b) In the same spectral sequence, since H i (A) are cohomologically trivial, H i (A) are of projective dimension 1, so ExtpG (H i (A), H q+i (B)) = 0 for all p ≥ 2. Therefore, on the position with p + q = 0, we have only 2 non-trivial groups E20,0 and E21,−1 , and both the difference maps are 0. This gives the desired short exact sequence. Lemma 3.3. The canonical exact triangle RHomZ (RΓc (Yét , Z), Z) → RHomZ (RΓc (Yét , Z), Q) → RHomZ (RΓc (Yét , Z), Q/Z) → is isomorphic to the exact triangle fS 0 [3] → Ψ0S [2] → XS ⊗Z Q[1] → Ψ 0 bS ) and Ψ fS is the image of ΨS in Ext3 (XS ⊗Z where Ψ0S is the image of ΨS in Ext2G (XS , U G bS ). Q/Z, U e S is the complex A → B → Proof. Suppose ΨS is the complex A → B and Ψ 46 XS ⊗Z Q. Recall that ΨS presents a class of Ext2G (XS , US ) and Ψ0S the image of ∼ bS ). Indeed , Ψ0 is the complex ΨS under the map Ext2G (XS , US ) − → Ext2G (XS , U S e 0 is the complex A0 → B → XS ⊗Z Q. Re(A ⊕ ÛS )/US =: A0 → B. Similarly, Ψ S qis e 0 by definition ([2] Prop. 3.1). Also, as Q is call that RHom(RΓc (Yét , Z), Q/Z) −→ Ψ S injective, H i (RHomZ (RΓc (Yét , Z), Q)) = HomZ (Hc−i (Yét , Z), Q). Thus, qis RHomZ (RΓc (Yét , Z), Q) −→ XS ⊗Z Q[1]. We claim that the following diagram commutes for a suitable choice of the left vertical quasi-isomorphism. RHomZ (RΓc (YO ét , Z), Q)[−3] qis XS ⊗Z Q[−2] / RHomZ (RΓc (Yét , Z), Q/Z)[−3] O (3.4) qis / e0 Ψ S. Since the both horizontal maps induce the canonical projection XS ⊗Z Q → XS ⊗Z Q/Z on H 2 , the right vertical quasi-isomorphism induces the identity map on H 2 and the left vertical one can be induced by any G-automorphism of XS ⊗Z Q, so there is a suitable choice of the quasi-isomorphism so that the above diagram commutes for H 2 groups. Let A be XS ⊗Z Q[−2] and B be RHomZ (RΓc (Yét , Z), Q/Z)[−3]. Clearly, H i (A) are Z[G]-cohomologically trivial. So we can apply lemma 3.2(b) to A and B, and it induces an exact sequence 0 = Ext1G (XS ⊗Z Q, H 1 (B)(= 0)) −→ HomD (A, B) −→ HomG (H 2 (A), H 2 (B)) → 0. Thus, HomD (A, B) ∼ = HomG (H 2 (A), H 2 (B)). Therefore, the commutative on H 2 implies the diagram (3.4) indeed commutes. By taking exact triangles of each row, e 0 )[−1] ≃ Ψ0 [2], we get the desired isomorphism and note that cone(XS ⊗Z Q[−2] → Ψ S S of exact triangles. Theorem 3.3. Let Y = Spec OF,S , where S is large enough so that Pic Y = 0 and is 47 stable under the action of G. Then qis RHomZ (τ ≤3 RΓc (YW , Z), Z)[−2] −→ ΨS , where ΨS is the Tate sequence representing the canonical class of Ext2G (XS , US ). Proof. Here, we use the same notation with previous lemma. Clearly, we have a canonical exact triangle bS /US [0] → . ΨS → Ψ0S → U On the other hand, by applying RHomZ (−, Z) to exact triangle (3.1), we have another exact triangle RHomZ (τ ≤3 RΓc (YW , Z), Z)[−2] → RHomZ (RΓc (Yét , Z), Z)[−2] → RHomZ (HomZ (US , Q), Z)[−1] qis The previous lemma ensures that Ψ0S −→ RHomZ (RΓc (Yét , Z), Z)[−2]. qis bS /US [0] and we have seen Also, observe that RHomZ (HomZ (US , Q), Z)[−1] −→ U that RHomZ (τ ≤3 RΓc (YW , Z), Z)[−2] has the same cohomologies as ΨS . We claim that there is an isomorphism between these two exact triangles, RHomZ (τ ≤3 RΓcO (YW , Z), Z)[−2] / Ψ0 S β / RHomZ (HomZ (US , Q), Z)[−1] O α δ ΨS / γ / Ψ0 /U b /U [0] S S S /. In the top exact triangle, we replaced RHomZ (RΓc (Yét , Z), Z)[−2] by Ψ0S . By taking the long exact sequence induced by the top exact triangle, we see that H 2 (β) = i ◦ p bS → U bS /US is the canonical projection and i ∈ Aut(U bS /US ). Clearly, where p : U qis bS /US [0], we may take α to be H 2 (γ) = p. As RHomZ (HomZ (US , Q), Z)[−1] −→ U the morphism induced by i. Then the right square commutes on H 2 . By lemma 3.2, bS /US is uniquely divisible and thus injective, H 2 (β) = H 2 (α)◦H 2 (γ) implies because U β = α ◦ γ. As α is a quasi-isomorphism, so is δ by property of derived categories. qis Remark. It is equivalent to say that τ ≤3 RΓc (YW , Z) −→ RHom(ΨS , Z)[−2] and this 48 is an evidence that, at least for open subschemes of spectra of number rings, the Weilétale cohomology can be obtained from usual cohomology theories without using the Weil groups. 49 Chapter 4 The Construction of R Gm Throughout this chapter, we assume that F is a totally imaginary number field, X = Spec OF , and U = Spec OL,S is any connected étale neighborhood of X. We also assume that K is a subfield of L such that the extension L/K is Galois and G := Gal(L/K). When P ic(U ) = 0 and S is a G-stable set containing all the archimedean places and those ramified in K(U )/Q, there exists a canonical Tate sequence: ΨU : Ψ0S → Ψ1S , which represents a canonical class of Ext2G (XS , US ). By the construction of Tate sequences, if U 0 = Spec OL0 ,S 0 is étale over U , then there is a morphism from ΨS to ΨS 0 that fits into the following morphism of complexes 0 0 / US / Ψ0 / Ψ1 / XS     /U 0 S S / Ψ0 0 S S /0 β / Ψ1 0 S /X 0 S / 0,
where β((av )v∈S ) = (aw := [Lw : Fv ] · av )w|v,w∈S 0 . In fact {XS } and {β} define an X ét -presheaf X , and we denote its associated f. In section 4.1, we construct a complex R Gm ∈ Sh(X ét ) which repsheaf by X 2 f, Gm ) = Z b (see below). The cohomologies of resents a canonical class of ExtX (X et
RΓ(Uét , R Gm ) and RHomZ τ ≤3 RΓc (UW , Z), Z [−2] are the same for arbitrary U . 50 Furthermore, in section 4.2, we prove that
qis RΓ(Uét , R Gm ) −→ RHom τ ≤3 RΓc (UW , Z), Z [−2], (4.1) in D(Z[G]), when S is stable under the action of G.
This implies RΓ(Uét , R Gm ) and RHomZ τ ≤3 RΓc (UW , Z), Z [−2] are canonically quasi-isomorphic in D(Z) for any U . Also, when U is small enough so that the Tate sequence exists, then by (4.1) and by the help of Theorem 3.3, one can see that RΓ(Uét , R Gm ) ≃ ΨU in D(Z[G]). Thus, one can conclude that the complex RΓ(Uét , R Gm ) generalizes the Tate sequences and its Z-dual defines Weil-étale cohomology of S-integers without truncations. 4.1 The Definition Let M be the sheaf sending a connected étale U → X to the group L w∈U Aw , where Aw is Q (resp. Z) if w - ∞ (resp. w | ∞), and the transition map M (U ) → M (V ) for an X-morphism V → U is M M Av −→ v∈U Aw w∈V av X [K(V )w : K(U )v ]av . 7→ w|v It is clear that the projections L an epimorphism of sheaves M → L v∈U Av  L v∈X iv,∗ Av . v∈U Av , for any étale U → X, define 51 We define F • to be the complex F P 0 / F1 Q, where F 0 = ker(M → L v∈X iv,∗ Av ) is the X-étale sheaf U 7→ M Aw , w∈U −U for any connected U → X and P P (U )((aw )) = w [K(U 1)w :Fv ] aw . P Note that if U doesn’t contain all the finite places of F , then (U ) is surjective. P This implies is an epimorphism in the category Sh(X ét ) as any U → X can be covered by some {Ui }, where Ui doesn’t contain all the finite places of K(Ui ), for all P i. Consequently, F • ≃ (ker )[0] in D(Sh(X ét )). a. The way that we define the transition maps of the sheaf F 0 makes P X a sub-presheaf of ker . In fact, we have the following exact sequence of Remark. presheaves 0 → X → ker X f → 0, → Br f is the presheaf U 7→ H 2 (U, Gm ). By taking associated sheaves, we see where Br P f∼ that X as a (H i (−, G )) = 0 for any sheaf G when i ≥ 1. = ker b. Let Mv = − lim →L ML,v where L runs over all finite extension of F and ML,v = L w Aw where the sum runs over all places w of L lying over v and the transition map is defined similarly to those of X . We denote Mv the sheaf associated to the Galois module Mv . Equivalently, in the level of sheaves, Mv = − lim → πL,∗ Av L v v where πL : Spec L −→ Spec F is the canonical morphism and L is the fixed L field of Dv ∩ Gal(L/F ). Clearly, M = v∈X α∗ Mv , where α : Spec F → X. 52 Moreover, consider the Cartesian diagram of schemes Spec Lv  αL / u(Lv ) πL Spec F  0 πL / X, α 0 where u(Lv ) = Spec OLv . Hence α∗ lim lim lim −→ πL,∗ Av = − → α∗ πL,∗ Av = − → πL,∗ αL,∗ Av = L L 0 v lim −→L πL,∗ Av . (Note that αL,∗ Av is the constant Av on Spec(L ).) L Lemma 4.1. The sheaf F 0 is cohomologically trivial. Proof. Since F 0 fits into the short exact sequence 0 → F0 → M α∗ Mv → M iv,∗ Av → 0, (4.2) v∈X v∈X which is derived from the short exact sequence of sections: 0→ M v∈U −U Av → M v∈U Av → M Av → 0. v∈U Since there is no real place and Av = Q when v - ∞, the cohomology of iv,∗ Av concentrates at degree zero. Also, the above exact sequence remains exact when we pass to global sections. Thus, we only need to show that α∗ Mv is cohomologically trivial for any v. 0 Recall that α∗ Mv = − lim → πL,∗ Av . When v is finite, α∗ Mv is acyclic as Av = Q and L 0 inductive limit and πL,∗ both preserve acyclic sheaves (as πL0 is finite). F When v|∞, Iv is trivial, so Mv = IndG 1 Av and then Mv is cohomologically trivial. Therefore i∗w (Rq α∗ Mv ) = H q (Iw , Mv ) = 0 for q > 0. The Leray spectral sequence for α, H p (Uét , Rq α∗ Mv ) ⇒ H p+q (F, Mv ), for any étale U over X, is degenerated, i.e. H p (Uét , α∗ Mv ) = H p (F, Mv ) = 0 for p > 0. Thus the result follows. 53 Proposition 4.1. For any connected étale U → X,    ker Σ(U ) p = 0,   H p (Uét , F • ) = coker Σ(U ) = Q/Im(Σ(U )) p = 1,     0 p ≥ 2. In particular, H 0 (X ét , F • ) = 0, H 1 (X ét , F • ) = Q, H 0 ((X − v)ét , F • ) = 0 and H 1 ((X − v)ét , F • ) = Q/Av . Corollary 4.1. The canonical morphism Hv1 (X ét , F • ) → H 1 (X ét , F • ) is an inclusion of Av into Q. Now we define a complex R Gm , up to quasi-isomorphism, of Sh(X ét ) by an exact triangle Gm → R Gm → F • [−1] →, or equivalently an exact triangle γ R Gm → F • [−1] → − Gm [1] → . One can choose carefully a morphism γ ∈ HomD(Sh(X ét )) (F • [−1], Gm [1]) ∼ = Ext2X ét (F • , Gm ) so that R Gm has the cohomologies that we expected. Before computing the group Ext2X ét (F • , Gm ), we need to derive the following lemmas. Lemma 4.2. Let I be an inductive system of index that can be refined by an index which is isomorphic to Z, then the canonical morphism ExtsC (lim −→ Ai , B) −→ i s lim ←−i ExtC (Ai , B) is an isomorphism if any of the following conditions holds a. the projective system {Exts−1 C (Ai , B)} satisfies the Mittag-Leffler condition, b. Exts−1 C (Ai , B) is a finite group for all i. p Proof. Because I can be refined by Z, so lim lim −→ Ai = lim −→ Ai and ← − ExtC (Ai , B) = I Z I p ∼ ∼ lim ←−Z ExtC (Ai , B). Thus, we only need to consider the case that I = Z. When I = Z, 54 by [13], there is a spectral sequence r s r+s lim ←−I ExtC (Ai , B) ⇒ ExtC (lim −→I Ai , B). r This spectral sequence is certainly degenerated as we know that lim ←− is vanishing for any inductive system of abelian groups when r ≥ 2 (cf. [17] 3.5). Therefore, we get exact sequences 1 1 s−1 s s 0 → lim ←− ExtC (Ai , B) → ExtC (lim −→ Ai , B) → lim ←− ExtC (Ai , B) → 0, I I I for all s ≥ 0. 1 s−1 The Mittag-Leffler condition implies lim ←− ExtC (Ai , B) = 0 and condition (b) 1 s−1 also implies the vanishing of lim ←− ExtC (Ai , B) by [7] 2.3. Thus, the isomorphism is established under both conditions. Lemma 4.3. a. p (Q, Gm ) = ExtX ét where   0 p = 2, N Q 0  Z b q Qq p = 3, ZQ = Q0 q Qq = AQ /R is the restricted product of the Qp ’s with respect to the Zp ’s. b.    0 p = 2,   p ExtX (α∗ Mv , Gm ) = H 3 (X ét , Gm ) = Q/Z p = 3 and v | ∞, ét   N   lim H 3 (X, G ) = Z b m Z Q p = 3 and v - ∞. ←−n Proof. i a. Since Q = − lim → Z and H (X ét , Gm ) are finite groups for i = 1, 2, we have n p ExtpX (Q, Gm ) = lim ←− H (X ét , Gm ). ét n 3 As H 2 (X ét , Gm ) = 0, one sees that Ext2X ét (Q, Gm ) = 0. Also, ← lim − H (X ét , Gm ) = n 55 bN lim ←−n Q/Z = Hom(Q, Q/Z) = Z Z Q. 0 b. As Mv = − lim → πL,∗ Av , L p 0 (α∗ Mv , Gm ) = ExtpX (lim ExtX −→ πL,∗ Av , Gm ). ét L ét When v is an infinite place, Av = Z, and by the norm theorem, 0 ExtpX (πL,∗ Z, Gm ) = H p (u(Lv )ét , Gm ), ét for all p. Note that F is totally imaginary and u(Lv ) contains all the finite places of Lv , so H 1 (u(Lv )ét , Gm ) = P ic(u(Lv )), H 2 (u(Lv )ét , Gm ) = 0, and H 3 (u(Lv )ét , Gm ) = Q/Z. Since H 1 (u(Lv )ét , Gm ) and H 2 (u(Lv )ét , Gm ) are both finite groups, by Lemma 4.2, we actually get p p v ExtX (α∗ Mv , Gm ) = ← lim − H (u(L )ét , Gm ) L ét 2 for p = 2, 3. Clearly, ExtX (α∗ Mv , Gm ) = H 2 (u(Lv )ét , Gm ) = 0. Moreover, ét for any finite extension K over L, the transition map H 3 (u(K v )ét , Gm ) → 3 (α∗ Mv , Gm ) = H 3 (u(Lv )ét , Gm ) is an isomorphism. So, we conclude that ExtX ét H 3 (X ét , Gm ) = Q/Z. For the case v is a finite place of F , there is a slightly difference, as now Av = 1 Q=− lim lim → → n Z and so α∗ Mv = − L,n p ExtX (lim −→ ét for p = 2, 3. L,n 0 Z. We claim that πL,∗ 0 πL,∗ Z, Gm ) = ← lim − L,n H p (u(Lv )ét , Gm ), 0 Indeed, because Ext1u(Lv )ét (πL,∗ Z, Gm ) = H 1 (u(Lv )ét , Gm ) = 0 P ic(u(Lv )) is a finite groups, and Ext2u(Lv )ét (πL,∗ Z, Gm ) = H 2 (u(Lv )ét , Gm ) = 0, p by Lemma 4.2, one gets ExtX (α∗ Mv , Gm ) = ← lim − L,n ét H p (u(Lv )ét , Gm ) for p = 2, 3. Again, by Lemma 4.2, one get ExtpX (Mv , Gm ) = ← lim − ét L,n Extpu(Lv ) (Z, Gm ) 56 for p = 2, 3. Hence ExtpX (Mv , Gm ) = ét   0 p = 2,  lim H 3 (X , G ) p = 3. ét m ←−n Proposition 4.2.   0 ExtpX (F 0 , Gm ) = ét  Q p = 2, v|∞ H 3 (X ét , Gm ) = Q X ∞ Q/Z p = 3. Proof. Let Fv0 := ker(α∗ Mv → iv,∗ Av ). Then it arises a long exact sequence ∂ p−1 ∂p p p −−−→ ExtX (α∗ Mv , Gm ) → ExtpX (Fv0 , Gm ) → ExtX (iv,∗ Av , Gm ) −→ Extp+1 (α∗ Mv , Gm ) → . X ét ét ét ét 2 Since ExtX (α∗ Mv , Gm ) = 0 by the previous lemma, and it is easy to see that ét Ext4X ét (iv,∗ Av , Gm ) = 0, we have the following exact sequence, 2 (Fv0 , Gm ) → Ext3X ét (iv,∗ Av , Gm ) → Ext3X ét (α∗ Mv , Gm ) → Ext3X ét (Fv0 , Gm ) → 0. 0 → ExtX ét By the adjunction of iv,∗ and i!v and Lemma 4.2 (as Hv2 (X ét , Gm ) = 0),   H 3 (X , G ) = 0 v | ∞, ét m v 3 ExtX ét (iv,∗ Av , Gm ) = Q  lim H 3 (X , G ) = 0 Q v - ∞. q q ←−n v ét m 2 3 Therefore, ExtX (Fv0 , Gm ) = 0 and Ext3X ét (Fv0 , Gm ) = ExtX (α∗ Mv , Gm ) = Q/Z ét ét when v|∞. 3 We claim that the morphism Ext3X ét (iv,∗ Av , Gm ) → ExtX (α∗ Mv , Gm ) is an isoét morphism when v - ∞. For this, observe that there is a commutative diagram of 57 sheaves 0 α∗ Mv = − lim → πL,∗ Av O L /i A 6 v,∗ v n n nn nnn n n n nnn 0 Av = πF,∗ Av where Av → iv,∗ Av is the canonical morphism induced by adjunction of i∗v and iv,∗ . Applying Ext3 (−, Gm ) to the above diagram, we get another commutative diagram : Ext3 (α∗ Mv , Gm ) o i lim Hv3 (X ét , Gm ) ←− kn k kkk kkk k k k h ku kk  3 lim ←−n H (X ét , Gm ). b All the groups in the above diagram are isomorphic to Z N ZQ = Q q Qq . Note the h is an isomorphism as the natural morphism Hv3 (X ét , Gm ) → H 3 (X ét , Gm ) is an Q isomorphism ([4] Prop.3.2). Therefore, i is an injection from q Qq to itself. Since HomZ (Qp , Qq ) = 0 is p 6= q and Qq are fields, one sees that i has to be an isomorphism. It follows that Ext2X ét (Fv0 , Gm ) and Ext3X ét (Fv0 , Gm ) are vanishing. Consequently, Ext2X ét (F 0 , Gm ) = 0 and Ext3X ét (F 0 , Gm ) = M Ext3X ét (Fv0 , Gm ) = v∈X M v|∞ Proposition 4.3. There is a canonical isomorphism b Ext2X ét (F • , Gm ) ∼ = Z. Proof. The exact sequence 0 −→ ker X −→ F 0 −→ Q −→ 0, Q/Z 58 induces a long exact sequence 2 ExtX (F 0 , Gm )(= 0) → Ext2X ét (ker ét X M ∆ , Gm ) → Ext3X ét (Q, Gm ) − → H 3 (X ét , Gm ) →, v|∞ where ∆ is a copies of the canonical map π : Hom(Q, Q/Z) → Hom(Z, Q/Z). b As a consequence Ext2 (ker P, Gm ) = Clearly, ker ∆ = ker π = HomZ (Q/Z, Q/Z) = Z. X ét b ker∆ = ker(π) = Z. b Let γ be the class in Ext2X ét (F • , Gm ) that corresponds to the generator 1 of Z, and R Gm is the complex decided by the exact triangle γ R Gm → F • [−1] → − Gm [1] → . To compute the étale cohomology of the complex R Gm , we need to determine the ^γ coboundary morphisms ∂Ui : H i (Uét , F • ) −−→ H i+2 (Uét , Gm ), for any connected étale U → X. Lemma 4.4. For any connected étale U → X, let S := U − U , we have a. ∂U0 is the canonical projection M M Av /Z)Σ=0 , ( Av )Σ=0 → ( v∈S v∈S b. ∂U1 is an isomorphism if S is non-empty, and it is the canonical projection Q → Q/Z otherwise. c. ∂Ui is isomorphic for all i ≥ 2. In particular, all the ∂ i are surjective for all i ≥ 0. Proof. We first deal with the second part of (b). Consider the following commutative 59 diagram H 1 (X ét , ker O Ext2X ét (ker × P ) o P , Gm ) ^ / H 3 (X ét , Gm )  H (X ét , Q) × Ext3X ét (Q, Gm ) ^ / H 3 (X ét , Gm ) 1 0 lim −→ H (X ét , n Z) × ^ 3 1 lim ←− ExtX ét ( n Z, Gm ) / H 3 (X ét , Gm ) 0 It is easy to see that the bottom cup product sends pair (a, 1) to (a mod Z) and thus a ^ γ = a ^ 1 = a mod Z. This shows ∂ 1 : H 1 (X ét , F • ) → H 3 (X ét , Gm ) is the canonical projection Q → Q/Z. The general case that U = U follows from the commutative diagram ^γ H 1 (X ét , F • ) mod Z/ H 3 (X ét , Gm )  = H 1 (U ét , F • ) ^γ / H 3 (U  = ét , Gm ). For a Zariski open subset j : U ,→ X, set i : X − U ,→ X, we have the following commutative diagram 0 / H 0 (U ét , F • ) / 1 • v ∈U / Hv (X ét , F ) L ^γ 0 / H 2 (U  ét , Gm ) P / H 1 (X ét , F / ^γ  L 3 v ∈U / Hv (X ét , Gm ) • ) / H 1 (U ) /0 ét , Gm ) / 0, ét , F ^γ ^γ P / H 3 (X  ét , Gm ) • / H 3 (U  where the rows are exact sequences induced by the short exact sequence 0 −→ j! j ∗ F −→ F −→ i∗ i∗ F −→ 0, for any F ∈ Sh(X). P ^γ The morphism H 0 (Uét , ker ) −−→ H 2 (Uét , Gm ) is completely determined by P ^γ H 1 (X ét , ker ) −−→ H 3 (X ét , Gm ) because Σ restricts to an inclusion on Hv1 (X ét , F • ) = P Av . (This can be seen by put U to be X − v and use the fact H 0 ((X − v)ét , ker ) = 60 0). Therefore, by the commutativity, ^ γ : H 0 (Uét , F • ) → H 2 (Uét , Gm ) maps L L ( v∈U −U Av )Σ=0 to ( v∈U −U Av /Z)Σ=0 by taking modular by Z componentwise. For general i : U → X, set U a to be the largest open subscheme of U such that for any v ∈ U a , all the valuations of K(U ) above i(v) is contained in U a . We have the commutative diagram H 0 (Uét , F • ) ^γ / H 2 (U ét , Gm ) _  H 0 (Uéta , F • ) ^γ H 0 (i(U a )ét , F • ) ^γ _  / H 2 (U a , Gm ) étO O / H 2 (i(U a ) ét , Gm ) More precisely,  L Aw w∈U −U  Σ=0 L /  L Aw Σ=0 ^γ / v∈X−i(U a ) L Σ=0 Aw /Z w∈U −U a O L _   w∈U −U a  Σ=0 Aw /Z w∈U −U _   ^γ Av Σ=0 ^γ /  L O Σ=0 Av /Z v∈X−i(U a ) Note that the cup products are canonical on v and w, and we have seen above that the bottom cup product is canonical projection. By chasing diagram, one sees that the middle one is also canonical projection, and so is the top one as the injection is canonical. It follows that ∂U0 is the canonical projection. To see the first part of part (b), we assume S is non-empty. When U f * U , H 3 (Uét , Gm ) = 0 and H 1 (Uét , F • ) = 0 (Prop. 4.1). Therefore, we have ∂U1 = 0. For the case U f ⊆ U , note that U is then étale over X as all the infinite places are 61 complex. Therefore, we have the following commutation diagram π Q = H 1 (U ét , F • )  / H 1 (U ét , F ∂1 U  • ) = Q/Z 1 ∂U H 3 (Uét , Gm ) = Q/Z. Q/Z = H 3 (U ét , Gm ) As we have seen, in the very beginning of the proof, that ∂U1 and π are the canonical projections, ∂U1 has to be the identity map on Q/Z by the commutative. Part (c) is trivial because H i (Uét , F • ) = H i+2 (Uét , Gm ) = 0 for all i ≥ 2. Now, we are ready to compute the étale cohomology of the complex R Gm . Proposition 4.4. Let U be étale over X, S = U − U and L = K(U ), then  ×   OL,S       Pic(OL,S ) ⊕ XS   H p (Uét , R Gm ) = 0      Z (resp. 0)      0 p = 0, p = 1, p = 2, p = 3 and S = ∅ (resp. S 6= ∅), p ≥ 4. Hence, these cohomologies coincide with those of the Z-dual of τ ≤3 RΓc (UW , Z)[2] (c.f. Prop. 3.5). Proof. Consider the usual long exact sequence on cohomology induced by the exact γ triangle R Gm → F • [−1] → − Gm [1] → and use the result that ∂ i are surjective (Lemma 4.4), we get ∼ H 0 (Uét , Gm ) − → H 0 (Uét , R Gm ), 0 → H 1 (Uét , Gm ) → H 1 (Uét , R Gm ) → ker(∂ 0 ) → 0 is exact, H i (Uét , R Gm ) = ker(∂ i−1 ). × Thus, H 0 (Uét , R Gm ) = OL,S . Because ker ∂ 0 = (⊕v∈S Z)Σ=0 = XS is Z-free, the mid- dle exact sequence splits and so H 1 (Uét , R Gm ) = P ic(OL,S )⊕XS . Also, H 3 (Uét , R Gm ) = 62 0 (resp. Z) when S 6= ∅ (resp. S = ∅) and H i (Uét , R Gm ) = 0 for all i ≥ 4, by Lemma 4.4 (b) and (c). The computations of Prop.4.4 and 3.5 suggest that
qis Theorem 4.1. RΓ(Uét , R Gm ) −→ RHom τ ≤3 RΓc (UW , Z), Z [−2] in D(Z). In the next section, we shall prove the following more general quasi-isomorphism
qis RΓ(Uét , R Gm ) −→ RHom τ ≤3 RΓc (UW , Z), Z [−2], in D(Z[G]), when S is stable under the action of G. Note that when G is the trivial group, this is just Theorem 4.1 4.2 The Duality Theorem Throughout this section, we assume that F is a totally imaginary number field, X = Spec OF , U = Spec OL,S be any connected étale neighborhood of X. We also assume that K is a subfield of L such that the extension L/K is Galois and G := Gal(L/K). We require that S contains all the archimedean places and is stable under the action of G. Recall that F • (U ) ≃ ker P (U )[0] in D(Z[G]). To simplify the notation, in the P following, we denote by F • (U ) the complex ker (U )[0]. Theorem 4.2. Let U = Spec OL,S and A → B represents RHom(τ ≤3 RΓc (UW , Z), Z)[−2] in Ext2G (XS ⊕ Pic(U ), US ). Then RΓ(Uét , Gm ) can be represented by A → B → F • (U ). Moreover, there is an exact triangle in D(Z[G]), F • (U )[−2] → RΓ(Uét , Gm ) → RHom(τ ≤3 RΓc (UW , Z), Z)[−2] → . (4.3) 63 This exact triangle is the same as the exact sequence of complexes 0 −→ F • (U )[−2] −→ (A → B → F • (U )) −→ (A → B) −→ 0. Proof. O O O RHom(τ ≤3 RΓc (U , Z), Z)[−2] O W / RHom(RΓc (Uét , Z), Z)[−2] O / I0 [0] O RΓ(UétO , Gm ) / RΓ (U F • (U )[−2] / XS c  D ét , Z) [−3] α O N Z Q[−2] / L / / I2 [−2] /, / I0 [0] I [−2] O 2  RHom(RΓc (Uét , Z), Q)[−3] bS /US and I2 = L where I0 = U v∈S∞ Q/Z are both Z[G]-injective modules. All the rows are known exact triangles. The top is applying RHomZ (−, Z) to the exact triangle RΓc (Uét , Z) → RΓc (UW , Z) → HomZ (US , Q)[−2] →, the middle is (1.5) induced by the Artin-Verdier Duality Theorem, and the bottom is followed from the definition of F • (U ) and is indeed a short exact sequence. The middle column is obtained by applying HomZ (−, Z) → HomZ (−, Q) → HomZ (−, Q/Z) → to the complex RΓc (Uét , Z). The top right hand and bottom right hand squares of the above diagram are commutative because they are commutative on H 0 and H 2 respectively (see Lemma 3.2). Thus, by property of derived categories, there exist morphisms in D(Z[G]) so that the 64 first column is an exact triangle and the above 3 by 3 diagram is semi-commutative. f Suppose RHom(τ ≤3 RΓc (UW , Z), Z)[−2] is in the same class of A → − B as an element of the 2-extension group Ext2G (XS ⊕Pic(U ), US ). Then RHom(τ ≤3 RΓc (UW , Z), Z)[−2] f bS ), and is in the class of image, says A0 → − B, of A → B in Ext2G (XS ⊕ Pic(U ), U N RΓc (Uét , Z)D [−3] is in the same class of A0 → B → XS Z Q. In fact, one may bS )/US and set f 0 (a, x) = f (a). Note that RΓ(Uét , Gm ) is dechoose A0 to be (A ⊕ U termined by α, more precisely by H 0 (α) and H 2 (α). It’s easy to see that the exact triangle associated to the following exact sequence of complexes is in fact isomorphic to the middle row of the above diagram. 0 0 f  /B  / F (U ) f0  /B 0  A  0 A  I0  0  /0  0 / XS  N ZQ  /I 2  0 We conclude that RΓ(Uét , Gm ) can be represented by A → B → F (U ) and it induced the exact triangle F • (U )[−2] → RΓ(Uét , Gm ) → RHomZ (τ ≤3 RΓc (UW , Z), Z)[−2] → . Remark. By definition of the complex R Gm , RΓ(Uét , R Gm ) satisfies the following exact triangle F • (U )[−2] → RΓ(Uét , Gm ) → RΓ(Uét , R Gm ) → . (4.4) 65 One expects that exact triangles (4.3) and (4.4) are isomorphic, which would imply the quasi-isomorphism RHom(τ ≤3 RΓc (UW , Z), Z)[−2] ≃ RΓ(Uét , R Gm ). Theorem 4.3. For any U = Spec OL,S , one has a quasi-isomorphism RHom(τ ≤3 RΓc (UW , Z), Z)[−2] ≃ RΓ(Uét , R Gm ), in D(Z[G]). Proof. We first show that HomD(Z[G]) (F • (U )[−2], RΓ(Uét , Gm )) ≃ HomG (F • (U ), H 2 (Uét , Gm )) (4.5) Observe that there is an exact sequence 0 → XSf ⊗Z Q → F • (U ) → M Z → 0. S∞ Since XSf ⊗Z Q and L ∼ Z[Gal(F/Q)]) are G-cohomologically trivial, so is S∞ Z ( = F • (U ). By lemma 3.2(b), there is an exact sequence 0 / Ext1 Z[G] (F • (U ), H 1 (Uét , Gm )) / HomD(Z[G]) (F • (U )[−2], RΓ(Uét , Gm )) −→ HomG (F • (U ), H 2 (Uét , Gm )) Note that Ext1G (F • (U ), H 1 (Uét , Gm )) = Ext1G (XSf ⊗Z Q, Pic(U )) = 0. Hence the isomorphism (4.5). By Lemma 4.4 and Theorem 4.2, we see that the induced homomorphisms on H 2 , F • (U ) → H 2 (Uét , Gm ), for (4.3) and (4.4) are both the canonical projections. Thus, exact triangles (4.3) and /0 66 (4.4) have to be isomorphic, in virtue of isomorphism 4.5. Consequently, RHomZ (τ ≤3 RΓc (UW , Z), Z)[−2] ≃ RΓ(Uét , R Gm ), in D(Z[G]). Corollary 4.2. Suppose that Pic(Spec(OF,S )) = 0 and S is stable under the action of G, then we have the following quasi-isomorphism RΓ((Spec(OF,S ))ét , R Gm ) ≃ ΨS , where ΨS is the canonical Tate sequence associated to S. Proof. This follows from Theorem 4.3 and Theorem 3.3. Remark. a. Theorem 4.3 allows us to recover the truncated Weil-étale cohomol- ogy groups Hcp (UW , Z) by the hyper étale cohomology of the complex R Gm . Also, one might find corresponding Weil-étale cohomology axioms in terms of H p (Uét , R Gm ). b. The reason that it is hard to generalize Lichtenbaum’s prototype to higher dimensional arithmetic schemes X is that there are no Weil groups for higher dimensional fields. However, Theorem 4.3 shows us a probability to generalize Lichtenbaum’s prototype, because we do not use Weil groups when defining the complex R Gm . 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