On Selmer Groups of Geometric Galois Representations Tom Weston

On Selmer Groups of Geometric Galois Representations

Tom Weston

Department of Mathematics, Harvard University, Cambridge, Mass- chusetts 02140 E-mail address: [email protected]

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Dedicated to the memory of Annalee Henderson

Contents

Introduction ix Acknowledgements xii Notation and terminology xv Fields xv Characters xv Galois modules xv Schemes xv Sheaves xvi Cohomology xvi K-theory xvi

Part 1. Selmer groups and deformation theory 1 Chapter 1. Local cohomology groups 3 1. Local finite/singular structures 3 2. Functorialities 4 3. Local exact sequences 5 4. Examples of local structures 6 5. Ordinary representations 7 6. Cartier dual structures 8 7. Local structures for archimedean fields 9 Chapter 2. Global cohomology groups 11 1. Selmer groups 11 2. Functorialities 13 3. The global exact sequence 13 4. A finiteness theorem for Selmer groups 14 5. The Kolyvagin pairing 16 6. Shafarevich-Tate groups 18 7. The Bockstein pairing 20 Chapter 3. Annihilation theorems for Selmer groups 21 1. Partial geometric Euler systems 21 2. The key lemmas 22 3. The annihilation theorem 25 4. Right non-degeneracy of the Bockstein pairing 27 5. A δ-vanishing result 28 Chapter 4. Flach systems 31 1. Minimally ramified deformations 31

v vi CONTENTS

2. Tangent spaces and Selmer groups 34 3. Good primes 36 4. Flach systems 38 5. Cohesive Flach systems 39 6. Cohesive Flach systems of Eichler-Shimura type 40

Chapter 5. Flach systems of Eichler-Shimura type 43 1. The map on diﬀerentials 43 2. The Tate pairing 45 3. A special case 47 4. A matrix computation 52 5. Computation of Ξ in the non-diagonal case 53 6. Computation of Ξ in the general case 54

Part 2. Construction of cohesive Flach systems 55

Chapter 6. The Flach map 57 1. The coniveau spectral sequence in ´etalecohomology 57 2. The localization sequence 59 3. Grothendieck’s purity conjecture 61 4. The coniveau spectral sequence in K-theory 62 5. Deﬁnition of the Flach map 64 6. Functoriality and passage to the limit 66 7. Functoriality II 67

Chapter 7. Local analysis of the Flach map 71 1. Overview 71 2. Local behavior I 72 3. Local behavior II 72 4. Local behavior III 74 5. The divisor map 75 6. The cycle map 77 7. Relations with Galois cohomology 78 8. Functoriality and passage to the limit 78 9. Example : Schemes over global ﬁelds 79 10. Local behavior at places over l 80

Chapter 8. Flach classes for correspondences 83 1. Algebraic correspondences 83 2. Correspondences and operations on ´etalecohomology 84 3. Composition of correspondences 86 4. Marked varieties 88 5. Divisors and compositions 89 6. The Leibniz relation 90 7. Algebras of correspondences 92 8. Derivations in the self-adjoint case 93 9. Local diagrams in the self-adjoint case 94 10. Derivations in the general case 95 11. Untwistings and cycle classes 97 12. Derivations modulo η 97 CONTENTS vii

Chapter 9. Construction of geometric Euler systems 99 1. Divisorial liftings of cycles 99 2. Construction of partial Euler systems 100 3. Partial Euler systems on products 101 4. Construction of Flach systems in the self-adjoint case 102 5. Construction of Flach systems in the general case 105 6. Construction of cohesive Flach systems 106

Part 3. Examples 107

Chapter 10. The modular curve X0(N) 109 1. The geometry of X0(N) 109 2. The modular unit ∆ 113 3. The divisor of fp in positive characteristic 115 4. The cohesive Flach system 116

Chapter 11. The modular curve X1(N) 117 1. The geometry of X1(N) 117 2. Admissible markings 120 3. The cohesive Flach system 121 Chapter 12. Kuga-Sato varieties 123 1. The geometry of Kuga-Sato varieties 123 2. Admissible markings 125 3. The cohesive Flach system 126 4. Applications 126

Appendix 129 Appendix A. Edge maps of spectral sequences 131 1. Notation for filtered complexes 131 2. Edge maps 132 3. Edge maps in spectral sequences of filtered complexes 133 4. Edge maps in Grothendieck spectral sequences I 134 5. Edge maps in Grothendieck spectral sequences II 135 6. Boundary maps of exact sequences of filtered complexes 136 7. Boundary maps of Grothendieck spectral sequences 137 8. Edge maps of exact couples 138 Appendix B. Gorenstein linear algebra 141 1. Definitions 141 2. Gorenstein traces and congruence elements 142 3. Gorenstein duality 143 4. Gorenstein pairings 144 5. Skew-symmetric Gorenstein pairings 145 6. Bilateral derivations 146 7. Torsion modules 148 Bibliography 151

Introduction

Fix a squarefree integer N and let f be a rational weight 2 newform for Γ0(N). Let H be the l-adic representation associated to f for some l 7; H is a free ≥ Zl-module of rank 2. Let QS denote the maximal extension of Q unramified away from Nl and set GQS = Gal(QS/Q). Flach proved the following theorem regarding the deformation theory of H. (See [Wes00, Appendix A] for the proof that the set of primes satisfying Flach’s conditions has density 1; this set can be given quite explicitly.) Theorem 0.1 ([Fla92]). Fix f as above and assume that f does not have complex multiplication. Then the set of primes l such that the universal deformation ring of the residual representation ρ : GQ AutF (H/lH) is isomorphic to S → l Zl[[T1,T2,T3]] has density 1. This result was extended to the case of newforms defined over arbitrary number fields in [Fla95] and [Maz]. In this case the l-adic representation H is free of rank 2 over a certain completion A of a Hecke algebra; this ring A is a reduced, finite, flat, local, Gorenstein Zl-algebra and contains a Hecke operator Tp for every prime p. Let k denote the residue field of A. Mazur observed that Flach’s construction can be used to obtain results on the Taylor-Wiles deformation problem for almost all l 7 not dividing N. ≥ Theorem 0.2 ([Maz]). Let f be a newform of weight 2 for Γ0(N). Let l 7 be a prime not dividing N and let H be the l-adic representation associated to≥f. Assume that the natural map GQ Autk(H A k) is surjective. Let R be the → ⊗ minimally ramified universal deformation ring for H A k. Then R is a finite A-algebra and the natural map R A induces an isomorphism⊗ of differentials → ΩR R A ΩA. ⊗ → In this thesis I extend the results of Flach and Mazur to the case of (most) newforms of weight at least 2 for Γ1(N). I also show that the “geometric Euler system” used to prove these results has a very rich algebraic structure and that the isomorphism it yields in deformation theory is essentially canonical. The main result in this context can be phrased as follows. Theorem 0.3 (Theorem XII.3.1 and Theorem IV.6.2). Let f be a newform of weight k 2 for Γ1(N). Let l max 7, k be a prime not dividing N and let H be the l-adic≥ representation (over≥ an appropriate{ } completion A of a Hecke algebra) associated to f. Assume that f can be “cleanly realized” in the cohomology of the universal elliptic curve with level N-structure (see Chapter XII for precise condi- 0 tions). Set T = End H(1) and assume that the natural map GQ Autk(H A k) A → ⊗ is surjective. Let R be the minimally ramified universal deformation ring for H A k (see Section IV.1). Then R is a finite A-algebra and the natural map R A⊗in- → duces an isomorphism of differentials ΩR R A ΩA. Furthermore, the inverse of ⊗ → ix x INTRODUCTION this isomorphism is characterized by the fact that the identification 1 ΩA ΩR R A = HomZ H (Q,T ∗[η]), Ql/Zl → ⊗ ∼ l f (see Section IV.2) identifies the differential of Tp A with 12 times the image under the Bockstein pairing ∈ − 1 1 H (Q, T/ηT ) HomZ H (Q,T ∗[η]), Ql/Zl f → l f 1 of the cohomology class cp Hf (Q, T/ηT ) obtained via the Flach construction. 1 ∈ Here H (Q, ) is a Selmer group (see Section II.1), T ∗ is the Cartier dual of T , f · and η is the congruence element (see Appendix B.2) for the Gorenstein Zl-algebra A. However, it is natural to hope that the method of proof of these results is as interesting as the results themselves. For this reason we proceed in as much generality as we can. Let X be a nonsingular algebraic variety over a global field 2m F and let H be a quotient of the étalecohomology group H (XFs , Zl(m + 1)) for some m. I give a “general” method (contingent on the existence of appropriate 0 geometric data on X X) for the production of geometric Euler systems for EndA H; these in turn yield× corresponding annihilators of certain Selmer groups. These annihilators yield results on the deformation theory of the Galois representation H/lH. Somewhat more generally one can hope to use appropriate geometric data on X itself to control the Selmer group of H; this could then possibly be related to the Bloch-Kato conjectures. The required geometric data (in the deformation theory case) does exist for modular curves and Kuga-Sato varieties. It seems likely that it exists in the case of Hilbert modular surfaces as well. One can give “explicit” conditions for the existence of this data in general, although at this point these are not particularly useful. We now discuss the contents of this thesis in more detail. The first five chapters concern Selmer groups and geometric Euler systems in Galois cohomology. The material of the first three chapters is presented in a fair amount of generality; Chap- ters IV and V are focused on the specific Taylor-Wiles deformation problem. The material of the first two chapters is essentially standard, although our presentation is a synthesis of several others. Chapter I concerns local conditions on Galois cohomology. We define such conditions in full generality in preparation for later results. We especially focus on the functorial aspects of these conditions. We also include the explicit computation of the “natural” local condition in the case of ordinary representations. Chapter II is the globalization of Chapter I: we use local conditions at every place to define Selmer groups of Galois modules. After establishing appropriate functorialities we turn to the definition of two pairings which play a crucial role in our work. The first, the Kolyvagin pairing, combines local and global information. We prove that in a certain sense this pairing evaluates how far a collection of local cohomology classes are from arising from a global cohomology class. The second pairing, the Bockstein pairing, is a pairing between Selmer groups which is of independent interest in many circumstances. In Chapter III we define the notion of a partial geometric Euler system and prove the corresponding annihilation theorems for Selmer groups via the Kolyvagin pairing and the Tchebatorev density theorem. We also give an application of these INTRODUCTION xi methods to prove the non-degeneracy of the Bockstein pairing in the presence of an Euler system. These results are all based on the ideas of Thaine and Kolyvagin, as refined by Flach, Mazur and Rubin. Our presentation is marginally more general than others, but otherwise is well-known. Chapter IV concerns Mazur’s notions of full geometric Euler systems. We begin by considering the deformation theory of certain rank 2 Galois representations over Zl-algebras A and explain the connection with Selmer groups. We then define the notion of a Flach system, which is a slightly strengthened partial geometric Euler system. The real refinement comes with Mazur’s cohesive Flach systems, which combine Flach systems with some additional global algebraic structure. This additional structure results in much more precise deformation theoretic consequences. We then refine this further with our new notion of a cohesive Flach system of Eichler-Shimura type; these are cohesive Flach systems with sharply specified local behavior. One can pass via the Bockstein pairing and a cohesive Flach system to a certain complicated pairing between the differentials ΩA and their dual. In Chapter V we give a computational proof that in the case of a cohesive Flach system of Eichler- Shimura type this pairing is nothing more than a scalar multiple of the canonical duality pairing. This is mostly straightforward except for the computation of the local invariant map and a certain matrix lemma. The second part of this thesis concerns the production of geometric Euler systems for the étalecohomology of varieties over number fields. The central tool, which we call the Flach map, originated in the work of Flach. Our description is a generalization of that of [Fla95] to higher dimensional varieties. [Maz] provided an alternate description in the low-dimensional case; this description does not make direct use of algebraic K-theory. We have not adopted this approach as it does not seem to easily generalize to higher dimensions. The Flach map is defined in Chapter VI, after some preliminaries on the coniveau spectral sequence in étalecohomology and algebraic K-theory. We define the Flach map for smooth separated schemes X over any perfect field and over some non-perfect fields as well; it is a map from certain pairs of cycles and functions on X to the Galois cohomology of the étalecohomology of X. The description of these cycles and functions is most straightforward over local and global fields. In Chapter VII we give the fundamental local description of the Flach map for a variety X over a global field. Specifically, we prove that at places of good reduction one can test a Galois cohomology class against an appropriate unramified local condition via a certain divisor map to the Chow groups of X. (Flach proved a similar result for products of elliptic curves in [Fla92]; he offered a result for products of modular curves in [Fla95], although the proof there is incomplete.) The proof of this theorem makes use of certain results from higher algebraic K- theory; the statement, however, does not involve any explicit K-theory. Assuming a detailed knowledge of the geometry of X, this makes it possible to use the Flach map to generate partial geometric Euler systems; this is the subject of Chapter IX. We also give a result of Flach concerning the local behavior of the Flach map to l-adic cohomology at places above l. Chapter VIII concerns certain algebraic structures on the Flach map for products X X. After some preliminary discussion of algebraic correspondences, we prove the× fundamental Leibniz relation of Mazur and Beilinson. Their argument xii INTRODUCTION for this relation for curves used a clever reduction to a trivial case. We instead present a more direct proof which is valid in arbitrary dimension. We then explain how to use the Leibniz relation to generate a system of Galois cohomology classes which have the algebraic structure of a cohesive Flach system. This involves the theory of Gorenstein rings and bilateral derivations as developed in Appendix B. Chapter IX combines the results of Chapter VII and Chapter VIII to give some sample theorems for the production of geometric Euler systems. In all cases we must assume the existence of appropriate geometric data. The ideas of Section IX.3 were inspired by [?] and [?], although these papers do not explicitly appear anywhere in the discussion. The last three chapters are the applications of the methods we have developed to modular forms. Chapter X concerns the case of the modular curve X0(N). We make some auxiliary hypotheses on the Hecke algebra to simplify the exposition; with these in place the construction is straightforward. (The existence of the cohesive Flach system in this case is due to Mazur.) These restrictions are removed via the more general results on X1(N) in Chapter XI. In Chapter XII we use the realization of Galois representations for modular forms of higher weight in the cohomology of open Kuga-Sato varieties to construct the cohesive Flach system. This requires some slight extensions of the results of Chapters VII and VIII to this setting. Otherwise the construction is completely analogous to the previous cases. The constructions of Chapters XI and XII are new, although in some cases the deformation theoretic conclusions are weaker than those of [?] (for the weight 2 case) and upcoming work of Diamond (for the higher weight case). We include two appendices. The first concerns compatibilities of edge maps of spectral sequences; this is mostly well-known but is included here for lack of an adequate reference. Appendix B is a discussion of the linear algebra of modules over Gorenstein rings. We also give an introduction to the theory of bilateral derivations.

Acknowledgements This thesis could not have been written without a tremendous amount of help from my advisor, Barry Mazur. He has always been extremely willing to share his ideas and insights with me; I can only hope that his point-of-view is visible in this work. There are many people who (for no obvious reason) invested a lot of time in helping me through this thesis. I must especially thank Brian Conrad, Mark Dickin- son, Matthew Emerton and Robert Pollack for their constant help, both answering my questions and working through confusing points with me. I would also like to thank Fred Diamond, Benedict Gross, Joe Harris, Karl Rubin and Richard Tay- lor for numerous helpful conversations, both mathematical and otherwise. I have been very fortunate to have had the opportunity to learn mathematics from many diﬀerent people; in particular, I would like to thank Matt Baker, Keith Conrad, Jordan Ellenberg, Wee-Teck Gan, Tom Graber, Tomas Klenke, Adam Logan, Elena Mantovan, Adi Ofer, David Pollack, David Savitt, Jason Starr and Sam Williams for their help. I, like all Harvard mathematics graduate students, have been extremely fortunate to have the support of a very helpful and dedicated administrative staﬀ. I would especially like to thank Donna D’Fini and Irene Minder for all of their help over the last four years. ACKNOWLEDGEMENTS xiii

My parents have always been extremely supportive (in every way) throughout my education; I hope that they know how much I have appreciated it even if I didn’t always make it clear. My brothers Michael and Matthew have been my best friends for my entire life, and talking to them has often been the best way to get away from this thesis for a while. I also want to thank Jessica Sidman; she has put up with an inordinate amount of whining over the last year and a half. She somehow has still managed to answer my algebraic geometry questions and make this entire process much more enjoyable than it had any right to be. Finally, I must thank P.J.; his remarkable aid at a pivotal moment will always be appreciated.

Notation and terminology

Fields Any time a field F appears in the text we assume that it is accompanied by a fixed choice of separable algebraic closure Fs. We further assume that for any inclusion of fields F, F 0 these choices are made in such a way that Fs is a subfield → of Fs0. We write GF for the absolute Galois group Gal(Fs/F ) of F and cd F for the Galois cohomological dimension of F . By a global field we will mean a finite extension of Q or Fp(t) (for some prime p) and by a local field we will mean a finite extension of Qp or Fp((t)) (for some prime p). If k is a finite field, by a Frobenius element Fr for k we will always mean a geometric Frobenius element, normalized by Fr(x)#k = x for all x k. If X is a scheme over k, we let Fr denote the k-linear Frobenius morphism of∈ X. If F is a global field and v is a place of F , we write Fr(v) for a geometric Frobenius element of GF .

Characters

For any field F of characteristic different from l, we let Zl(1) denote the Tate module of the l-power roots of unity. The natural map GF AutZ Zl(1) will → l be called the cyclotomic character and denoted ε. We write Zl( 1) for the dual − HomZl (Zl(1), Zl) of Zl(1). Let F be a global field and χ : GF A× a character. If v is a place of F at → 1 which χ is unramified, we write χ(v) for χ(Fr(v)). Note that ε(Fr(v)) = (#kv)− .

Galois modules

If F is a field and T is a topological abelian group with an action of GF , we will always assume that the GF -action is continuous for the profinite topology on GF and the given topology on T . If T is such a GF -module, we write F (T ) for the fixed field of the kernel of the map GF Aut T ; we call F (T ) the splitting field of → T over F . If T is a Zl-module with a Zl-linear action of GF and n > 0, then we write T (n) for the n-fold tensor product of T with Zl(1); if n < 0, we write T (n) for the n -fold tensor product of T with Zl( 1). | | − Schemes If x is a point of a scheme X, we write k(x) for its residue field and define its codimension, denoted codimX x, to be the dimension of the local ring X,x. If Z is an arbitrary closed subset of X, we define k(Z) to be the product of theO residue fields of the minimal points of Z, and we define codimX Z to be the least codimension of any point of Z. We write Xp for the set of points of X of codimension exactly p. If x is a point of X, we will writex ¯ for the reduced closed subscheme of X defined

xv xvi NOTATION AND TERMINOLOGY on the closure of x . If X is a scheme over a base S and S0 S is any morphism, { } → we will write XS0 for the fibre product X S S0. If Y and Z are two subschemes of a third scheme X, then by the scheme-theoretic× intersection of Y and Z (in X) we mean the fibre product Y X Z. By a variety over a field×F we will mean a reduced irreducible separated scheme of finite type over Spec F . If X and Y are varieties over a field F , then we write X Y for the fiber product X Spec F Y . × × Sheaves All sheaves other than structure sheaves are assumed to be sheaves for the étaletopology unless otherwise specified; locally constant sheaves are assumed to be locally constant for the étaletopology. If i : U X is an open immersion and → is a sheaf on X, we will usually just write for the pullback sheaf i∗ on U. If F is any torsion sheaf, we write (m) for itsFmth Tate twist: F F F m (m) = µ⊗ . F F ⊗ ∞ If i : X0 X is a morphism and is a Tate twist of a constant sheaf on X we will → F usually still write for the pullback sheaf i∗ on X0. F F Cohomology All spectral sequences are assumed to be cohomological. All cohomology is either étaleor Galois; we will attempt to be careful as to which is which, even when they coincide. If L/K is an extension of fields and T is a topological Gal(L/K)- module, we write Hi(L/K, T ) for the cohomology group Hi(Gal(L/K),T ), computed with continuous cochains. If L is a separable algebraic closure of K, then we just write Hi(K,T ) for these cohomology groups. If is an l-adic étalesheaf on a scheme X, then we write Hi(X, ) for the inverseF limit of the étalecohomology groups Hi(X, /ln ). If is an étalesheafF i F F F of Ql-vector spaces, then we write H (X, ) for the tensor product of Ql with i n F H (X, 0/l 0) where 0 is any l-adic subsheaf of with 0 Ql = ; this F F F F F ⊗ F definition is independent of the choice of 0. F K-theory

We write Ki and Ki0 for Quillen’s K-groups for the category of locally free sheaves and the category of all coherent sheaves respectively; these functors agree on regular schemes. We will write i(X) for the Zariski sheaf of K-groups on X. We take all K-groups to vanish forK negative indices. Part 1

Selmer groups and deformation theory

CHAPTER 1

Local cohomology groups

In this chapter we give the basic theory of finite/singular structures over local fields in preparation for the definition of Selmer groups in Chapter II.

1. Local ﬁnite/singular structures

Fix a prime l and let A be a finite, flat, local Zl-algebra. Let K be a local field with residue field k of characteristic p; we allow p = l. We write Kur for the maximal unramified extension of K, and we let K = Gal(Ks/Kur) denote the inertia group of K. I Let T be an A-module with an A-linear action of GK . We further assume that one of the following holds:

T is a finitely generated Zl-module and the GK -action on T is continuous • for the l-adic topology on T ; or T is a torsion Zl-module of finite corank (that is, T is isomorphic as a Zl- • r module to (Ql/Zl) T 0 for some r 0 and some Zl-module T 0 of finite ⊕ ≥ order) and the GK -action on T is continuous for the discrete topology on T . We will be working with cohomology with continuous cochains (see [Rub00, Ap- pendix B]) and these assumptions are necessary in order to insure that it is well behaved. In the second case, continuous cohomology agrees with the usual profinite/discrete cohomology. We will refer to T as above as l-adic GK -modules over A; if T satisfies the first condition we will say that it is finitely generated, and if it satisfies the second condition we will say that it is discrete. Note that T is both finitely generated and discrete if and only if T is finite. We require maps of l-adic GK -modules over A to be continuous, A-linear and GK -equivariant. (In fact, the continuity is a consequence of the A-linearity.) We will say that T is unramified if K acts trivially on T . I 1 Definition 1.1. The unramified subgroup Hur(K,T ) of the K-cohomology of T is 1 1 1 H (K,T ) = ker H (K,T ) H (Kur,T ) . ur →

1 1 K [Rub00, Lemma 1.3.2] identifies Hur(K,T ) with H (Kur/K, T I ) via inflation, 1 K 1 and this further identifies with H (k, T I ). Note also that Hur(K,T ) is naturally an A-module since the action of GK on T is A-linear. Definition 1.2. A local finite/singular structure on T consists of a choice 1 1 S of A-submodule Hf, (K,T ) H (K,T ). S ⊆ 1 1 1 We will write Hs, (K,T ) for the A-module quotient H (K,T )/Hf, (K,T ). We S 1 S write cs for the image of a cohomology class c H (K,T ) under the quotient map ∈ 3 4 1. LOCAL COHOMOLOGY GROUPS

1 1 1 1 H (K,T ) Hs, (K,T ). We call Hf, (K,T ) and Hs, (K,T ) the finite subgroup and the singular→ S subgroup of the K-cohomologyS of T respectively.S We will omit the structure from the notation if it is clear from context. S 1 The standard choice for Hf, (K,T ) (at least when p = l) is the unramified 1 S 6 1 subgroup Hur(K,T ). In this case we have the following description of Hs, (K,T ). S Lemma 1.3. Let T be an l-adic GK -module over A. Assume that p = l and that T is unramified. Let be the local finite/singular structure on T given6 by 1 1 S 1 Gk Hf, (K,T ) = Hur(K,T ). Then Hs, (K,T ) = T ( 1) . S S ∼ − Proof. We can write the inflation-restriction exact sequence as

1 1 1 Gk 2 0 H (k, T ) H (K,T ) H ( K ,T ) H (k, T ). → → → I → Since k has cohomological dimension 1 [Ser97, Chapter 2, Section 3], the last term vanishes. It follows that

1 1 Gk Hs, (K,T ) = H ( K ,T ) . S ∼ I Since T is unramiﬁed there is also an isomorphism

1 Gk H ( K ,T ) = HomG ( K ,T ). I ∼ k I T is a pro-l group, so any homomorphism K T must factor through the max- I → imal pro-l quotient of K . Letting π denote a uniformizer of K, this quotient 1/l∞ I is Gal(Kur(π )/Kur), which as a Gk-module identiﬁes with Zl(1); see [Fr¨o67, Section 8]. Thus

Gk HomG ( K ,T ) = HomG (Zl(1),T ) = T ( 1) k I ∼ k ∼ − as claimed.

2. Functorialities

Let f : T T 0 be a map of l-adic GF -modules over A. Assume also that T → and T 0 have local finite/singular structures and 0 respectively. Let S S 1 1 f : H (K,T ) H (K,T 0) ∗ → denote the map induced by f. We say that the structures , 0 are compatible with f if S S 1 1 f Hf, (K,T ) Hf, (K,T 0). ∗ S ⊆ S0 If this is the case, then there are natural maps 1 1 Hf, (K,T ) Hf, (K,T 0) S → S0 1 1 Hs, (K,T ) Hs, (K,T 0). S → S0 Note that unramified structures are always compatible. Let i : T 0 , T and j : T T 00 be an injection and a surjection of l-adic → GK -modules over A, respectively. Given a local finite/singular structure on T , S we define the induced local finite/singular structures i∗ and j on T 0 and T 00 by S ∗S 1 1 1 Hf,i (K,T 0) = i− Hf, (K,T ) ∗S ∗ S 1 1 Hf,j (K,T 00) = j Hf, (K,T ). ∗S ∗ S One checks easily that these structures are compatible with i and j, respectively. We will usually just write for i∗ or j if the maps are clear from context. S S ∗S 3. LOCAL EXACT SEQUENCES 5

Lemma 2.1. Let i : T 0 , T and j : T T 00 be maps of unramified l-adic → GK -modules over A. Let denote the unramified finite/singular structure on T . S Then i∗ (resp. j ) is the unramified structure on T 0 (resp. T 00). S ∗S Proof. This is an easy diagram chase; the proof for j requires the fact that ∗S k has cohomological dimension 1. 3. Local exact sequences

Let T be an l-adic GK -module over A with a given local finite/singular structure . Let S 0 T 0 T T 00 0 → → → → be an exact sequence of GK -modules, and give T 0 and T 00 the local finite/singular structures induced from . In this situation the long exact sequence of GK - cohomology splits into a “finite”S and a “singular” exact sequence.

Lemma 3.1. Let 0 T 0 T T 00 0 be an exact sequence of l-adic GK - → → → → modules over A. Let T have a ﬁnite/singular structure and let T 0 and T 00 have the induced structures. Then there are exact sequences S

0 0 0 0 / H (K,T 0) / H (K,T ) / H (K,T 00) /

1 1 1 Hf (K,T 0) / Hf (K,T ) / Hf (K,T 00) / 0 and 1 1 1 0 / Hs (K,T 0) / Hs (K,T ) / Hs (K,T 00) /

2 2 2 H (K,T 0) / H (K,T ) / H (K,T 00) / 0

Proof. We begin with the long exact sequence of GK -cohomology 0 0 0 (3.1) 0 / H (K,T 0) / H (K,T ) / H (K,T 00) /

1 1 1 H (K,T 0) / H (K,T ) / H (K,T 00)

Since T 0 and T 00 have the induced finite/singular structures we have a canonical sequence 1 1 1 (3.2) H (K,T 0) H (K,T ) H (K,T 00). f → f → f 0 1 The map H (K,T 00) H (K,T 0) comes from the exactness of → f 0 1 1 H (K,T 00) H (K,T 0) H (K,T ) → → 1 1 and the fact that H (K,T 0) contains the full inverse image of 0 H (K,T ). Com- f ∈ bining this with (3.1) and (3.2) yields the first sequence of the lemma; exactness is easily checked using the exactness of (3.1) and the definition of induced structures. For the second exact sequence, we begin with the exact sequence

1 1 1 H (K,T 0) / H (K,T ) / H (K,T 00) /

2 2 2 H (K,T 0) / H (K,T ) / H (K,T 00) / 0 The last map is a surjection by standard cohomological dimension results; see [Ser97, Section 5.3, Proposition 15]. The existence of the sequence 1 1 1 H (K,T 0) H (K,T ) H (K,T 00) s → s → s 6 1. LOCAL COHOMOLOGY GROUPS follows immediately from the compatibility of the ﬁnite/singular structures, and 1 2 1 1 the map Hs (K,T 00) H (K,T 0) exists since Hf (K,T 00) is the image of Hf (K,T ) → 1 2 and thus is in the kernel of H (K,T 00) H (K,T 0). This yields the sequence, and exactness is checked by an easy diagram→ chase and the fact that the map 1 1 H (K,T ) H (K,T 00) is surjective. f → f

4. Examples of local structures Following Bloch, Kato and others, we will consider several diﬀerent choices of local ﬁnite/singular structures, depending on the behavior of T as an K -module. I T arbitrary, p arbitrary: The weak structure is given by

1 1 Hf (K,T ) = H (K,T ). T arbitrary, p arbitrary: The strong structure is given by

1 Hf (K,T ) = 0. T unramiﬁed, p = l: The unramiﬁed structure is given by 6 1 1 Hf (K,T ) = Hur(K,T ).

For the rest of the definitions we first must define a certain Ql-vector space V . Assume for this that T is free over Zl (resp. is l-divisible). If T is finitely-generated n (resp. discrete), then set V = T Zl Ql (resp. V = (lim T [l ]) Zl Ql). We will 1 ⊗ ⊗ 1 define choices of Hf (K,V ); these give rise to corresponding←− choices of Hf (K,T ) by pulling back via the natural map T, V (resp. pushing forward via the natural → map V T ). T as above, p = l: The minimally ramified structure is given by 6 1 1 Hf (K,V ) = Hur(K,V ). (One checks as in Lemma 2.1 that this yields the unramified structure if T is unramified.) T as above, p = l: The exponential structure is given by

1 1 1 f=1 H (K,V ) = ker H (K,V ) H (K,V Q B ) , f → ⊗ l cris where f is the Frobenius endomorphism of Bcris. T as above, p = l: The crystalline structure is given by

1 1 1 H (K,V ) = ker H (K,V ) H (K,V Q Bcris) . f → ⊗ l T as above, p = l: The deRham structure is given by

1 1 1 H (K,V ) = ker H (K,V ) H (K,V Q BdR) . f → ⊗ l Here Bcris and BdR are the “big rings” of Fontaine; see [FI93] for an exposition and references. Of course, if T arises as a quotient of a free Zl-module or as a subgroup of an l-divisible Zl-module, then one can give T a local ﬁnite/singular structure induced by one of the above structures. 5. ORDINARY REPRESENTATIONS 7

5. Ordinary representations It will be useful to give an “explicit” characterization of the minimally ramified finite/singular structure on certain ramified rank 3 representations. Assume for this section that K does not have characteristic l.

Definition 5.1. Let T be an l-adic GK -module over A which is free of rank 2 as an A-module. We say that T is ordinary if K acts non-trivially on T and if there is an exact sequence I 0 A(1) T A 0 → → → → which is GL-equivariant for some unramiﬁed extension L of K. Lemma 5.2. Let T be an ordinary representation. Then the minimally ramiﬁed 0 and weak structures on EndA T (1) coincide.

Proof. Set B = A Ql and V = T Ql; by the definition of the minimally ramified structure, we must⊗ show that ⊗ 1 0 1 0 Hur(K, EndB V (1)) = H (K, EndB V (1)). By the inflation-restriction exact sequence, to show this it suffices to show that

1 0 Gk (5.1) H ( K , End V (1)) = 0. I B Let us recall some facts about the cohomology of K . First, it has cohomological dimension 1; see [Ser97, Chapter 2, Section 3.3(c)].I Secondly, by [Frö67, Section 8] the maximal pro-l quotient of K is isomorphic to Zl(1) as a Gk-module; since B(i) is unramified for any i, it followsI that there is an isomorphism 1 H ( K ,B(i)) = Hom( K ,B(i)) = HomZ (Zl(1),B(i)) = B(i 1) I ∼ I ∼ l ∼ − of Gk-modules. Since T is ordinary, there is a B-linear filtration (5.2) 0 B(1) V B 0 → → → → which is GL-equivariant, where L is a finite unramified extension of K. In particular, K is also the inertia group of L. ByI (5.2) we can choose a basis x, y of V such that γx = ε(γ)x γy = y + ν(γ)y for all γ GL; here ε is the cyclotomic character and ν : GL B is some map. ∈ → By definition V is actually ramified, so we know that ν( K ) = 0. I 6 Twisting (5.2) by B(1) and taking K -cohomology yields an exact sequence I α1 K α2 α3 α4 1 α5 0 B(2) V (1)I B(1) B(1) H ( K ,V (1)) B 0. → → → → → I → → K Using our basis of V , one finds that V (1)I ∼= B(2), so α1 is an isomorphism. Thus α2 is the zero map, so α3 is also an isomorphism. Now α4 = 0, so α5 is an 1 isomorphism. We conclude that H ( K ,V (1)) = B. I ∼0 Using our basis of V one can compute EndB V (1) completely explicitly; one finds a GL-equivariant filtration 0 V (1) End0 V (1) B 0. → → B → → 8 1. LOCAL COHOMOLOGY GROUPS

The long exact sequence in K -cohomology and our computations above yield an exact sequence I β β β β β 1 0 K 2 3 4 1 0 5 0 B(2) (End V (1))I B B H ( K , End V (1)) B( 1) 0. → → B → → → I B → − → 0 K One computes directly that (EndB V (1))I = B(2), so β1 is an isomorphism. Thus β2 = 0 and β3 is an isomorphism. Now β4 = 0, so 1 0 H ( K , End V (1)) = B( 1) I B ∼ − as GL-modules. B( 1) has no GL-invariants, so this yields (5.1) as desired. − 6. Cartier dual structures

Let T be an l-adic GK -module over A. We define the Cartier dual T ∗ of T to be HomZl (T, µl∞ (Ks)) with the induced A-module structure (via the A-module g 1 structure on T ). We give T ∗ a GK -action by f(t) = gf(g− t) for f T ∗, g GK ∈ ∈ and t T . T ∗ is also an l-adic GK -module over A; if T is finitely generated, then ∈ T ∗ will be discrete, and if T is discrete, then T ∗ will be finitely generated. For any ideal a of A, there are canonical identifications: a a (T/ T )∗ ∼= T ∗[ ] a a T [ ]∗ ∼= T ∗/ T ∗ a a ( T )∗ ∼= T ∗/T ∗[ ] a a (T/T [ ])∗ ∼= T ∗. One easily checks that (T ∗)∗ is canonically isomorphic to T . The cohomology groups of T and T ∗ are related by Tate local duality. Theorem 6.1 (Tate local duality). Cup product, Cartier duality and the invariant map of local class field theory yield a perfect A-hermitian pairing i 2 i 2 2 H (K,T ) Z H − (K,T ∗) H (K,T Z T ∗) H (K, µl ) ' Ql/Zl. ⊗ l → ⊗ l → ∞ −→ 1 1 Furthermore, if p = l and T is unramified, then Hur(K,T ) and Hur(K,T ∗) are exact orthogonal complements6 under this pairing with i = 1. Proof. See [Mil86, Corollary I.2.3] and [Rub00, Chapter 1, Section 4]. Ru- bin proves his result only for the characteristic 0 case and does not state it in exactly this form, but the general case is the same; one replaces his dimension counts with rank counts in the free case and cardinality counts in the finite case and then combines the two results. The fact that the pairing is A-hermitian is clear since GK acts A-linearly. Given a local finite/singular structure on T , we can use Theorem 6.1 to define S 1 a local finite/singular structure ∗ on T ∗: we define H (K,T ∗) to be the exact f, ∗ 1 S S orthogonal complement of Hf, (K,T ) under Tate local duality. We call this the S Cartier dual local finite/singular structure on T ∗. Tate local duality restricts to yield perfect pairings 1 1 Hf, (K,T ) Zl Hs, (K,T ∗) Ql/Zl S ⊗ S∗ → 1 1 Hs, (K,T ) Zl Hf, (K,T ∗) Ql/Zl. S ⊗ S∗ →

If is the weak (resp. strong, resp. minimally ramiﬁed) structure, then ∗ is the strongS (resp. weak, resp. minimally ramiﬁed) structure; see [Rub00, ChapterS 7. LOCAL STRUCTURES FOR ARCHIMEDEAN FIELDS 9

1, Section 4]. In particular, if T is unramified and is the unramified structure, S then ∗ is also the unramified structure. To make similar statements for the more S subtle structures when p = l, we need to assume that T is free over Zl and that T Z Ql is deRham. In this case, if is the crystalline (resp. exponential, resp. ⊗ l S deRham) structure, then ∗ is the crystalline (resp. deRham, resp. exponential) structure; see [BK90, PropositionS 3.8].

7. Local structures for archimedean fields We now consider the archimedean case. Let K denote either R or C and let 1 T be a GK -module. If K = C, then H (K,T ) is trivial, so there is no structure to assign. If K = R and l = 2, the same is true of H1(K,T ), as it is 2-torsion and 2 is invertible acting on6 T . The only interesting case is K = R and l = 2. In this case one can define weak and strong structures as before. If T is free or divisible, we can also attempt to define a minimally ramified structure. However, since H1(K,V ) = 0, there is no choice for the structure on V . Thus this just gives rise to the weak (resp. strong) structure on T which are finitely generated (resp. discrete).

CHAPTER 2

Global cohomology groups

We begin this chapter by defining global finite/singular structures and Selmer groups. We then turn to the definitions of the local/global Kolyvagin pairing and the global Bockstein pairing. The Kolyvagin pairing will be of fundamental importance to the annihilation theorems of Chapter III, while the Bockstein pairing is of independent interest and will be studied more closely in Chapter V.

1. Selmer groups

Let F be a global field and let MF denote the set of places of F . Recall that if F is a number field, then MF consists of both archimedean and non-archimedean places, while if F is a function field, then MF consists only of non-archimedean places; see [Cas67, Section 3 and Section 12] for details. For every place v we fix now and forever embeddings Fs , Fv,s; these induce injections GF , GF , and → v → changing the choice of embedding changes these injections by conjugation. Let kv denote the residue field of Fv and let v = Gal(Fv,s/Fv,ur) denote the inertia group I of Fv. Let A be a finite, flat, local Zl-algebra with maximal ideal m and residue field k. We assume that l does not equal the characteristic of F . Let Σl denote the set of places of F above l;Σl is empty if F is a function field.

Definition 1.1. An l-adic GF -module over A is an A-module T endowed with an A-linear action of GF such that the action of GFv on T is unramiﬁed for almost all v and such that one of the following holds:

T is a finitely generated Zl-module and the GF -action on T is continuous • for the l-adic topology on T ; or T is a torsion Zl-module of finite corank (that is, T is isomorphic as a Zl- • r module to (Ql/Zl) T 0 for some r 0 and some Zl-module T 0 of finite ⊕ ≥ order) and the GF -action on T is continuous for the discrete topology on T . In the first case we say that T is finitely generated and in the second case we say that T is discrete. We will say that a set of places Σ of F is sufficiently large for T if it contains Σl, all archimedean places and all places where T is ramified; by the definition of l-adic GF -modules, there exist finite sets of places which are sufficiently large for T . For every place v there is a canonical restriction map 1 1 resv : H (F,T ) H (Fv,T ); → resv is initially determined by our embedding Fv , Fv,s, but by [Ser79, Chapter 7, Proposition 3] is actually independent of this choice.→ If c H1(F,T ), then we write ∈ 11 12 2. GLOBAL COHOMOLOGY GROUPS cv for its image under resv. We have the following fundamental lemma regarding these maps. 1 Lemma 1.2. Let T be a discrete l-adic GF -module over A and let c H (F,T ) 1 ∈ be a cohomology class. Then cv lies in Hur(Fv,T ) for almost all v.

Proof. Letc ˜ : GF T be a cocycle representing c; since T is discrete as a → GF module and GF is compact, there is some finite extension F 0 of F such that c˜ factors through Gal(F 0/F ). Now let Σ be a finite set of places of F containing all archimedean places and all places where F 0/F is ramified.c ˜v : GF T v → factors through Gal(Fv0 /Fv); this extension is unramified away from Σ, so so cv is an unramified cocycle for v / Σ. This proves the lemma. ∈ The global analogue of a local finite/singular structure is given by specifying local finite/singular structures at every place.

Definition 1.3. Let T be an l-adic GF -module over A.A finite/singular 1 structure on T consists of choices of local finite/singular structures Hf, (Fv,T ) S 1 1 S for all places v of F such that Hf, (Fv,T ) = Hur(Fv,T ) for almost all v. S Let Σ be a finite set of places of F . We will say that a finite/singular structure is unramified away from a set of places Σ if the local finite/singular structures at Sv are unramified for v / Σ. ∈ If T is free over Zl or l-divisible, then the structures considered in [BK90] and [FPR94] are those which are minimally ramified away from Σl; they consider various possibilities for the structures at Σl. A finite/singular structure determines a Selmer group, which will be our central object of study. 1 Definition 1.4. The Selmer group Hf, (F,T ) of T (with the finite/singular structure ) is the kernel of the map S S 1 1 H (F,T ) Hs, (Fv,T ); → Y S v MF ∈ that is,

1 1 1 Hf, (F,T ) = c H (F,T ) cv Hf, (Fv,T ) for all v , S ∈ | ∈ S the set of global cohomology classes which are everywhere locally ﬁnite. See [Rub00, Chapter 1, Section 6] for interpretations of Selmer groups in terms of ideal class groups, global units and rational points on abelian varieties. 1 Definition 1.5. The Kolyvagin group Hs, (F,T ) is deﬁned to be the quotient 1 1 S H (F,T )/Hf, (F,T ). S There is a natural map

1 1 Hs, (F,T ) Hs, (Fv,T ); S → Y S v MF ∈ if T is a discrete GF -module, then Lemma I.1.2 shows that the image of this map 1 actually lands in vHs, (Fv,T ). ⊕ S 3. THE GLOBAL EXACT SEQUENCE 13

2. Functorialities

Let f : T T 0 be a map of l-adic GF -modules over A. Assume also that T → and T 0 have finite/singular structures and 0 respectively. We say that these S S structures are compatible with f if the local finite/singular structures at Fv are compatible with f for every place v of F ; in this case there is an induced map 1 1 Hf, (F,T ) Hf, (F,T 0) S → S0 of Selmer groups. Let i : T 0 , T and j : T T 00 be an injection and a surjection of l-adic → GF -modules over A, respectively. Given a finite/singular structure on T , we S define the induced finite/singular structures i∗ and j on T 0 and T 00 by assigning S ∗S the induced local finite/singular structures for every place v. By Lemma I.2.1 i∗ and j really are unramified almost everywhere, as required, and they are visiblyS compatible∗S with . This construction applies in particular to maps of the form S T [a] , T and T T/aT , where a is an ideal of A; we will always assume that finite/singular→ structures on such modules are induced as above. We will usually just write for i∗ or j if the maps are clear from context. S S ∗S Definition 2.1. If T is an l-adic GF -module over A with a finite/singular structure , we define the Cartier dual T ∗ of T to be the l-adic GF -module over A S HomZl (T, µl∞ ) with a finite/singular structure ∗ given by the local Cartier dual finite/singular structure at every place of F . S Note that Theorem I.6.1 and our assumption that T is ramified at only finitely many places insures that the structure ∗ really is unramified almost everywhere. S 3. The global exact sequence In this section we give the global analogue of the first local exact sequence of Lemma I.3.1.

Lemma 3.1. Let 0 T 0 T T 00 0 be an exact sequence of l-adic GF - → → → → modules over A. Let be a ﬁnite/singular structure on T and let T 0 and T 00 have the induced ﬁnite/singularS structures. Then there is an exact sequence 0 0 0 1 1 1 0 H (F,T 0) H (F,T ) H (F,T 00) H (F,T 0) H (F,T ) H (F,T 00) → → → → f → f → f Proof. Exactness at the H0-terms follows from the long exact sequence in GF -cohomology. The existence and exactness of the remaining maps follows from the commutative diagram 0 0 0 0

0 1 1 1 H (F,T 00) ____ / Hf (F,T 0) ____ / Hf (F,T ) ____ / Hf (F,T 00)

0 1 1 1 H (F,T 00) / H (F,T 0) / H (F,T ) / H (F,T 00)

1 1 1 0 / H (Fv,T 0) / H (Fv,T ) / H (Fv,T 00) Q s Q s Q s 14 2. GLOBAL COHOMOLOGY GROUPS

Here all columns are exact, as are the bottom two rows (using Lemma I.3.1 for the singular row). The desired maps and exactness follow from an easy diagram chase. 1 1 The map H (F,T ) H (F,T 00) need not be surjective in general, although f → f as we will see later one can often force surjectivity by modifying the ﬁnite/singular structures. As a consequence of Lemma 3.1, we have the following useful result.

Lemma 3.2. Suppose that T is an l-adic GF -module over A and that α A is GF GF 1 1 ∈ such that (αT ) = (T/αT ) = 0. Then Hf (F,T [α]) injects into Hf (F,T ), and 1 under this identiﬁcation it identiﬁes with Hf (F,T )[α]. Proof. Consider the exact sequences 0 T [α] T α αT 0 → → −→ → and 0 αT T T/αT 0. → → → → By Lemma 3.1 the row and column in the commutative diagram

(T/αT )GF

GF 1 1 1 (αT ) / Hf (F,T [α]) / Hf (F,T ) / Hf (F, αT ) LL LLLα LLL LL& 1 Hf (F,T )

GF 1 1 are exact. It follows that if (αT ) = 0, then Hf (F,T [α]) injects into Hf (F,T ). If GF 1 1 (T/αT ) = 0 as well, then Hf (F, αT ) injects into Hf (F,T ), and the rest of the lemma follows. 4. A ﬁniteness theorem for Selmer groups

Let Σ be a finite subset of MF . We define the weak Σ-finite/singular structure Σ on T to be the finite/singular structure on T which is unramified away from Σ andS weak at Σ. Note that H1 (F,T ) is simply the set of cohomology classes which f, Σ are unramified at all places vS / Σ but are unrestricted for v Σ. In particular, if is any other finite/singular structure∈ on T which is unramified∈ away from Σ, thenS H1 (F,T ) H1 (F,T ). f, f, Σ SWe have⊆ theS following cohomological interpretation of Selmer groups for the weak Σ-finite/singular structure.

Lemma 4.1. Let T be an l-adic GF -module over A and let Σ be a finite set of places of F sufficiently large for T . Then H1 (F,T ) H1(F /F, T ) f, Σ = Σ S ∼ where FΣ is the maximal extension of F unramified outside of Σ. Proof. See [Was97, Proposition 6]. The interpretation of the weak structure yields the following fundamental finiteness result, which is really just a slight generalization of the weak Mordell-Weil theorem. 4. A FINITENESS THEOREM FOR SELMER GROUPS 15

Proposition 4.2. Let T be a finite l-adic GF -module over A. Then the Selmer 1 group Hf, (F,T ) is finite for any finite/singular structure . S S Proof. Let Σ be a finite set of places of F which is sufficiently large for T and such that is unramified away from Σ. Since H1 (F,T ) H1 (F,T ), it is f, f, Σ S S ⊆ S enough to show that H1 (F,T ) is finite. f, Σ S Since T is finite we can choose a finite Galois extension F 0 of F such that GF 0 acts trivially on T . Enlarge Σ to contain all places of F which are ramified in F 0/F , and let Σ0 be the set of places of F 0 lying above places of Σ. One sees easily that the finite/singular structures Σ and Σ0 are compatible in the sense that there is a commutative diagram S S 1 / H1 (F ,T ) H (F,T ) v s, Σ v ⊕ S

1 / 1 H (F 0,T ) v0 Hs, (Fv0 ,T ) ⊕ SΣ0 0 We now get an induced map on the kernels of the horizontal maps, which are just the Selmer groups:

Hf, Σ (F,T ) Hf,S (F 0,T ). S → Σ0 Let ker be the kernel of this map; it sits in an exact commutative diagram 0 0

1 0 / ker / H (F 0/F, T )

/ H1 (F,T ) / 1 / H1 (F ,T ) 0 f, Σ H (F,T ) v s, Σ v S ⊕ S

/ 1 / 1 / 1 0 Hf,S (F 0,T ) H (F 0,T ) v0 Hs, (Fv0 ,T ) Σ0 ⊕ SΣ0 0 1 It is clear from the cocycle description that H (F 0/F, T ) is ﬁnite; thus ker is ﬁ-

nite. Since GF 0 acts trivially on T , it follows immediately from Lemma 4.1 that 1 H (F 0,T ) identifies with Hom(Gal(F 00/F 0),T ), where F 00 is the maximal abelian f, Σ extensionS 0 of F unramified outside of Σ and of exponent #T . By [Sil86, Chapter 8, Proposition 1.6], F 00/F 0 is a finite extension, so this Hom-group is finite. The proposition follows, as H1 (F,T ) lies between two finite groups. f, Σ S We have the following version of this result when T is infinite.

Proposition 4.3. Let T be a finitely generated (resp. discrete) l-adic GF - 1 module over A. Then the Selmer group Hf, (F,T ) is finitely generated (resp. of S finite corank) over Zl for any finite/singular structure . S Proof. Let Σ be a finite set of places of F which is sufficiently large for T and such that is unramified away from Σ. Since Zl is noetherian it suffices to show 1 S that H (FΣ/F, T ) is finitely generated (resp. of finite corank). Let G denote the Galois group of FΣ/F . By [Sil86, Chapter 8, Proposition 1.6], the quotient of G 16 2. GLOBAL COHOMOLOGY GROUPS by Gm is finite for every m. Since G is profinite, this implies that G is topologically finitely generated. The proposition now follows from the fact that the Zl-module of continuous maps from G to T is finitely-generated (resp. of finite corank). See [Rub00, Appendix B, Proposition 1.7] for a slightly more general statement.

5. The Kolyvagin pairing In this section we will compile the Tate local dualities over all places of F to define a global pairing which will be of fundamental importance in our annihilation theorems for Selmer groups. Let T be an l-adic GF -module over A and assume that we are given a finite/singular structure on T . Let ∗ be the Cartier dual finite/singular structure S S on T ∗; we omit both of these structures from our notation for the remainder of the section. For every place v of F , let , denote the perfect Tate local pairing h· ·iv 1 1 H (Fv,T ) Z H (Fv,T ∗) Ql/Zl. s ⊗ l f → We define the Kolyvagin pairing

1 1 , : Hs (Fv,T ) Zl Hf (F,T ∗) Ql/Zl h· ·i v ⊕MF ⊗ → ∈ as follows:

(cv), d = cv, dv . h i X h iv v MF ∈ That this is well-deﬁned is immediate from the fact that (cv) is an element of the direct sum and thus zero for almost all v. In order to prove our main theorem regarding this global pairing we will need the following standard result on the Brauer group of a global ﬁeld. We include a cohomological proof for lack of an adequate reference. 2 2 Lemma 5.1. For any c H (F,Fs×), the restriction cv H (Fv,Fv,×s) vanishes for almost all v. ∈ ∈

Proof. Since Fs× is a discrete GF -module, there is some finite Galois extension 2 F 0/F such that c lies in H (F 0/F, F 0×). Letc ˜ : Gal(F 0/F ) Gal(F 0/F ) F 0× be × → some choice of cocycle representing c. Gal(F 0/F ) is finite, soc ˜ takes on only finitely many values. Let Σ be the subset of MF of all archimedean places, all places where F 0/F is ramified and all places at which elements of F 0× in the image ofc ˜ have non-trivial valuation; Σ is finite. Fix v / Σ; we will show that cv = 0. Let v0 be the place of F 0 lying over v, via ∈ our fixed embedding Fs , Fv,s. Since the image ofc ˜ has trivial v-adic valuation, → the cohomology class cv lies in the image of the natural map

2 2 H F 0 /Fv, × H (F 0 /Fv,F 0 ×). v0 F 0 v0 v0 O v0 → We will show that the source of this map vanishes. Consider the exact sequence of Gal(F /Fv)-modules v00

0 U1 × k× 0 F 0 → → O v0 → → where U1 is the group of units of F 0 congruent to 1 modulo the maximal ideal O v0 and k is the residue ﬁeld of F 0 . The long exact sequence in cohomology together O v0 5. THE KOLYVAGIN PAIRING 17

i with the fact that H (F /Fv,U1) = 0 for i 1 (see [Ser79, Chapter 12, Section v00 3, Lemma 2]) shows that ≥

2 2 H F 0 /Fv, × = H (F 0 /Fv, k×). v0 F 0 ∼ v0 O v0

Since F /Fv is unramified, the computation of the cohomology of a finite cyclic v00 group (see [Ser79, Chapter 8, Section 4]) and the fact that the norm is surjective on a finite field shows that this last cohomology group is trivial. This completes the proof.

For a proof of Lemma 5.1 using the cohomology of the ideles, see [?, Section 7, Proposition 7.3 and Section 9.6]. (Note that Tate doesn’t actually prove that the maps he is considering are the restriction maps, but it is not diﬃcult to check this.) For a proof in terms of division algebras, see [Pie82, Chapter 18, Section 5]. We are now in a position to prove the following consequence of global class ﬁeld theory. For any l-adic GF -module T , consider the map

(5.1) H1(F,T ) H1(F ,T ). Y s v → v

1 We deﬁne the compactly supported cohomology Hc (F,T ) to be the A-submodule of H1(F,T ) which has image under (5.1) in the direct sum rather than the direct product:

1 1 (5.2) Hc (F,T ) Hs (Fv,T ). → ⊕v 1 That is, Hc (F,T ) consists of those global cohomology classes which are locally 1 1 unramiﬁed almost everywhere. Note that Hc (F,T ) = H (F,T ) by Lemma 1.2 if T is discrete.

Proposition 5.2. Let T be an l-adic GF -module over A. Then the image of 1 (5.2) is orthogonal to all of Hf (F,T ∗) under the Kolyvagin pairing.

Proof. Consider ﬁrst the commutative diagram

1 1 1 1 H (F,T ) Z H (F,T ∗) / H (Fv,T ) Z H (Fv,T ∗) ⊗ l Qv ⊗ l

2 2 H (F,T Z T ∗) / H (Fv,T Z T ∗) ⊗ l Qv ⊗ l

2 2 H F, µl (Fs) / H Fv, µl (Fv,s) ∞ Qv ∞

∼= 2 2 H (F,F ) / H (Fv,F × ) s× Qv v,s Here all horizontal maps are restriction maps and the vertical maps are cup product, Cartier duality and the map on cohomology coming from the inclusion of the l-power roots of unity into the multiplicative group. It follows from the commutativity of 18 2. GLOBAL COHOMOLOGY GROUPS this that the diagram

1 1 1 1 (5.3) H (F,T ) Z H (F,T ∗) / vH (Fv,T ) Z H (Fv,T ∗) c ⊗ l f ⊕ s ⊗ l f

2 2 H (F,F ) / vH (Fv,F × ) s× ⊕ v,s is commutative as well. Here we are using Lemma 5.1 to insure that the bottom map is well-defined. By [?, Section 9.6] and [Mil86, Appendix A, Theorem 7], there is an exact sequence 2 2 (5.4) H (F,Fs×) H (Fv,Fv,×s) Q/Z 0 → ⊕v → → 2 where the last map is the summation map; here H (Fv,Fv,×s) is identified via local 1 class field theory with Q/Z, 2 Z/Z or 0 in the usual way. 1 1 Fix c H (F,T ) and d H (F,T ∗). Following c d clockwise around (5.3) ∈ c ∈ f ⊗ and then mapping it by summation to Q/Z yields the global pairing (cv,s), d by definition. Following c d around (5.3) in the counter-clockwise directionh showsi ⊗ 2 that it maps to Q/Z via H (F,Fs×); by (5.4) this map vanishes, which completes the proof.

6. Shafarevich-Tate groups To deﬁne a pairing between Selmer groups we will need some basic facts on Shafarevich-Tate groups. If Σ is any set of places of F , let FΣ be the maximal extension of F unramiﬁed away from Σ.

Definition 6.1. Let T be an l-adic GF -module over A and let Σ be an arbitrary set of places of F . The ﬁrst Σ-Shafarevich-Tate group of T is

1 1 GF 1 XΣ(F,T ) = ker H FΣ/F, T Σ H (Fv,T ) . →v ⊕Σ ∈ 1 By [Mil86, Chapter 1, Theorem 4.10(a)], XΣ(F,T ) is finite for any set of places Σ and any finite l-adic GF -module T . If Σ is sufficiently large for T , then by Lemma 4.1 the inflation map 1 1 H (FΣ/F, T ) , H (F,T ) → 1 identifies H (FΣ/F, T ) with Hf, Σ (F,T ); thus we can also write S 1 1 1 XΣ(F,T ) = ker Hf, (F,T ) H (Fv,T ) S →v ⊕Σ ∈ (6.1) = ker H1(F,T ) H1(F ,T ) H1 (F ,T ) . v s,ΣΣ v →v ⊕Σ ×v ⊕ /Σ ∈ ∈ 1 That is, a cohomology class lies in XΣ(F,T ) if and only if it is unramified away from Σ and is actually zero at all places of Σ. If Σ Σ0 are sufficiently large for T , then by (6.1) there is a canonical inclusion ⊆ X1 (F,T ) , X1 (F,T ). Σ0 → Σ Write 1(F,T ) for 1 (F,T ). We will need the following lemma. X XMF 6. SHAFAREVICH-TATE GROUPS 19

Lemma 6.2. Let T be a ﬁnite l-adic GF -module over A and suppose that X1(F,T ) = 0. 1 Then there is some ﬁnite set of places Σ such that XΣ(F,T ) = 0.

Proof. Choose a finite set of places Σ0 which is sufficiently large for T . By our remarks above, X1 (F,T ) is finite. Since X1(F,T ) = 0, for every x X1 (F,T ) Σ0 ∈ Σ0 there is some place vx such that xv = 0. Taking Σ to contain Σ0 and all of the x 6 places vx and using (6.1) proves the lemma.

Definition 6.3. Let T be an l-adic GF -module over A and let Σ be an arbitrary set of places of F . The second Σ-Shafarevich-Tate group of T is

2 2 GF 2 XΣ(F,T ) = ker H FΣ/F, T Σ H (Fv,T ) . →v ⊕Σ ∈ We have the following fundamental relationship between X1 and X2.

Proposition 6.4. Let T be a ﬁnite l-adic GF -module over A and let Σ be a ﬁnite set of places containing Σl. Then there is a perfect pairing 1 2 X (F,T ) X (F,T ∗) Ql/Zl. Σ ⊕ Σ → Proof. See [Mil86, Chapter 1, Theorem 4.10].

Consider now a surjection T T 00 of l-adic GF -modules over A. Assume that T 00 is given a ﬁnite/singular structure induced by one on T . In general there is no reason to expect the induced map on Selmer groups 1 1 H (F,T ) H (F,T 00) f → f to be surjective. However, if a certain Shafarevich-Tate group vanishes, we can obtain a partial result.

Lemma 6.5. Let 0 T 0 T T 00 0 be an exact sequence of finite l-adic → → → → GF -modules over A. Assume that T has a finite/singular structure and let T 0 1S and T 00 have the induced finite/singular structures. Suppose that X (F,T 0∗) = 0. 1 Then for any x00 Hf, (F,T 00), there is a finite set of places Σ and an element ∈ S x H1(F,T ) such that ∈ 1 1 x maps to x00 under the map H (F,T ) H (F,T 00); • 1 → xv Hf, (Fv,T ) for all v / Σ. • ∈ S ∈ Proof. Let Σ be a finite set of places which is sufficiently large for T and such 1 2 that XΣ(F,T 0∗) = 0. By Proposition 6.4 this implies that XΣ(F,T 0) = 0. Let 0 be the finite/singular structure on T which agrees with away from Σ and whichS S has the weak structure at all places of Σ. We will also write 0 for the induced S finite/singular structures on T 0 and T 00. The long exact sequence in GF -cohomology and Lemma 4.1 yield an exact sequence 1 1 2 (6.2) Hf, (F,T ) Hf, (F,T 00) H (FΣ/F, T 0). S0 → S0 → 1 1 1 Since x00 Hf, (F,T 00), its restriction to H (Fv,T 00) lies in Hf, (Fv,T 00) for ∈ S S all places v. For v Σ, consider the image of x00 under the natural map ∈ 1 2 2 Hf, (F,T 00) H (FΣ/F, T 0) H (Fv,T 0). S0 → → 20 2. GLOBAL COHOMOLOGY GROUPS

1 By Lemma I.3.1, Hf, (Fv,T 00) is annihilated by the boundary map S 1 2 H (Fv,T 00) H (Fv,T 0), → 2 2 so x00 maps to 0 in H (Fv,T 0). This shows that x00 maps into XΣ(F,T 0) 2 ⊆ H (FΣ/F, T 0). But this group vanishes, so by the exactness of (6.2) x00 pulls back 1 to an element x of Hf, (F,T ). This x satisfies the required conditions. S0 7. The Bockstein pairing Let α β 0 T 0 T T 00 0 → −→ −→ → be an exact sequence of finite l-adic GF -modules over A. Let T have a fixed finite/singular structure and let T 0 and T 00 have the induced finite/singular struc- 1 tures. Assume also that X (F,T 0∗) vanishes. Under these hypotheses we will define the Bockstein pairing 1 1 , α,β : H (F,T 00) H (F,T 0∗) Ql/Zl {· ·} f ⊗ f → which we will study in much more detail later. 1 1 1 Fix x00 H (F,T 00) and y0 H (F,T 0∗). Since we assumed that X (F,T 0∗) = ∈ f ∈ f 0, by Lemma 6.5 there is a finite set Σ of places of F and an element x H1(F,T ) ∈1 such that xv,s = 0 for v / Σ and such that x maps to x00 under the map H (F,T ) 1 ∈ → H (F,T 00). For each v Σ, consider the diagram ∈ (7.1) x / x00

H1(F,T ) / H1(F,T ) x 00 _00

1 1 1 0 / Hs (Fv,T 0) / Hs (Fv,T ) / Hs (Fv,T 00) 0

x xv0 / v,s / 0

Here the bottom row is the exact sequence of Lemma I.3.1. Since xv00 lies in 1 1 Hf (Fv,T 00) and x maps to x00, the image of xv,s in Hs (Fv,T 00) is zero; thus by 1 (7.1) there is an element xv0 Hs (Fv,T 0) which maps to xv,s. Set xv0 = 0 for v / Σ. We define ∈ ∈ x00, y0 α,β = (x0 ), y0 Ql/Zl. { } h v i ∈ We must check that this definition is independent of the choice of places Σ and the choice of lifting x. Since for any fixed lifting x we can always freely enlarge Σ, it is enough to check this for two different liftings of x00 for the same Σ. However, 1 the difference of these liftings lies in H (F,T 0), and Proposition 5.2 now gives the desired independence. 1 The Bockstein pairing can be defined without the assumption on X (F,T 0∗); see [FPR94, Chapitre 2, Section 1.4]. CHAPTER 3

Annihilation theorems for Selmer groups

We now turn to the deﬁnition of partial geometric Euler systems and the corresponding annihilation theorems for Selmer groups. The same methods will also yield a non-degeneracy result for the Bockstein pairing.

1. Partial geometric Euler systems

Let A and F be as in Chapter II. Let T be a finitely generated l-adic GF -module over A with a finite/singular structure . If C is an A-submodule of H1(F,T ) and S 1 v is a place of F , we write Cv,s for the image of C in Hs (Fv,T ). Definition 1.1. Let be a (possibly infinite) set of places of F and let η be L v an ideal of A.A partial (geometric) Euler system C v of depth η for T (with the structure ) at is an assignment of A-submodules{ } ∈LCv H1(F,T ) for each v such thatS L ⊆ ∈ L v Cw,s = 0 for all places w = v; • 1 v 6 H (Fv,T )/C is killed by η. • s v,s v 1 If in addition C vanishes in Hs (Fv, T/ηT ) for all v , we say that the partial Euler system has strict depth of η. In this case the image∈ ofL each Cv in H1(F, T/ηT ) 1 v lies in Hf (F, T/ηT ), and we define the Euler module Φ of C v to be the A- { } ∈L submodule of H1(F, T/ηT ) generated by the image of Cv for all v . f ∈ L The next result explains how partial Euler systems behave under pushforward.

Lemma 1.2. Let j : T T 00 be a surjection of l-adic GF -modules over A and let T 00 have the finite/singular structure induced by . Assume that T admits a v S partial Euler system C v of depth η. Let d be an ideal of A which annihilates the cokernel of { } ∈L 1 1 Hs, (Fv,T ) Hs, (Fv,T 00) S → S v for every v . Then j C v L is a partial Euler system for T 00 of depth ηd. ∈ L { ∗ } ∈ Proof. That j Cv is supported only at v is immediate from the definition of ∗ the induced finite/singular structure on T 00. The assertions at v are clear from the definitions. The following result is the first step in the proof of our annihilation theorems for Selmer groups. For a set of places , define L

1 1 1 H (F,T ∗) = ker H (F,T ∗) H (Fv,T ∗)! . L Y → v 1 1 Note that X (F,T ∗) H (F,T ∗). L ⊆ L 21 22 3. ANNIHILATION THEOREMS FOR SELMER GROUPS

Lemma 1.3. Let T be a finitely generated l-adic GF -module over A with a v finite/singular structure . Suppose that T admits a partial Euler system C v of depth η. Then S { } ∈L 1 1 ηHf (F,T ∗) H (F,T ∗). ⊆ L Proof. Fix v . By the definition of a partial Euler system, Cv maps to ∈ L 1 v 0 in every singular cohomology group except for Hs (Fv,T ). In particular, C 1 v 1 ⊆ H (F,T ). In addition, for c C and d H (F,T ∗), the Kolyvagin pairing c ∈ ∈ f (cw), d is simply the Tate local pairing at v: h i (cw), d = cv, dv h i h iv v 1 1 Proposition II.5.2 now shows that Cv,s and the image of Hf (F,T ∗) Hf (Fv,T ∗) are orthogonal under the Tate local pairing at v. Since this is a perfect→ pairing and 1 v 1 η kills Hs (Fv,T )/Cv,s, it follows immediately that η kills the image of Hf (F,T ∗) 1 in Hf (Fv,T ∗). This is the statement of the lemma. 1 Note that we can not conclude from Lemma 1.3 that ηHf (F,T ∗) is contained 1 in X (F,T ∗) because of possible bad behavior at the bad places for T ∗. L 2. The key lemmas In this section we prove the key lemmas for the annihilation theorems for Selmer groups. Let T be a finite l-adic GF -module over A with a finite/singular structure . Let F 0 = F (T ) be the splitting field of T ; it is a finite Galois extension of F and S we set ∆ = Gal(F 0/F ). Note that ∆ injects into AutA T . If τ is any element of ∆, we define τ to be the set of non-archimedean places of L F which are unramified in the extension F 0/F and which have Frobenius conjugate to τ over F 0. That is,

τ = v MF v non-archimedean, F 0 /Fv unramified, there L { ∈ | v exists v0 MF such that v0 v and Fr (v0) = τ . ∈ 0 | F 0/F } By the Tchebatorev density theorem τ has positive density in MF . L We also make the analogous definitions for T [m]: let Fm = F (T [m]) be its splitting field and set ∆m = Gal(Fm/F ); it injects into Autk(T [m]). We will say that an element of ∆m or ∆ is a non-scalar involution if it acts in that way (as an A-linear endomorphism) on T [m] or T , respectively. The next lemma shows that it is easy to lift non-scalar involutions.

Lemma 2.1. Assume l = 2. Suppose that there is a non-scalar involution τm 6 in ∆m. Then there exists a non-scalar involution τ in ∆ lifting τm.

Proof. Note that the map AutA T Autk(T [m]) is surjective with kernel an → l-group. Since ∆ and ∆m inject into these groups and ∆ surjects onto ∆m, we see that we can lift τm to some element τ0 in ∆ which has order 2 times a power of l. Taking an appropriate l-power of τ0 we obtain an involution τ ∆. Since l = 2, τ ∈ 6 will still reduce to τm in ∆m and thus is still non-scalar

Lemma 2.2. Let T be a ﬁnite l-adic GF -module over A. Suppose that the following conditions hold: l = 2; • 6 T [m] is absolutely irreducible as a GF -module over k; • There is a non-scalar involution τm ∆m. • ∈ 2. THE KEY LEMMAS 23

Let τ ∆ be some non-scalar involution lifting τm. Let be a set of places coﬁnite ∈ 1 1 L1 in τ . Then H (F,T ) H (∆,T ); here we regard H (∆,T ) as a subgroup of L L ⊆ H1(F,T ) via inﬂation. In particular, X1 (F,T ) H1(∆,T ). L ⊆ Proof. Consider the exact sequence

1 GF GF ∆ 1. → 0 → → → This yields an inflation-restriction exact sequence 1 1 ab (2.1) 0 H (∆,T ) H (F,T ) Hom∆(G ,T ). → → → F 0 Chasing through the definitions, one finds that δ ∆ acts on g Gab as follows: F 0 ˜ δ ˜ ˜ ∈1 ∈ let δ be any lifting of δ to GF , and set g = δgδ− . One checks immediately that ab this is a well-defined action and yields an element of GF . Homomorphisms in ab ab 0 Hom∆(G ,T ) are equivariant for this action of ∆ on G and the natural action F 0 F 0 of ∆ on T . 1 ab Let c H (F,T ) satisfy cv = 0 for all v . Let ϕ : GF T be the image ∈ L ab ∈ L 0 → of c in Hom∆(G ,T ). To prove the lemma we must show that ϕ = 0. F 0 Let F 00 be the fixed field of the kernel of ϕ and set Γ = Gal(F 00/F 0); we have an exact sequence 1 Γ Gal(F 00/F ) ∆ 1 → → → → and a commutative diagram ϕ ab / GF T 0 A @ AA ÑÑ AA ÑÑ AA ÑÑ ϕ A 0 ÑÑ Γ In particular, Γ is a finite abelian l-group since it injects into T . Let τ be our fixed non-scalar involution in ∆. Since Γ has odd order we can lift τ to an involutionτ ˜ in Gal(F 00/F ) as in the proof of Lemma 2.1. Let g be any element of Γ and considerτg ˜ Gal(F 00/F ). By the Tchebatorev density theorem there exists an unramified place∈ v of F such that Fr (v ) =τg ˜ . Setting 00 00 F 00/F 00 v0 = v00 F and v = v00 F , we have the following situation: | 0 | (2.2) F v _ 00 00 Γ

Gal(F /F ) 00 F 0 v0

∆ _ F v By standard properties of Frobenius elements we have

Fr (v0) = Fr (v00) F =τg ˜ F = τ F 0/F F 00/F | 0 | 0 (so in particular v τ ) and ∈ L (2.3) Fr (v ) = Fr (v )deg(v0/v) = (˜τg)2. F 00/F 0 00 F 00/F 00

Here by deg(v /v) we mean the degree of the local field extension F /Fv; it is 2 0 v00 since Fr (v ) = τ has order 2 and v /v is unramified. Note also that by the F 0/F 0 0 Tchebatorev density theorem there are infinitely many possible choices of such v00; 24 3. ANNIHILATION THEOREMS FOR SELMER GROUPS in particular, we can avoid any finite set of places of τ and therefore assume that L v . Thus, by hypothesis, cv = 0. ∈ LWe claim that since c = 0 we have ϕ(Fr (v )) = 0. To see this begin with v F 00/F 0 00 the commutative diagram

1 resv 1 c H (F,T ) / H (Fv,T ) ∈

/ Hom(G ,T ) ϕ Hom(GF 0 ,T ) F 0 ∈ v0 Since ϕ factors through Γ = Gal(F /F ), ϕ G factors through Gal(F /F ), 00 0 F v0000 v00 | v0 which is generated by Fr (v ). Since c = 0 we0 have ϕ = 0, which we now F 00/F 0 00 v GF | v0 see says precisely that ϕ(Fr (v )) = 0, as claimed. 0 F 00/F 0 00 Combining (2.3) with this, we conclude that (2.4) ϕ(˜τgτg˜ ) = 0. 2 1 Sinceτ ˜ = 1, we can writeτg ˜ τg˜ =τg ˜ τ˜− g; by the deﬁnition of the action of ∆ on Gab this is nothing other than τ g g. Since ϕ is ∆-equivariant (2.4) now implies F 0 that · (2.5) τϕ(g) = ϕ(g). − (2.5) holds for all g Γ, so if we let Ψ be the A-submodule of T generated by ∈ ϕ(Γ), then we have Ψ T −. Here by T − we mean the 1 eigenspace for the action of τ on T . Note also that⊆ Ψ is stable under the action− of ∆, as ϕ is ∆-equivariant and the action of ∆ is A-linear. Consider Ψ[m] T −[m] T [m]. Since τm acts as a non-scalar, the second inclusion is strict. As⊆ Ψ[m] is⊆ ∆-stable and T [m] is irreducible as a ∆-module, this implies that Ψ[m] = 0. By Lemma B.7.3 this implies that Ψ = 0; thus ϕ = 0, which is what we were trying to prove. Note that it is implicit in the above proof that T [m] has dimension at least 2 over k, as otherwise all k-linear automorphisms of T [m] are scalar. To get a result for the one dimensional case one can mimic the above proof with τ = 1; this is a special case of the next result. We now give an alternate version of Lemma 2.2 which is due to Rubin. In fact, Lemma 2.2 is a special case of this result, but we include it separately as the proof is of independent interest.

Lemma 2.3. Let T be a ﬁnite l-adic GF -module over A. Suppose that the following conditions hold:

T [m] is absolutely irreducible as a GF -module over k; • There is a τm ∆m such that T [m]/(τm 1)T [m] = 0. • ∈ − 6 Let τ be any lifting of τm to ∆. Let be a set of places cofinite in τ . Then L1 L H1 (F,T ) H1(∆,T ). In particular, X (F,T ) H1(∆,T ). L ⊆ L ⊆ Proof. The proof is similar in spirit to that of Lemma 2.2. As before, by 1 (2.1) it is enough to show that for any c H (F,T ) such that cv = 0 for all v , the associated homomorphism ϕ : Gab ∈T is trivial. Choose also a representative∈ L F 0 → cocyclec ˜ : GF T for the cohomology class c. Let F 00 be some finite extension → of F 0 through whichc ˜ factors; ϕ necessarily factors through Gal(F 00/F 0), which we 3. THE ANNIHILATION THEOREM 25 denote by Γ. That is, we now have mapsc ˜ : Gal(F 00/F ) T and ϕ :Γ T , the first a cocycle and the second a homomorphism. Note that→ϕ need not be→ injective on Γ. Fix some liftingτ ˜ of τ to Gal(F 00/F ). Choose also some g Γ. By the Tchebatorev density theorem we can find a place v of F such that∈ Fr (v ) = 00 00 F 00/F 00 τg˜ . Let v0 and v be the restriction of v00 to F 0 and F respectively. (This is the same basic set-up as in (2.2).) We have Fr (v ) = τ, so that v . As before, F 0/F 0 τ we can assume that v avoids any finite set and therefore we can take∈ Lv to lie in ; L thus cv = 0.

c˜ Gal(F 00 /Fv ) is a coboundary, since cv = 0. (Here we are also using the fact that | v00 inﬂation maps are injective on H1 in order to insure thatc ˜ really is a coboundary for Gal(F 00 /Fv).) Thus in particularc ˜(FrF /F (v00)) (FrF /F (v00) 1)T ; that is, v00 00 ∈ 00 − (2.6)c ˜(˜τg) (˜τg 1)T = (τ 1)T. ∈ − − Taking g = 1 shows thatc ˜(˜τ) (τ 1)T . Returning to the case of arbitrary∈ − g Γ, by the cocycle relation we have ∈ c˜(˜τg) =c ˜(˜τ) +τ ˜c˜(g) =c ˜(˜τ) + τc˜(g). This lies in (τ 1)T by (2.6), so combined with the fact thatc ˜(˜τ) (τ 1)T , this shows that τc˜(g−) lies in (τ 1)T . Since (τ 1)˜c(g) trivially lies in here,∈ we− conclude that − − c˜(g) (τ 1)T ∈ − for all g Γ. Thus∈ the image ofc ˜, and therefore the image of ϕ, lies in (τ 1)T . Letting Ψ be the A-submodule of (τ 1)T generated by ϕ(Γ), the proof continues− as before, using the fact that − (τm 1)T [m] = T [m] − 6 and the absolute irreducibility of T [m] to show that Ψ = 0.

3. The annihilation theorem The following theorem is essentially due to Flach (see [Fla92, Proposition 1.1]), although the ideas go back to Thaine and Kolyvagin and this presentation is due to Mazur. For any l-adic GF -module T , let δ = δ(T ) be the A-annihilator of the 1 cohomology group H (∆,T ∗).

Theorem 3.1. Let T be a finite l-adic GF -module over A with a finite/singular structure . Suppose that one of the following two conditions hold: S l = 2 and there is a non-scalar involution τm ∆m; or • 6 ∈ there is a τm ∆m such that T ∗[m]/(τm 1)T ∗[m] = 0. • ∈ − 6 Let τ be an appropriate lifting of τm to ∆. (That is, τ is a non-scalar involution in the first case and is an arbitrary lifting in the second case.) Assume also that

T ∗[m] is absolutely irreducible as a GF -module over k; • v T admits a partial Euler system C v of depth η for some set of places • { } ∈L coﬁnite in τ . L L 1 Then δη annihilates the Selmer group Hf (F,T ∗). Proof. This is immediate from Lemma 1.3 and Lemma 2.2 or Lemma 2.3 (applied to T ∗), as appropriate. 26 3. ANNIHILATION THEOREMS FOR SELMER GROUPS

We now let T be an arbitrary finitely generated l-adic GF -module over A. For any ideal a of finite index in A, T ∗[a] is finite; we let Fa be its splitting field and we set ∆a = Gal(Fa/F ). Let δ = δ(T ) be the largest ideal of A which annihilates each of the groups 1 n H (∆ln ,T ∗[l ]) for sufficiently large n. Let d = d(T ) be the largest ideal of A which annihilates the cokernel of

1 1 n H (Fv,T ) H (Fv, T/l T ) s → s for n greater than some ﬁxed n0 and all v . ∈ L As always, if l = 2 we can lift a non-scalar involution τm ∆m to non-scalar 6 ∈ involutions τn ∆ln for each n. ∈ Corollary 3.2. Let T be a ﬁnitely generated l-adic GF -module over A. Sup- pose that one of the following two conditions hold:

l = 2 and there is a non-scalar involution τm ∆m; or • 6 ∈ There is a τm ∆m such that T ∗[m]/(τm 1)T ∗[m] = 0. • ∈ − 6 Let τn be appropriate liftings of τm to each ∆ln . If T ∗[m] has rank one over k, assume further that n T ∗/T ∗[l ] has no GF -invariants for suﬃciently large n. • Assume also that

T ∗[m] is absolutely irreducible as a GF -module over k; • v T admits a partial Euler system C v of depth η for some set of places • { } ∈L coﬁnite in τ for some n. L L n 1 Then δdη annihilates the Selmer group Hf (F,T ∗).

1 Proof. Let c H (F,T ∗) be any element. Since T ∗ is discrete, c factors ∈ f through some finite extension F 0/F . Letc ˜ : Gal(F 0/F ) T ∗ be some cocycle → representing c. Gal(F 0/F ) is finite, soc ˜ takes on only finitely many values. In m particular, its image must lie in T ∗[l ] for some m. This means that c lies in the image of the map

1 m 1 (3.1) H F,T ∗[l ] H (F,T ∗). → We can also assume that m is sufficiently large so that m T ∗/T ∗[l ] has no GF -invariants (this is automatic when T ∗[m] has rank • at least two, since T ∗[m] is absolutely irreducible); T admits a partial Euler system of depth η for some set of places cofinite • L in τm ; L1 m δH (∆lm ,T ∗[l ]) = 0; • 1 1 m d annihilates the cokernel of H (Fv,T ) H (Fv, T/l T ) for all v . • s → s ∈ L m GF It follows from the fact that (T ∗/T ∗[l ]) = 0 and the long exact sequence 1 in cohomology that (3.1) is injective. Since c lies in Hf (F,T ∗), it follows from our 1 m definition of the induced structure that c actually lies in the image of Hf (F,T ∗[l ]). By Lemma 1.2 the partial Euler system for T induces one of depth dη for T/lmT , 1 m so Theorem 3.1 shows that δdη annihilates Hf (F,T ∗[l ]); thus it must annihilate c as well. 4. RIGHT NON-DEGENERACY OF THE BOCKSTEIN PAIRING 27

4. Right non-degeneracy of the Bockstein pairing

Let T be a finitely generated l-adic GF -module over A. Assume: The hypotheses of Corollary 3.2 are satisfied; • v The partial Euler system C v is of strict depth η; • η is principal, generated by{ a non-zero} ∈L divisor (we also write η for a fixed • generator); 1 H (Fη/F, T ∗[η]) = 0; • 1 1 The groups Hf (Fv,T ) are divisible by η, in the sense that if ηc Hf (Fv,T ) • 1 1 ∈ for some c H (Fv,T ), then c H (Fv,T ); ∈ ∈ f Note that the last hypothesis is satisfied in the case of any of the local finite/singular structures we described in Section I.4, except possibly for the strong structure. Let 1 v Φ Hf (F, T/ηT ) denote the Euler module of the partial Euler system C v . ⊆ 1 { } ∈L By Lemma 2.2 or Lemma 2.3, the assumption that H (Fη/F, T ∗[η]) = 0 implies 1 that X (F,T ∗[η]) = 0. Thus we can define the Bockstein pairing L 1 1 (4.1) , η : H F, T/ηT H (F,T ∗[η]) Ql/Zl {· ·} f ⊗ f → associated to the exact sequence η 0 T/ηT T/η2T T/ηT 0; → −→ −→ → here the first map is multiplication by η and the second map is the natural quotient map. (One checks easily that the divisibility hypothesis above insures that the 2 2 injection T/ηT , T/η T and the surjection T/η T T/ηT induce the same finite/singular structure→ on T/ηT .) Proposition 4.1. Let T be as above. The restriction 1 , η :Φ H F,T ∗[η] Ql/Zl {· ·} ⊗ f → 1 of the Bockstein pairing (4.1) is right non-degenerate. That is, if y H (F,T ∗[η]) ∈ f is such that x, y η = 0 for all x Φ, then y = 0. In particular, if also { } ∈ δ(T ) = d(T ) = A; • GF (ηT ∗) = 0; • GF (T ∗/ηT ∗) = 0; • then the Bockstein pairing induces an injection 1 H (F,T ∗) , HomZ (Φ, Ql/Zl). f → l

1 Proof. Let y be any element of H (F,T ∗[η]) and let c Φ be the image of f ∈ a classc ˜ Cv H1(F,T ) for some v . We compute the Bockstein pairing for ∈ ⊆ ∈ L these elements. To compute the pairing c, y η we ﬁrst must lift c via the map { } H1(F, T/η2T ) H1(F, T/ηT ) → to an element of H1(F, T/η2T ) with singular restriction 0 away from some ﬁnite subset of places Σ. But this is easy; we simply take Σ = v and lift c to the image ofc ˜ in H1(F, T/η2T ), which we also denote byc ˜. The{ next} step is to pull back 1 2 c˜v,s H (Fv, T/η T ) under the injection ∈ s 1 1 2 (4.2) η : H (Fv, T/ηT ) H (Fv, T/η T ). s → s 28 3. ANNIHILATION THEOREMS FOR SELMER GROUPS

1 We denote this pull back by η c˜v,s. The pairing c, y η is now nothing other than 1 1 { }1 the Tate pairing of c˜v,s H (Fv, T/ηT ) and yv H (Fv,T ∗[η]): η ∈ s ∈ f 1 (4.3) c, y η = η c˜v,s, yv . { } D Ev Since Cv has strict depth η we can ﬁnd classes c Φ such that the associated 1 2 ∈ classesc ˜v,s generate ηHs (Fv, T/η T ) as an A-module. But this is just the image of 1 1 1 Hs (Fv, T/ηT ) under (4.2), so we see that the classes η c˜v,s generate Hs (Fv, T/ηT ) as an A-module. 1 Now assume that y Hf (F,T ∗) is orthogonal to all of Φ. (4.3) shows that yv 1∈ is orthogonal to all of H (Fv, T/ηT ) under the Tate pairing for all v . Since the s ∈ L Tate pairing is perfect, this implies that yv = 0 for all v . Thus by Lemma 2.2 or Lemma 2.3 we have y = 0. This proves the non-degeneracy.∈ L 1 For the last injection, the non-degeneracy shows that Hf (F,T ∗[η]) injects into 1 1 HomZl (Φ, Ql/Zl). Lemma II.3.2 shows that Hf (F,T ∗[η]) = Hf (F,T ∗)[η], and now Corollary 3.2 completes the proof.

5. A δ-vanishing result In this section we give a proof of the following basic result on cohomology of the general linear group. Proposition 5.1. Let A be an artin local ring with finite residue field k of characteristic l = 2 and let H be a free A-module of finite rank n. If n = 2 (resp. n > 2) then assume6 that #k = 5 (resp. l is at least 5 and does not divide n + 1). Then 6 1 0 H GLn(A), End H = 0. A Proof. Let V be a free k-module of rank n. We first prove that 1 0 (5.1) H GLn(A), End V = 0 k by induction on the length of A; here V is considered as a GLN (A)-module via the reduction map GLN (A) GLN (k). A has length 1 precisely→ when A = k, and in this case [DDT97, Lemma 2.48] (for n = 2) and [CPS75, Table 4.5] (for n > 2) show that 1 0 H SLn(k), End V = 0. k Since the index of SLn(k) in GLn(k) is prime to l, (5.1) follows immediately from this. In the general case of (5.1), let m be the largest integer such that mm = 0, and consider the surjection 6 m GLn(A) GLn(A/m ). → Let U denote the kernel, so that we have a short exact sequence m (5.2) 0 U GLn(A) GLn(A/m ) 0 → → → → m Note that U = 1 + m Mn(k) as a subgroup of GLn(A). In particular, U is a GLn(k)-module and has the following decomposition into irreducible constituents: (5.3) U = (End0 V )r kr; ∼ k ⊕ here r is the dimension of mm as a k-vector space. 5. A δ-VANISHING RESULT 29

Associated to (5.2) we have an inﬂation-restriction sequence / 1 0 / 1 m 0 / 0 H (GLn(A), Endk V ) H (GLn(A/m ), Endk V )

0 δ 2 m 0 m / HomGLn(A/m )(U, Endk V ) H (GLn(A/m ), Endk V ) Since A/mm has smaller length than A, by induction to prove (5.1) it suffices to show that δ is injective. Note that 0 0 Hom m (U, End V ) = Hom (U, End V ) GLn(A/m ) k ∼ GLn(k) k m 0 since the GLn(A/m )-actions on both U and Endk V factor through GLn(k). Consider the exact sequence (5.2) as an element 2 m c H (GLn(A/m ),U). ∈ 0 Given an f HomGLn(k)(U, Endk V ), by [HS53, Theorem 4] the image of f under δ is nothing∈ other than f c; that is, δ(f) is the the pushforward of (5.2) by f. To show that δ is injective,∗ we must show that this pushforward splits only if f is trivial. One checks easily that none of the subextensions of f c corresponding to copies 0 ∗ of Endk V in (5.3) split (for example, it suffices to exhibit two commuting matrices m in GLn(A/m ) which lift to non-commuting matrices in GLn(A)). If f = 0, then 0 6 by Schur’s lemma there must be at least one factor of Endk V in U which maps 0 isomorphically to Endk V under f. But then the subextension of f c corresponding 0 ∗ to this factor of Endk V does not split either, so δ(f) = 0. This shows that δ is injective, and thus completes the proof of (5.1). 6 Lifting the result from V to H is easy: simply use the long exact sequence in GLn(A)-cohomology associated to the short exact sequence 0 End0 (mH/mt) End0 (H/mt) End0 (V ) 0 → A → A → A → and an induction on t. As an immediate corollary we have the following δ-vanishing result. We return now to the case of a finite, flat, local Zl-algebra A.

Corollary 5.2. Let H be an l-adic GF -module over A which is free of rank n as an A-module and let T = End0 H. Assume that l = 2 and that the Galois A 6 representation GF AutA H is surjective. If n = 2 (resp. n > 2), then assume that #k = 5 (resp. →l 5 does not divide n + 1). Then δ(T ) = 0. 6 ≥ Proof. Note that the surjectivity of GF AutA H implies the surjectivity → of GF AutA H∗[a] for all ideals a of A. The corollary thus follows immediately → from Proposition 5.1.

CHAPTER 4

Flach systems

We begin this chapter by setting up the deformation theory of certain rank 2 Galois representations and relating it to Selmer groups. We then turn to the deﬁnitions and study of various notions of geometric Euler systems.

1. Minimally ramified deformations We now turn to applications of the theory of the previous chapters to the deformation theory of Galois representations. We will consider deformation problems very similar to those considered in [Wil95, Chapter 1]. Let l be an odd prime and let A be a reduced, finite, flat, local Zl-algebra with maximal ideal m and residue field k. Let W (k) denote the Witt vectors of k; A is canonically a W (k)-algebra. We now require F to be a number field with at least one real embedding. We further require that Fv is absolutely unramified for every v Σl. ∈ Let H be a free A-module of rank 2 with a continuous A-linear action of GF . Fix an integer k > l. We make the following assumptions on this Galois representation:

H A k is absolutely irreducible; • ⊗ H is unramified away from a finite set of places Σ containing Σl; • For every v Σ Σl, H is minimally ramified at v (see below); in partic- • ∈ − ular, the inertia coinvariants H v are free over A for every v / Σl; I ∈ H is crystalline of weight k at every place v of Σl (see below); • There is a free W (k)-module W of rank 1 (with a chosen generator ξ) • with a continuous W (k)-linearf action of GF and a A-hermitian, Galois equivariant, perfect pairing

ψ : A˜ A H Z H Zl; ⊗ ⊗ l → here A˜ = W A; ⊗W (k) Every complexf conjugation element in GF acts on W as multiplication by • 1. f − The W (k)-algebra A is generated by the traces of Fr(v) acting on H v for • I all v / Σl. ∈ We will say that such a Galois representation is of Taylor-Wiles type of weight k.

Let χ : GF W (k)× denote the inverse of the character of W ; H has determinant → χ over A. Set H˜ = A˜ A H. By Lemma 4.1 the existence offψ implies that A is a ⊗ Gorenstein Zl-algebra. Fix a Gorenstein trace tr : A Zl. → For each v / Σl we deﬁne the Hecke operator Tv A to be the trace of Fr(v) ∈ ∈ acting on H v . For v / Σ, we will write χ(v) for χ(Fr(v)); Fr(v) has characteristic polynomial I ∈ 2 x Tvx + χ(v) − for its action on H.

31 32 4. FLACH SYSTEMS

We say that H is minimally ramiﬁed at a place v if the image of the inertia group v in GL2(A) is conjugate to one of the two subgroups I 1 b a 0 ( ); b A , ( ); a A× . { 0 1 ∈ } 0 1 ∈

Note in particular that in either case the inertia coinvariants H v of H are free over A of rank 1, as asserted above. See [Maz97, Section 29] for moreI details. For our crystalline conditions we must consider the integral theory of [FL82]; see also [BK90, Section 4]. Set

0 Dcris(H) = H Fv,Bcris Z H . ⊗ l We say that H is crystalline at a place v Σl if H arises from a strongly divisible ∈ lattice D in Dcris(H) via the Tate module functor; see [BK90, Theorem 4.3]. This implies in particular that

dimFv Dcris(H) = rankZl H so that H Zl Ql is crystalline in the usual sense. D is endowed with a decreasing filtration F⊗iD by direct summands and we say that H is of weight k if 2 i 0; i  ≤ rank F F D = 1 1 i k 1; O v  ≤ ≤ − 0 k i.  ≤ For an example of a representation of Taylor-Wiles type of weight 2, let E be an elliptic curve over F with good reduction away from Σ Σl and multiplicative − reduction at every place of Σ Σl. Then the l-adic Tate module TlE is a repre- − sentation as above with A = Zl and χ cyclotomic; see [DDT97, Section 2.2] for the case F = Q. More generally, representations coming from modular forms or abelian varieties with real multiplication are often of this form. We will consider these examples in more detail later. We are interested in deformations of H A k which satisfy the same local conditions as H. Specifically, let denote the⊗ category of local noetherian W (k)- algebras with residue field k; a mapC between two rings in is assumed to be local and to induce the identity map on k. C Following Diamond, if B is an object of and H0 is a free B-module of rank 2 C with a B-linear action of GF , we say that H0 is minimally ramified if

H0 is unramified away from Σ; • For every v Σ Σl, H0 A k is minimally ramified at v; • ∈ − ⊗ H0 is crystalline at every place in Σl in the sense of Fontaine-Laffaille and • the filtration on the associated Dieudonnémodule D satisfies F 0D = D and F kD = 0; H0 has determinant χ. • It is for the crystalline condition above for which we must assume that k < lm as otherwise the integral theory is not known. Let Sets denote the category of sets.

Definition 1.1. For any ring B in , by a minimally ramified lifting of H A k C ⊗ we will mean a minimally ramified free B-module H0 of rank 2 with a continuous action of GF together with an isomorphism α0 : H0 B k = H A k. We consider ⊗ ∼ ⊗ two such pairs (H0, α0), (H00, α00) to be isomorphic if there is an isomorphism β : 1 H0 ∼= H00 as B[GF ]-modules such that α00βα0− is the identity map. We define 1. MINIMALLY RAMIFIED DEFORMATIONS 33 a minimally ramified deformation of H A k to B to be an isomorphism class of ⊗ minimally ramified liftings of H A k to B. We define a covariant functor ⊗ D : Sets C → by letting D(B) be the set of minimally ramified deformations of H A k to B. If ⊗ f : B B0 is a morphism in , we let f : D(B) D(B0) be the map induced by → C ∗ → B B0 corresponding to f. We call D the minimally ramified deformation functor · ⊗ of H A k. ⊗ Results of Mazur and Ramakrishna show that the functor D is representable by a W (k)-algebra R; that is, for any B , there is a functorial isomorphism ∈ C D(B) = HomW (k) alg(R, B). ∼ − This isomorphism is therefore realized by a universal minimally ramified deformation (HR, αR) D(R) such that given a W (k)-algebra map f : R B, the pair ∈ → (HR R B, αR f) (with B being regarded as an R-algebra via f) represents the corresponding⊗ isomorphism⊗ class in D(B). See [Maz90, Section 1] for the basic representability result and [Maz97, Chapters 5 and 6] for a discussion of the extra deformation conditions. (Standard properties of crystalline representations show that the conditions at Σl are categorical conditions in the sense of [Maz97, Section 25]; see [Fon82, Théorèmeof Section 5.2] and [Fon94, Section 0.1].) We should comment that deformation functors are perhaps most naturally studied as a functor from the category of inverse limits of artinian local rings with residue fields k. (The inverse limits are computed in the category of topological rings.) See [dSL97] and [Dic00] for expositions. This is the category in which the universal deformation ring R initially lives. In our cases R will always be noetherian, so we will not concern ourselves with this distinction. Note that (H, id) is an element of D(A); we therefore have a canonical map π : R A such that H = HR R A. We will always regard A as an R-algebra → ∼ ⊗ via π. For every place v / Σl, Fr(v) is defined acting on the inertia coinvariants ∈ ˆ HR, v up to conjugation, and we define Hecke operators Tv R as the trace of Fr(v) I ∈ acting on HR, v ; these are all well-defined as before due to the minimal ramification I hypotheses. Note that π(Tˆv) = Tv; since we assumed that the Tv generate A as a W (k)-algebra, this implies that π is surjective. In order to study the ring R, we will primarily consider a variant of the functor D which takes into account our initial representation H. Let (A) be the category of local noetherian W (k)-algebras with residue field k equippedC with a local homomorphism f to A inducing the identity map on k: B / / k

f A / / k A morphism in (A) must respect these maps. We deﬁne theC modiﬁed deformation functor

DA : (A) Sets C → as follows: Given an element f : B A of (A), we let DA(f : B A) be the inverse image of (H, id) D(A) under→ the mapC f : D(B) D(A) induced→ by f. ∈ ∗ → 34 4. FLACH SYSTEMS

That is, DA(f : B A) consists of the deformations of H A k to B which are “congruent to H” via→ the augmentation homomorphism to A⊗. Since we assumed that A is generated by the Hecke operators Tv,[Maz97, Section 20, Proposition 4] shows that DA is represented by π : R A for the same → ring and universal deformation (HR, αR) as D. Given our deﬁnition of morphisms in (A), this means that for any object f : B A in (A), DA(f : B A) consists C → C → of those elements of HomW (k) alg(R, B) which yield π on composition with f. − 2. Tangent spaces and Selmer groups

The Zariski tangent A-module tDA to the deformation functor DA is deﬁned to be

tDA = DA(A[]). Here A[] = A[]/(2) and we take the map A[] A to be the natural map given by 0; in fact this is the only such map since→A is reduced. Since R represents 7→ DA, this means that tDA consists of those elements of HomW (k) alg(R, A[]) which map to our distinguished homomorphism π : R A on composition− with the map → A[] A. tDA does not encode information about torsion, however, so we shall work→ with certain other tangent spaces which carry more information. For any 2 ideal a of A, let Aa be the ring A[]/(a, ) = A A/a; we consider Aa as an object of (A) by using the 0 map as the augmentation⊕ to A. The following C 7→ well known result connects the set DA(Aa) to the module of continuous differentials ΩR = ΩR/W (k) of R. Proposition 2.1. For any ideal a of A, there is a canonical isomorphism of A-modules DA(Aa) = HomA(ΩR R A, A/a). ∼ ⊗ Proof. The universal property of tensor products implies that there is a natural identification of HomA(ΩR R A, A/a) with HomR(ΩR, A/a). This in turn ⊗ identifies with the set DerW (k)(R, A/a) of W (k)-linear derivations from R to A/a. Given such a derivation ω : R A/a, we obtain a homomorphism fω : R Aa → → by fω(r) = π(r) + ω(∂r); fω obviously yields π on mapping down to A. We therefore have defined a map DerW (k)(R, A/a) DA(Aa) which is easily checked to be a map of A-modules. This map has an obvious→ inverse, sending an appropriate 1 homomorphism f : R Aa to the derivation given by ωf (∂r) = (f(r) π(r)); → − this proves that it is an isomorphism. It is a fundamental result of deformation theory that the tangent module also has an interpretation as a certain Selmer group. We first need to introduce a Galois representation associated to H. Set 0 T = EndA H(1).

T is a free A-module of rank 3 with a continuous A-linear action of GF . The isomorphism of Corollary B.5.3 yields an isomorphism of A[GF ]-modules 2 T = A˜(1) A Sym H. ∼ ⊗ A Lemma 2.2. The canonical pairing 0 0 End H(1) End H A(1) Zl(1) A ⊗ A → → sending f g to tr of the usual trace of fg is a perfect pairing. In particular, it ⊗ 0 identiﬁes T ∗ with End H Z Ql/Zl. A ⊗ l 2. TANGENT SPACES AND SELMER GROUPS 35

Proof. That the pairing End0 H End0 H A A ⊗ A → sending f g to the trace of fg is perfect is a standard fact and is easily checked. ⊗ The fact that composition with tr yields a perfect pairing to Zl follows easily from Lemma B.3.1; twisting now gives the result. Let be the finite/singular structure on T which is minimally ramified away S from Σl (in particular, unramified away from Σ) and crystalline at all places of Σl. Since H is assumed to be crystalline at all places above l, it follows easily that T is crystalline at all such places. T is therefore deRham as well and the discussion in Section I.5 shows that the structure ∗ on T ∗ also has the minimally ramified S structure away from Σ and the crystalline structure at all places of Σl. For any 0 ideal a of A, we also let EndA(H/aH) have the induced structure coming from its natural injection into T ∗. With respect to these finite/singular structure, we have the following result. This is essentially a standard isomorphism of deformation theory, and was proved in many cases in [Wil95, Propositions 1.2 and 1.3]. Proposition 2.3. For any ideal a of A, there is an isomorphism 1 0 H F, End (H/aH) = DA(Aa). f A ∼ Proof. We recall briefly the definition of this isomorphism; see [Maz97, Sec- tion 21] for more details. We have a distinguished element ρ0 of DA(Aa) coming from the natural injection A, Aa. Given any other deformation ρ0 : GF → → GL2(Aa), (we have implicitly chosen a basis here in order to go from the deformation to an actual homomorphism) define a cocycle cρ0 : GF EndA(H/aH) by → 1 1 cρ (σ) = (ρ− (σ)ρ0(σ) 1). 0 0 − (The expression in parentheses is divisible by since ρ0 is congruent to ρ modulo

.) One checks that cρ0 really is a cocycle; that conjugating ρ0 changes cρ0 by a 1 coboundary; and that this map from DA(Aa) to H (F, EndA(H/aH)) is injective. 1 0 We must show that the image is precisely Hf (F, EndA(H/aH)). The restriction to trace zero matrices corresponds to our fixed determinant condition; see [Maz97, Section 24]. For the conditions at v / Σ, we first note that a representation of GF is unramified away from Σ if and only∈ if it factors through the maximal quotient of GF unramified away from Σ. This is the sort of Galois group to which deformation theory is usually applied, and Lemma II.4.1 shows that this notion of unramified is 1 0 compatible with our local conditions Hf (Fv, EndA(H/aH)). For v Σ Σl, the compatibility of our deformation condition and our cohomological∈ condition− is contained in [Maz97, Proposition of Section 29]. This leaves the crystalline conditions at v Σl; these are dealt with via the interpre- 1 0 ∈ tation of H (Fv, EndA(H/aH)) as extensions [Maz97, Section 22] together with 1 0 [BK90, Lemma 4.5], which shows that the elements of H (Fv, EndA(H/aH)) which correspond to crystalline extensions are precisely those which lie in the subgroup 1 0 Hf (Fv, EndA(H/aH)). Combining Proposition 2.1 and Proposition 2.3, we obtain an isomorphism 1 0 n n (2.1) H F, End (H/l H) = HomA ΩR R A, A Z Z/l Z f A ∼ ⊗ ⊗ l 36 4. FLACH SYSTEMS for any positive integer n. One checks easily that the formation of this Selmer group commutes with direct limits, so we obtain an isomorphism 1 H (F,T ∗) = HomA ΩR R A, A Z Ql/Zl . f ∼ ⊗ ⊗ l Our choice of Gorenstein trace tr and Lemma B.3.2 now yield an isomorphism 1 (2.2) H (F,T ∗) = HomZ (ΩR R A, Ql/Zl) f ∼ l ⊗ which will be fundamental to what follows.

3. Good primes

Let τ GF be a ﬁxed choice of complex conjugation; such a τ exists since we assumed∈ that F has at least one real embedding. By our assumption on the character χ, the determinant of τ acting on H is 1. It follows that τ acts as a non-scalar involution on H, and one checks directly− from this that τ also acts as a non-scalar involution on H∗, T and T ∗. Let Fm denote the splitting ﬁeld of H A k. We let denote the set of places ⊗ L of F with Frobenius on H A k conjugate to τ. Fix a place v . Since Fr(v) is ⊗ ∈ L conjugate to τ in Fm, we have the relation (3.1) Fr(v)2 1 0 (mod m); − ≡ this is the minimal polynomial of Fr(v) acting on H A k since Fr(v) is a non-scalar. We also have the characteristic polynomial ⊗ 2 Fr(v) Tv Fr(v) + χ(v) = 0 − for the action of Fr(v) on H. We conclude from (3.1) that if v , then ∈ L (3.2) Tv 0 (mod m); χ(v) 1 (mod m). ≡ ≡ − We also have a factorization 2 x Tvx + χ(v) (x 1)(x + 1) (mod m) − ≡ − coming from (3.1). A is complete for the m-adic topology, so Hensel’s lemma shows that this lifts to a factorization 2 x Tvx + χ(v) = (x α)(x β) − − − in A[x] with

α β 1 (mod m); α + β = Tv; αβ = χ(v). ≡ − ≡ Lemma 3.1. Let v be a place in . There is a direct sum decomposition (depending on v) L H = Hα Hβ ⊕ where Hα and Hβ are free of rank 1 over A, and Fr(v) acts on Hα as the scalar α and on Hβ as the scalar β. Proof. Set

Hα = (Fr(v) β)H; − Hβ = (Fr(v) α)H. − Since (Fr(v) α)(Fr(v) β) = 0, we see that Fr(v) acts on Hα as multiplication by − − α and on Hβ as multiplication by β. Furthermore,

(3.3) Hα Hβ = 0, ∩ 3. GOOD PRIMES 37 since Fr(v) acts on any element of this intersection simultaneously as 1 and 1 modulo m, and 2 / m. − Let h be an arbitrary∈ element of H. Setting

hα = (Fr(v) β)h; − hβ = (Fr(v) α)h − − we have hα + hβ = (α β)h. Since α β 2 (mod m), it is a unit in A; this shows − − ≡ that Hα + Hβ = H. Combined with (3.3) this proves that we have a direct sum decomposition H = Hα Hβ. ⊕ It remains to show that both Hα and Hβ are free over A of rank 1. Suppose ﬁrst that Hα A k = 0. Then Fr(v) acts on all of H A k as the scalar 1; thus Fr(v) has trace⊗ congruent to 2 modulo m, which contradicts⊗ (3.2). We− conclude − that Hα A k = 0. In the same way we see that Hβ A k = 0. Since H A k is ⊗ 6 ⊗ 6 ⊗ a two-dimensional vector space over k, it follows that Hα A k and Hβ A k are ⊗ ⊗ both one dimensional over k. Nakayama’s lemma now implies that Hα and Hβ are cyclic as A-modules. Since Hα and Hβ are cyclic A-modules, we have

dimQ Hα Z Ql dimQ A Z Ql l ⊗ l ≤ l ⊗ l dimQ Hβ Z Ql dimQ A Z Ql. l ⊗ l ≤ l ⊗ l However, we also have

dimQ (Hα Hβ) Z Ql = 2 dimQ A Z Ql. l ⊕ ⊗ l l ⊗ l It follows that

dimQ Hα Z Ql = dimQ Hβ Z Ql = dimQ A Z Ql. l ⊗ l l ⊗ l l ⊗ l Since A is torsion-free over Zl, this implies that Hα and Hβ are both free of rank 1 over A, as claimed. The preceding result gives a very explicit characterization of the Galois representation T at v and makes it possible to compute the singular quotient 1 ∈ L Hs (Fv,T ). 1 Lemma 3.2. For all v , Hs (Fv,T ) is a free A-module of rank 1. For any 1 ∈ L ideal a of A, Hs (Fv,T/aT ) is a free A/a-module of rank 1. In particular, the natural map 1 1 H (Fv,T ) H (Fv,T/aT ) s → s is surjective.

Proof. By Lemma 3.1 we can ﬁx an A-basis x, y of H such that Fr(v)(x) = αx and Fr(v)(y) = βy; we have αβ = χ(v) and α β 1 (mod m). This last congruence implies that α2 and β2 are not equal to≡χ( −v), since≡ χ(v) 1 (mod m) by (3.2). ≡ − 0 0 1 Gkv Gkv By Lemma I.1.3, Hs (Fv,T ) = T ( 1) = (EndA H) . Gkv acts on EndA H ∼ − ∼ α 0 as conjugation by Fr(v); that is, as conjugation by the matrix 0 β . This sends a a b matrix c a to − 2 2 α 1 αβa α b a χ(v) b = 2 ! . χ(v) β2c αβa β c a − χ(v) − 38 4. FLACH SYSTEMS

2 2 0 Gk Since α and β do not equal χ(v), it follows that (EndA H) v is free of rank 1 1 0 over A, generated by 0 1 . 1 − The result for Hs (Fv,T/aT ) is proven in the same way, and the surjectivity follows immediately. 4. Flach systems In this section we introduce the weakest of the structures on the cohomology of Galois representations of Taylor-Wiles type which we will consider. We make the additional assumptions:

T A k is absolutely irreducible; • 1⊗ H (F (T ∗[a])/F, T ∗[a]) = 0 for every ideal a of ﬁnite index in A. • v Definition 4.1. Let η be a non-zero divisor in A.A Flach system c v of v { } ∈L v depth η for T is a weak Euler system C v of strict depth η such that each C is a cyclic A-module, generated by cv { H}1(∈LF,T ). v ∈ Let c v be a Flach system for T and let Φ denote the Euler module of { } ∈L v the associated partial Euler system C v . Note that essentially by deﬁnition { } ∈L the set contains the set τ of places associated to T ∗ in Chapter III, Section 2. L L v Lemma 3.2 and Corollary III.5.2 insure that T and C v satisfy the hypotheses of Corollary III.3.2 with δ = d = A; thus { } ∈L 1 ηHf (F,T ∗) = 0. 1 In addition, Lemma III.2.2 shows that X (F,T ∗[η]) = 0. The assumption that GF GF T A k is irreducible as a GF -module implies that (ηT ∗) = (T ∗/ηT ∗) = 0, so that⊗ by Lemma II.3.2 we have 1 1 (4.1) H (F,T ∗) = H F,T ∗[η] . f f Combining all of this, we see that the Bockstein pairing as in Section III.4 exists: 1 1 H (F,T ∗) Z H (F, T/ηT ) Ql/Zl. f ⊗ l f → Proposition III.4.1 shows that it induces an injection 1 H (F,T ∗) , HomZ (Φ, Ql/Zl). f → l Note that this pairing depends on the choice of the generator η. The isomorphism (2.2) and Pontrjagin duality now yield a surjection

(4.2) Φ ΩR R A. ⊗ In particular, ΩR R A is η-torsion since Φ is by definition. In the case that η is a unit, this completely⊗ determines R. Lemma 4.2. Let π : R A be a surjection of finite W (k)-algebras with residue → field k and suppose that ΩR R A = 0. Suppose also that W (k) injects into A. Then π is an isomorphism, and both⊗ of R and A are isomorphic to W (k).

Proof. Since ΩR R A = 0 and R is local, Nakayama’s lemma implies that ⊗ ΩR = 0. Consider the map W (k) R. Reducing modulo l yields a map k R/lR. → → R/lR is automatically flat over k, and ΩR/lR/k = ΩR/W (k) R/lR = 0; thus R/lR is an étalelocal k-algebra with residue field k. But the only⊗ such algebra is k itself. We conclude that k R/lR is an isomorphism, and then by Nakayama’s lemma that W (k) R is a→ surjection. Since the map W (k) A is an injection, this → → implies that W (k) = R = A, as claimed. 5. COHESIVE FLACH SYSTEMS 39

To apply Lemma 4.2 in our case we need to know that R is a finite W (k)- algebra. A standard argument in deformation theory shows that R is surjected onto by a power series ring in dimFl D(k[ε]) variables. But D(k[ε]) is isomorphic 1 0 to H (F, End (H A k)); the assumption that η is a unit and the discussion above f k ⊗ implies that the latter group is trivial. Thus R is surjected onto by W (k), so it is definitely finite. We state our result in this case as a proposition.

Proposition 4.3. If T admits a Flach system of depth η A×, then the natural maps ∈ W (k) R A → → are all isomorphisms. In order to obtain analogous results when η is not a unit we will need to impose more structure on our Flach system.

5. Cohesive Flach systems We continue with the assumptions of the previous section. Definition 5.1. A cohesive Flach system of depth η for T is a collection of v 1 classes c H (F,T ) for all v / Σl such that ∈v ∈ c v is a Flach system of depth η; •{v } ∈L 1 c vanishes in Hs (Fw,T ) for all v and all w = v; • v 1 6 c vanishes in Hs (Fw, T/ηT ) for all places v and w; • 1 v The map Θ : A H (F, T/ηT ) sending Tv to c is well-deﬁned and is a • (continuous) derivation.→

The third condition and the fact that the Tv generate A as a W (k)-module 1 imply that the image of Θ actually lands in Hf (F, T/ηT ). Note also that Θ is automatically W (k)-linear since W (k) is unramiﬁed over Zl. Θ induces an A-linear map 1 h :ΩA H (F, T/ηT ) → f with image im Θ. Thus we have a surjection

(5.1) ΩA im Θ. Of course, there is an injection (5.2) Φ , im Θ. → We also have the surjection (4.2) induced by the Bockstein pairing:

(5.3) Φ ΩR R A. ⊗ Lastly, we have a surjection

(5.4) ΩR R A ΩA ⊗ coming from the surjection π : R A. The existence of these four maps and [Mat86, Theorem 2.4] imply that all of them are isomorphism. We define the Flach automorphism Ξ:ΩA ΩA → to be the composition of (5.1), the inverse of (5.2), (5.3) and (5.4). Returning to the isomorphism (5.4), we see that this means that the surjection π : R A induces an isomorphism on differentials. Such a map is said to be an evolution→ . In [EM97], it is shown that for many classes of rings A (for example, 40 4. FLACH SYSTEMS local complete intersections), A admits no non-trivial evolutions. In this case, one can conclude that π is an isomorphism, and therefore that the Galois representation H is the universal minimally ramified deformation of H A k. ⊗ 6. Cohesive Flach systems of Eichler-Shimura type In order to give an explicit description of the map Ξ introduced in the previous section, we will need an assumption about the behavior of the Flach classes cv in 1 Hs (Fv,T ). We assume from now on that v does not lie in Σ. 1 Recall that by Lemma I.1.3 we have a canonical identification of Hs (Fv,T ) G 0 G with T ( 1) kv = (End H) kv . Explicitly, this isomorphism is realized as follows: − ∼ A 1 ur H (Fv,T ) = HomG Gal(Fv,s/F ),T s ∼ kv v ur 1/l∞ ur = HomG Gal(F (λ )/F ),T ∼ kv v 0 v where λ0 is a fixed uniformizer of Fv and the second isomorphism comes from the ur 1/l∞ ur ur fact that Gal(Fv (λ0 )/Fv ) is the maximal pro-l quotient of Gal(Fv,s/Fv ). Fix n a l∞-root λ of λ0, in the sense of fixing a compatible l -th root λn for each n. We ur ur 1/l∞ will write Fv (λ) for Fv (λ0 ). Fix also a generator ζ of Zl(1), and let ζn be the corresponding choice of generator of µln . These choices determine a generator τ ur ur of Gal(Fv (λ)/Fv ) by requiring τ(λ) = ζλ, where this equation has the obvious ur ur meaning in terms of inverse limits. With these choices, Gal(Fv (λ)/Fv ) ∼= Zl(1) as Gkv -modules, where τ corresponds to ζ. (We have indefinitely suspended the use of τ for a fixed complex conjugation.) Let Fr(v) denote the endomorphism of H induced by Frobenius. We define the 1 verschiebung Ver(v) to be the endomorphism given by χ(v) Fr(v)− . v Definition 6.1. A cohesive Flach system c v /Σl is said to be of Eichler- { } ∈ v 1 Shimura type of weight 2w if for each v / Σ, the class cv,s Hs (Fv,T ) is given by ∈ ∈ cv : Gal(F ur(λ)/F ur) T v,s v v → τ j wjη(Fr(v) Ver(v)) ζ 7→ − ⊗ v To check that cv,s really is a cocycle it suffices by Lemma I.1.3 to check that

Fr(v) Ver(v) is Gkv -equivariant; this is clear since Gkv is abelian. Note that Fr(v) −Ver(v) really does have trace zero. One checks easily that this definition is independent− of the choice of ζ and λ. In the case that a cohesive Flach system is of Eichler-Shimura type we can compute the Flach automorphism completely explicitly. Theorem 6.2. Assume that T admits a cohesive Flach system of Eichler- Shimura type of depth η and weight 2w. Then the Flach automorphism Ξ:ΩA → ΩA is multiplication by 2w. We will prove this result in the next chapter. Theorem 6.2 can be regarded as a sort of reciprocity law for the Bockstein pairing 1 1 H F, T/ηT Z H F,T ∗[η] Ql/Zl. f ⊗ l f → By (2.2) and (4.1) the second term identifies with HomZ (ΩR R A, Ql/Zl) and l ⊗ then via the evolution π with HomZl (ΩA, Ql/Zl). The first term admits a map from ΩA coming from the cohesive Flach system. In this context, Theorem 6.2 is precisely the following characterization of the Bockstein pairing. 6. COHESIVE FLACH SYSTEMS OF EICHLER-SHIMURA TYPE 41

Corollary 6.3. The pairing

ΩA Z HomZ (ΩA, Ql/Zl) Ql/Zl ⊗ l l → induced from the Bockstein pairing by the maps described above is 2w times the canonical duality pairing.

CHAPTER 5

Flach systems of Eichler-Shimura type

In this chapter we give the proof of Theorem IV.6.2.

1. The map on diﬀerentials We begin by recalling the details of the construction of the map Ξ. Fix a power ln of l such that η divides ln in A; such a power exists since η is a non zero-divisor by Corollary B.2.2. By the existence of the cohesive Flach system and the irreducibility of T A k, we have that ⊗ 1 1 1 n H (F,T ∗) = H F,T ∗[η] = H F,T ∗[l ] . f f f

For any Zl-module M we will write M ∨ for its Pontrjagin dual HomZl (M, Ql/Zl); n n when M is l -torsion this can be identified with HomZl (M, Z/l Z). We recall the definition of the map Ξ in seven steps. We can and will work at finite levels since everything here is ln-torsion. (1) Let 1 ξ1 :ΩA H (F, T/ηT ) → f be the A-linear map induced by Θ and the universal property of ΩA. We v have ξ1(∂Tv) = c . (2) We have a Bockstein pairing

1 1 , η : H (F, T/ηT ) Z H F,T ∗[η] Ql/Zl {· ·} f ⊗ l f → as in Section II.4. For the Flach classes cv H1(F, T/ηT ), we computed ∈ f in (III.4.3) that this pairing is given by

v 1 v c , κ η = η cv,s, κv . { } D Ev 1 Here κ H (F,T ∗[η]) and ∈ f 1 1 , : H (Fv, T/ηT ) Z H Fv,T ∗[η] Ql/Zl h· ·iv s ⊗ l f → is the restriction of the usual Tate pairing, which we will consider in more detail in the next section. Note that in the Tate pairing we can compute on the level of ln-torsion rather than η-torsion. The Bockstein pairing thus yields a map

1 1 n ξ2 : H (F, T/ηT ) H F,T ∗[l ] ∨. f → f (3) The pairing of Lemma IV.2.2 yields an isomorphism

0 n n ξ3 : End (H/l H) ' T ∗[l ]. A −→ 43 44 5. FLACH SYSTEMS OF EICHLER-SHIMURA TYPE

(4) Let ρR : GF GL2(R) → be a representative of the universal minimally ramified deformation of H A k. Recall that our hypotheses on H imply that there is a unique ⊗ map π : R A such that π ρR yields the deformation H. π satisfies → ◦ π(Tˆv) = Tv for all v. We now use the isomorphism of (IV.2.1): n 1 0 n ξ4 : HomA(ΩR R A, A/l A) H F, End (H/l H) ; ⊗ → f A here as always ΩR = ΩR/W (k). We recall (with some additional motiva- n tion) how ξ4 is defined: Let an A-linear map ω :ΩR R A A/l A be ⊗ → given. Let I be the kernel of the diagonal map ∆ : R W (k) R R. There 2 ⊗ → is a well-known isomorphism ΩR ∼= I/I , under which ∂r corresponds to the residue class of r 1 1 r. ω thus defines an A-linear map ⊗ − ⊗ n ν1 : I A/l A → by sending r 1 1 r to ω(∂r 1). The exact⊗ sequence− ⊗ ⊗ 1 I R R R 1 → → ⊗W (k) → → splits as an exact sequence of R-algebras (where we let R act on the right factor in R W (k) R) via the map sending r R to 1 r. This yields an isomorphism⊗ ∈ ⊗ (1.1) R R = R I ⊗W (k) ∼ ⊕ of R-algebras, where r s corresponds to (rs, r s 1 rs). Restricting (1.1) to the left factor⊗ of R we obtain a map ⊗ − ⊗

ν2 : R R I → ⊕ sending r to (r, r 1 1 r). ⊗ − ⊗ Now let A0 be the ring n n 2 A0 = A A/l A = A[]/(l , ). ⊕ We deﬁne a map π0 : R A0 → by composing ν2 with π ν1. Thus ⊕ π0(r) = π(r) + ω(∂r 1). ⊗ This in turn induces a representation

ρ0 : GF GL2(A0) → by ρ0 = π0 ρR; here ρR is a representative of the universal minimally ramified deformation◦ as above. We now define a cocycle 0 n κ0 : GF End (H/l H) ω → A 1 1 σ (ρ− (σ)ρ0(σ) 1) 7→ A − where ρA = π ρR. We take κω0 to be the image of ω under ξ4. The discussion in Section◦ IV.2 shows that this map is independent of the choice of ρR and respects the finite/singular structure. 2. THE TATE PAIRING 45

In terms of our computations above, we ﬁnd that if we write

aˆ ˆb ρR(σ) = cˆ dˆ

(so thata, ˆ ˆb, c,ˆ dˆ are functions of σ) then we have

π(â) + ω(∂aˆ) π(ˆb) + ω(∂ˆb) ρ0(σ) = . π(ˆc) + ω(∂cˆ) π(dˆ) + ω(∂dˆ) From here one computes that ˆ ˆ ˆ ˆ ˆ ˆ 1 π(d)ω(∂aˆ) π(b)ω(∂cˆ) π(d)ω(∂b) π(b)ω(∂d) κ0 (σ) = − − . ω π(âdˆ ˆbcˆ) π(â)ω(∂cˆ) π(ˆc)ω(∂aˆ) π(â)ω(∂dˆ) π(ˆc)ω(∂ˆb) − − − (5) The Gorenstein trace tr induces an isomorphism

n n ξ5 : HomA(ΩR R A, A/l A) HomZ (ΩR R A, Z/l Z) ⊗ → l ⊗ sending ω to tr ω; see Lemma B.3.2. (6) There is a double◦ duality isomorphism

ξ6 :ΩR R A = (ΩR R A)∨ ∨ ⊗ ∼ ⊗ sending z ΩR R A to the evaluation at z map on (ΩR R A)∨. (7) There is a∈ natural⊗ map ⊗

ξ7 :ΩR R A ΩA ⊗ → sending dr a to a∂π(r). ⊗ The map

Ξ:ΩA ΩA → is deﬁned to be the composition

ξ1 1 ξ2 ΩA H (F, T/ηT ) −−−−→ f −−−−→ 1 n H(ξ3)∨ 1 0 n ξ4∨ H (F,T ∗[l ])∨ H F, End (H/l H) ∨ f −−−−−→ f A −−−−→ (ξ 1) ξ 1 n 5− ∨ n 6− HomA(ΩR R A, A/l A)∨ HomZ (ΩR R A, Z/l Z)∨ ⊗ −−−−→ l ⊗ −−−−→ ξ7 ΩR R A ΩA ⊗ −−−−→ Note that the cohesive Flach system enters only into the very ﬁrst map ξ1; the remaining maps are at most dependent on the choice of Gorenstein trace tr, although one checks easily that the composite does not depend on that choice.

2. The Tate pairing In order to explicitly compute the map Ξ we will need to work with the Tate pairing. Let M be a ﬁnite GFv -module of exponent m and let M ∗ = HomZ(M, µm) be its Cartier dual. Recall that the Tate pairing is the map

1 1 H (Fv,M) H (Fv,M ∗) Q/Z ⊗ → 46 5. FLACH SYSTEMS OF EICHLER-SHIMURA TYPE deﬁned as the composition of

1 1 cup 2 Cartier H (Fv,M) H (Fv,M ∗) H (Fv,M Z M ∗) ⊗ −−−−→ ⊗ −−−−→ 2 2 val H (Fv, µm) ' H (L/Fv,L×) −−−−→ −−−−→ 2 δ 1 eval H (L/Fv, Z) H (L/Fv, Q/Z) Q/Z. ←−−−− −−−−→ Here L is the unique unramiﬁed extension of Fv of degree m. We recall in more detail the various maps involved. For yet more details, see [Ser79, Chapter 13, Section 3] or [Mil86, Chapter 1, Sections 1-2]. (1) The ﬁrst map is simply cup product. One computes from the explicit

formulas given in [AW67] that if f : GFv M and g : GFv M ∗ are → 2 → cocycles, then a cocycle representing f g H (Fv,M Z M ∗) is given by ∪ ∈ ⊗

σ1 (2.1) (σ1, σ2) f(σ1) g(σ2). 7→ ⊗ (2) The next map is induced by Cartier duality between M and M ∗. Com- bined with the first map, the image of the pair f, g is the cocycle 1 (σ1, σ2) σ1g(σ2) σ− f(σ1) µm 7→ 1 ∈ by definition of the adjoint action. (3) The third map is the inverse of the isomorphism on m-torsion induced by the isomorphism 2 ur ur 2 2 H (Fv /Fv,Fv ×) H (Fv,Fv,×s) H (Fv, µ ) → ← ∞ where the first map is inflation and the second is induced by the inclusion µ , Fv,×s. (See [Ser79, Chapter 13, Section 3].) This map seems to be quite∞ → difficult to describe explicitly; in our computation we will get lucky. (4) The next map is simply the map induced by the valuation map on L×; this has the obvious interpretation on cocycles. (5) The next map is the connecting homomorphism in the long exact sequence of Gal(L/Fv)-cohomology coming from the short exact sequence 0 Z Q Q/Z 0. → → → → It is an isomorphism since Q is divisible. One computes from the construction of the connecting homomorphism that if f : Gal(L/Fv) Q/Z is a 1-cocycle, then δ(f) is the 2-cocycle given by → ˜ ˜ ˜ δ(f)(σ1, σ2) = f(σ1σ2) f(σ1) f(σ2) Z − − ∈ ˜ where f : Gal(L/Fv) Q is any lifting of f. (Here we have used the fact → that Gal(L/Fv) acts trivially on Q.) (6) The last map is evaluation at Fr(v) Gal(L/Fv); this makes sense as Q/Z 1 ∈ has trivial action, so that H (L/Fv, Q/Z) ∼= Hom(Gal(L/Fv), Q/Z). 2 In our computation we will follow the Tate pairing as far as H (L/Fv,L×). Let e us compute the images in this group of m Q/Z in order to have something to e ∈ compare with. m corresponds under eval to the homomorphism fe : Gal(L/Fv) i ei → Q/Z given by fe(Fr(v) ) = m . Let : Q/Z Q be the map sending x Q/Z to {·} → ˜ ∈ the uniquex ˜ Q such that 0 x˜ < 1 and x x˜ (mod Z). Let fe : Gal(L/Fv) Q ∈ ˜≤ i ei≡ → be the lifting of fe given by fe(Fr(v) ) = . { m } 3. A SPECIAL CASE 47

Under the map δ we now obtain the 2-cocycle

˜ i j (i+j)e ie je δ(fe)(Fr(v) , Fr(v) ) = n m o − m − m 2 in H (L/Fv, Z). Let λ0 be our ﬁxed uniformizer of Fv. Using that Gal(L/Fv) acts trivially on λ0 and that λ0 is a uniformizer in L, we see that the cocycle 2 Ce H (L/Fv,L×) given by ∈ (i+j)e ie je i j m m m (2.2) Ce(Fr(v) , Fr(v) ) = λ0{ }−{ }−{ } ˜ e 1 maps to δ(fe) under v. Thus Ce corresponds to m Q/Z under eval δ− ; this is the cocycle we will use for comparison later. ∈ ◦ In our computation, we will actually consider the induced pairing

1 1 H (Fv,M) H (Fv,M ∗) µm. s ⊗ f → 1 To compute with this pairing, one must first lift the cocycle in Hs (Fv,M) to a 1 cocycle in H (Fv,M); the rest of the definition is the same. The fact that the other cocycle is finite implies that the pairing is independent of the choice of lifting.

3. A special case

3.1. Additional hypotheses. In this section we compute Ξ(∂Tv) with some additional simplifying hypotheses; this computation will still contain most of the content of the general case, but it is significantly simpler algebraically. We make two assumptions. First, assume that the action of Fr(v) on H is diagonal with respect to a fixed basis x, y; that is, Fr(v) acts on H by a matrix u 0 0 v with u, v A. In particular, uv = χ(v) and u + v = Tv. (We apologize for the use of v in two∈ completely different ways; we hope that it will cause no confusion.) It follows from the definition of a cohesive Flach system of Eichler-Shimura type that v in this case that the cocycle cv,s is given by cv : Gal(F ur(λ)/F ur) T v,s v v → j 1 0 (3.1) τ wjη(u v) 0 1 ζn 7→ − − ⊗ n with the notation of Chapter IV.7; ζn is the primitive l -root of unity induced by ζ. The second simplifying assumption is that the map π : R A is an isomor- → phism; that is, A is the universal minimally ramified deformation ring of H A k and H is the universal deformation. Of course, we will identify R with A via⊗π.

3.2. Preliminaries. To compute Ξ(Tv), we begin by computing the image of n ∂Tv in HomA(ΩA, A/l A)∨. (Recall that R = A, so that ΩR R A = ΩA.) So let ⊗ n ω :ΩA A/l A → be a ﬁxed map; we will compute its image in Z/lnZ under

n (3.2) ξ∨ H(ξ3)∨ ξ2 ξ1(∂Tv) HomA(ΩA, A/l A)∨. 4 ◦ ◦ ◦ ∈ 48 5. FLACH SYSTEMS OF EICHLER-SHIMURA TYPE

3.3. ξ4. Using the deﬁnition of ξ4, the image of ω under (3.2) is the same as the image under

1 0 n (3.3) H(ξ3)∨ ξ2 ξ1(∂Tv) H (F, End (H/l H))∨ ◦ ◦ ∈ f A of the cohomology class represented by the cocycle

0 n κ0 : GF End (H/l H) → A given by dω(∂a) bω(∂c) dω(∂b) bω(∂d) κ (σ) = 1 . 0 ad bc aω(∂c) −cω(∂a) aω(∂d)− cω(∂b) − − − Here σ GF and ∈ a b ρA(σ) = GL2(A). c d ∈

3.4. ξ3. Using the deﬁnition of ξ3, the image of κω under (3.3) is the same as the image under 1 n ξ2 ξ1(∂Tv) H (F,T ∗[l ])∨ ◦ ∈ f of the cohomology class represented by the cocycle

n 0 n κ : GF T ∗[l ] = HomZ End (H/l H)(1), µln → l A 0 n n = HomZ End (H/l H), Z/l Z l A n 0 n where T ∗[l ] is identiﬁed with EndA(H/l H) via Lemma IV.2.2. Explicitly, we ﬁnd that

α β (3.4) κ(σ) γ α − dω(∂a) bω(∂c) dω(∂b) bω(∂d) = tr trace 1 − − ad bc aω(∂c) cω(∂a) aω(∂d) cω(∂b) · − − − α β γ α − = tr 1 αdω(∂a) αbω(∂c) + γdω(∂b) γbω(∂d)+ ad bc − − − βaω(∂c) βcω(∂a) αaω(∂d) + αcω(∂b) . − −

(Keep in mind that we have both a Gorenstein trace tr : A Zl and the usual trace of linear algebra.) →

v 3.5. ξ2. Using the deﬁnition of ξ2 and its explicit expression for c = ξ1(∂Tv), we ﬁnd that the desired element of Z/lnZ is the value of the Tate pairing

1 v η cv,s, κv ; D Ev

1 n here we are identifying the image ln Z/Z with Z/l Z. It remains to compute this. 3. A SPECIAL CASE 49

3.6. The Tate pairing : preliminaries. To begin with, note that κv factors ur through Gal(Fv /Fv), as it is unramified at v. It follows that we are only interested in κ(Fr(v)i). Using the fact that ui 0 ρ (Fr(v)i) = A 0 vi which has determinant χ(v)i, we find that (3.4) simplifies to

i α β i i i i i (3.5) κ(Fr(v) ) γ α = tr χ(v)− α v ω(∂u ) u ω(∂v ) . − − Next, note that i i i i 1 i 1 u ∂v = u v − i∂v = χ(v) − ui∂v and similarly i i i 1 v ∂u = χ(v) − vi∂u. We therefore can write (3.5) as

i α β 1 κ(Fr(v) ) γ α = tr iχ(v)− α (vω(∂u) uω(∂v)) . − − n Setting K = Fv(H/l H) (so that K/Fv is unramified), we see that κ factors through Gal(K/Fv). For cv, we computed in (3.1) that cv : Gal(F ur(λ)/F ur) T/lnT v,s v v → j 1 0 τ wjη(u v) 0 1 ζn 7→ − − ⊗ ln v ur ur Since τ goes to 0 under this map, c factors through Gal(Fv (λn)/Fv ); here by n λn we mean the l -th root of λ0 determined by our earlier choice of λ. In order to compute the Tate pairing of κ and cv we first must descend cv to a cocycle over Fv. We can do this over the field K(λn) as follows: let G = Gal(K(λn)/Fv). Denote by ϕ the element of G which acts as Frobenius on K and fixes λn, and denote by τ the element of G which is the identity on K and sends λn to ζnλn. Then ϕ and τ generate G with the relations

n ϕ[K:Fv ] = τ l = 1, τϕ = ϕτ ε(v).

ε(v) n 1 v Note that τ makes sense as τ is l -torsion. We can represent η cv,s by the map θv : G T/lnT → i j i 1 0 ϕ τ wε(v) j(u v) 0 1 ζn. 7→ − − ⊗ v v One easily checks that this really is a cocycle and that ηθ restricts to cv,s in 1 n Hs (K(λn)/Fv, T/l T ). Via inﬂation we can represent κv by the cocycle n κv : G T ∗[l ] → given by

i j α β 1 (3.6) κv(ϕ τ ) γ α = tr iχ(v)− α(vω(∂u) uω(∂v)) . − − 50 5. FLACH SYSTEMS OF EICHLER-SHIMURA TYPE

3.7. The Tate pairing : cup product. We can now compute the Tate pairing. The ﬁrst step is to form the cup product

v 2 n n θ κv H G, T/l T Z T ∗[l ] . ∪ ∈ ⊗ l

v i j i0 j0 By (2.1), we see that θ κv sends the pair (ϕ τ , ϕ τ ) G G to ∪ ∈ × i j v i j ϕ τ i0 j0 n n θ (ϕ τ ) κ (ϕ τ ) T/l T Z T ∗[l ]. ⊗ v ∈ ⊗ l 2 Under Cartier duality this maps to the cocycle C H (G, µln ) given by ∈ i j C(ϕiτ j, ϕi0 τ j0 ) = κϕ τ (ϕi0 τ j0 ) θv(ϕiτ j) v i j ϕ τ i0 j0 i 1 0 = κv (ϕ τ ) wε(v) j(u v) 0 1 ζn. − −

i0 j0 Recall that κv(ϕ τ ) is a map

End0 (H/lnH) Z/lnZ. A → In particular, ϕiτ j acts trivially on both the domain and the range. Thus by the deﬁnition of the adjoint Galois action we ﬁnd that

i j i0 j0 1 i C(ϕ τ , ϕ τ ) = tr i0χ(v)− wε(v) j(u v)(vω(∂u) uω(∂v)) ζn − −

If we let C0 : G G µln be the cocycle × → i j i0 j0 i C0(ϕ τ , ϕ τ ) = ε(v) i0jζn, then we conclude that ω maps to

1 (3.7) tr ω(wχ(v)− (v∂u u∂v)) I − where I is the image of C0 under the invariant map

2 n H (K(λn)/Fv, µln ) Z/l Z. →

3.8. ξ5 and ξ6. At this point, thankfully, we get the maps ξ5 and ξ6 for free. Speciﬁcally, suppose that we began with

n ω0 :ΩA Z/l Z → n 1 and wished to compute its image in Z/l Z under (ξ− )∨ ξ1(∂Tv). By the deﬁ- 5 ··· nition of ξ5 this would be the same as the image under ξ4∨ ξ1(∂Tv) of the unique n ··· ω :ΩA A/l A such that tr ω = ω0. But by (3.7) this is visibly just → ◦ 1 (3.8) ω0 wχ(v)− (u v)(v∂u u∂v) I. − − n Similarly, in HomZl (ΩA, Z/l Z)∨ (3.8) is just the evaluation at

1 (3.9) wχ(v)− (u v)(v∂u u∂v)I − − map, so that (3.9) is the ﬁnal image of ∂Tv in ΩA. It remains, then, to compute I and to simplify our expression. 3. A SPECIAL CASE 51

3.9. Computation of I. We begin by computing I; it is the image of C0 under the maps

2 2 n H (K(λn)/Fv, µln ) H (L/Fv,L×) Z/l Z, → → n where L is the unique unramiﬁed extension of Fv of degree l . We ﬁrst need to modify C0 by a coboundary to get it to factor through Gal(L/Fv) and to take values in L×. We can do this using the cochain f : G K(λn)× given by → i j i f(ϕ τ ) = λnh i where i is the unique integer in 0, 1, . . . , ln 1 which is congruent to i modulo ln. Theh i coboundary formula states{ that − }

i j i0 j0 i j i0 j0 i j i0 j0 C0(ϕ τ , ϕ τ )f(ϕ τ ϕ τ ) C0∂f(ϕ τ , ϕ τ ) = i j ϕ τ f(ϕi0 τ j0 )f(ϕiτ j)

i j i0 j0 i+i0 j00 One computes easily that ϕ τ ϕ τ = ϕ τ for some j00, so that we can compute this as

i ε(v) i0j i+i0 h i i j i0 j0 ζn λn C0∂f(ϕ τ , ϕ τ ) = i j i0 i ϕ τ (λnh i)λnh i i ε(v) i0j i+i0 ζ λh i = n n i j i0 i0 i ϕ (ζnh iλnh i)λnh i i ε(v) i0j i+i0 ζn λnh i = i ε(v) j i0 i0 + i ζn h iλnh i h i

i+i0 i i0 = λhn i−h i−h i.

2 This, however, is simply the inﬂation to H (K(λn)/Fv,K(λn)×) of the cocycle 2 C1 H (L/Fv,L×) of (2.2). Since C1 was deﬁned to map to 1 under the invariant ∈ map, we see that C0 does as well. Thus I = 1.

3.10. Diﬀerentials and Hecke operators. We conclude by (3.9) that

1 Ξ(∂Tv) = wχ(v)− (u v)(v∂u u∂v) ΩA. − − ∈ It remains to simplify this expression. Using that uv = χ(v) we ﬁnd that

(u v)(v∂u u∂v) = u χ(v) χ(v) ∂u u∂ χ(v) − − − u u − u = u χ(v) χ(v) ∂u + χ(v) ∂u − u u u = u χ(v) 2χ(v) ∂u − u u = 2χ(v) ∂u χ(v) ∂u − u2 = 2χ(v)(∂u + ∂v) . Thus we conclude that

Ξ(∂Tv) = 2w∂(u + v) = 2w∂Tv. This completes the proof of Theorem IV.6.2 in this case. 52 5. FLACH SYSTEMS OF EICHLER-SHIMURA TYPE

4. A matrix computation The key to removing both of the assumptions of the previous computation is the following matrix lemma. Lemma 4.1. Let R be a ring, S an R-algebra and M an S-module. Let ∂ : S M be an R-linear derivation. Let → a b T = GL2(S) c d ∈ be a matrix with determinant δ = ad bc and trace t = a + d. Assume that δ lies in the image of R in S. Let e be a positive− integer, and write

e AB T = GL2(S). CD ∈ Then

(4.1) 2eδe∂t = (2bC aD + dD)∂A + ( 2cD aC + dC)∂B+ − − − − ( 2bA + aB dB)∂C + (2cB + aA dA)∂D − − − Proof. We prove (4.1) by induction on e, with the case e = 0 being trivial. Suppose then that we know (4.1) for e. We have AB a b aA + cB bA + dB (4.2) T e+1 = T eT = = . CD c d aC + cD bC + dD Let Z be the value of the expression on the right in the statement of the lemma for e + 1. After some simpliﬁcation, one ﬁnds from (4.2) that

Z = tbC + ( 2δ + td)D ∂(aA + cB) + tcD + (2δ ta)C ∂(bA + dB)+ − − − tbA + (2δ td)B ∂(aC + cD) + tcB + ( 2δ + ta)A ∂(bC + cD). − − − Expanding this out with the product rule one obtains

(4.3) Z = δtD + 2δ(bC aD) ∂A + δtC + 2δ(dC cD) ∂B+ − − − δtB + 2δ(aB bA) ∂C + δtA + 2δ(cB dA) ∂D+ − − − δ0 (td 2δ)∂a tc∂b tb∂c + (ta 2δ)∂d − − − − e where δ0 = AD BC = δ . Now, since ad− bc = δ R and ∂ is R-linear, we have − ∈ a∂d + d∂a b∂c c∂b = 0. − − Similarly, (4.4) A∂D + D∂A B∂C C∂B = 0. − − Using these relations (4.3) simpliﬁes to

(4.5) Z = δ (2bC 2aD)∂A + (2dC 2cD)∂B+ − − (2aB 2bA)∂C + (2cB 2dA)∂D 2δ0δ(∂a + ∂d). − − − Multiplying (4.4) by tδ yields δ (aD + dD)∂A + ( aC dC)∂B + ( aB dB)∂C + (aA + dA)∂D = 0. − − − − 5. COMPUTATION OF Ξ IN THE NON-DIAGONAL CASE 53

Adding this to (4.5), we ﬁnd that

Z = δ (2bC aD + dD)∂A + ( 2cD aC + dC)∂B+ − − − ( 2bA + aB dB)∂C + (2cB + aA dA)∂D 2δ0δ(∂a + ∂d). − − − − The induction hypothesis shows that this is just e e+1 Z = δ( 2eδ ∂t) 2δδ0(∂t) = 2(e + 1)δ ∂t. − − − This completes the induction.

5. Computation of Ξ in the non-diagonal case

We now explain how to compute Ξ(∂Tv) when Fr(v) is not necessarily diagonal. We continue to assume that π is an isomorphism. This computation is fundamentally the same as the previous special case, just a bit messier and with the simple computation of Section 3.10 replaced by the more elaborate computation of Lemma 4.1. The complication is that Fr(v) no longer acts diagonally. Write a b ρ (Fr(v)) = A c d

i ai bi ρA(Fr(v) ) = ci di for some fixed basis x, y of H. Note that a+d = Tv, ad bc = χ(v) and aidi bici = χ(v)i. The formula for the cocycle κ is exactly as computed− in (3.4), replacing− i a, b, c, d with ai, bi, ci, di for σ = Fr(v) . The Flach class is now cv : Gal(F ur(λ)/F ur) T v,s v v → j a d 2b τ wjη 2−c d a ζn. 7→ − ⊗ 1 v 1 v In order to compute the Tate pairing of η cv,s and κv we first must lift η cv,s to 1 n H (Fv, T/l T ). In fact, the same lifting v n θ : Gal(K(λn)/Fv) T/l T → i j i a d 2b ϕ τ wε(v) j 2−c d a ζn 7→ − ⊗ still works. v We now compute the cup product of κv and θ as cohomology classes for G. v Writing C = θ κv, one finds that ∪ C(ϕiτ j, ϕi0 τ j0 ) =

i i0 tr wε(v) jχ(v)− ( 2bci + adi ddi )ω(∂ai ) + (2cdi + aci dci )ω(∂bi )+ − 0 0 − 0 0 0 0 − 0 0

(2bai abi + dbi )ω(∂ci ) + ( 2cbi aai + dai )ω(∂di ) ζn. 0 − 0 0 0 − 0 − 0 0 0 Applying Lemma 4.1, we ﬁnd that this is simply

i j i0 j0 i C(ϕ τ , ϕ τ ) = 2wε(v) i0j tr(ω(∂Tv))ζn and from here the computation is identical to the earlier case; we conclude that

Ξ(∂Tv) = 2w∂Tv. 54 5. FLACH SYSTEMS OF EICHLER-SHIMURA TYPE

6. Computation of Ξ in the general case We now remove the assumption that π is an isomorphism. The computation in this case is essentially the same as in the previous case. First, recall that universality of R means that, fixing a universal deformation ρR, there is some basis of H with respect to which ρA = πρR. We can conjugate in GL2(A) from this basis to our fixed basis x, y; since π is surjective (and R is local) we can lift this conjugation to GL2(R). That is, we can conjugate ρR so as to assume that ρA = πρR where ρA is now the representation on our fixed basis x, y of H. To compute Ξ this time, we begin with n ω :ΩR R A A/l A ⊗ → and compute its image in Z/lnZ. Proceeding as before, we find that this is the n v n image under the Tate pairing of two cocycles κv : G T ∗[l ] and θ : G T/l T . Writing → → ˆ i aî bi ρR(Fr(v) ) = ˆ cî di and i ai bi ρA(Fr(v) ) = ci di we find that

i j α β κ(ϕ τ ) γ α − i ˆ ˆ = tr χ(v)− αdiω(∂aî) αbiω(∂cî) + γdiω(∂bi) γbiω(∂di)+ − − ˆ ˆ βaiω(∂cî) βciω(∂cî) αaiω(∂di) + αciω(∂bi) . − − The cocycle θv is given by v i j i a d 2b θ (ϕ τ ) = wε(v) j 2−c d a ζn − ⊗ where a b ρ (Fr(v)) = A c d as before. From these expressions the computation works out exactly as in the previous ˆ ˆ case, with ∂ai, ∂bi, ∂ci, ∂di replaced by ∂aî, ∂bi, ∂cî, ∂di respectively. Lemma 4.1 applies to show that

i j i0 j0 i ˆ C(ϕ τ , ϕ τ ) = 2wε(v) i0j tr(ω(∂aˆ + ∂d))ζn, where aˆ ˆb ρR(Fr(v)) = . cˆ dˆ

From here the computation is as before, with the fact that ξ7(∂aˆ) = ∂a and ˆ ξ7(∂d) = ∂d showing that Ξ is still multiplication by 2w. This completes the proof of Theorem IV.6.2. Part 2

Construction of cohesive Flach systems

CHAPTER 6

The Flach map

In this chapter we deﬁne the Flach map from algebraic K-theory to Galois cohomology; this map will be used in Chapter IX to generate geometric Euler systems.

1. The coniveau spectral sequence in étalecohomology We begin with the background material in algebraic geometry and algebraic K-theory which will be required for the remainder of this thesis. In particular, we will work in more generality then is actually needed for the definition of the Flach map. We begin with a spectral sequence in étalecohomology which will be used in the construction of the Flach map. The main reference for this construction is [Gro68, Section 10.1]; see also [CTHK97, Section 1], [Gil81, pp. 239–242] and [Fla95, Section 5.1]. Let X be a scheme of finite Krull dimension and let be a torsion étalesheaf on X. Let Y be a closed subscheme of X. For all i, p, defineF

Hi (X, )p = lim Hi (X, ) Y F Z F Z−→Y codim⊆X Z p ≥ i p/p+1 i (1.1) HY (X, ) = lim HZ Z (X Z0, ). F − 0 − F Z0−→Z Y codim⊆X⊆Z p ≥ codimX Z0 p+1 ≥

For each pair Z0 Z as in (1.1) we have the usual exact sequence ⊆ i i i i+1 HZ (X, ) HZ (X, ) HZ Z (X Z0, ) HZ (X, ) · · · → 0 F → F → − 0 − F → 0 F → · · · Taking the direct limit over all such pairs (for ﬁxed p) we obtain a long exact sequence (1.2) Hi (X, )p+1 Hi (X, )p Hi (X, )p/p+1 Hi+1(X, )p+1 · · · → Y F → Y F → Y F → Y F → · · · Set

p+q p D = HY (X, ) p,q⊕ F p+q p/p+1 E = HY (X, ) . p,q⊕ F

57 58 6. THE FLACH MAP

(1.2) yields an exact couple (in which we have labeled the maps by their (p, q)- bidegrees)

( 1,1) (1.3) − / D ` D @@ ~ @@ ~~ (1,0) @@ ~~(0,0) @ ~~ E This in turn yields the coniveau spectral sequence (see [Wei94, Section 5.9] or Section A.8) pq p+q p/p+1 p+q (1.4) E1,Y (X, ) = HY (X, ) HY (X, ). F p,q⊕ F ⇒ F n n p Here HY (X, ) appears as the direct limit of HY (X, ) as p goes to ; in fact, it already appearsF at the p = 0 term. F −∞ We can compute the E1-term of (1.4) a bit further. If Z1 Z2 = , then ∩ ∅ (1.5) Hi (X, ) Hi (X, ) Hi (X, ), Z1 Z2 = Z1 Z2 ∪ F ∼ F ⊕ F as one sees easily from the deﬁnition of cohomology with support and excision. For the case where Z1 and Z2 are not necessarily disjoint closed subschemes of X, we can rewrite (1.5) as

(1.6) Hi (X Z Z , ) Z1 Z2 Z1 Z2 1 2 = ∪ − ∩ − ∩ F ∼ Hi (X Z Z , ) Hi (X Z Z , ). Z1 Z1 Z2 1 2 Z2 Z1 Z2 1 2 − ∩ − ∩ F ⊕ − ∩ − ∩ F If Z1 and Z2 are also distinct, irreducible and of codimension p, then Z1 Z2 has ∩ codimension at least p + 1 since Z1 and Z2 each have a unique generic point. For p+q an arbitrary Z, splitting up each H (X Z0, ) into the pieces corresponding Z Z0 to the irreducible components of Z and− using− (1.6)F we ﬁnd that

i p/p+1 def i HY (X, ) = lim HZ Z (X Z0, ) F − 0 − F Z0−→Z Y codim⊆X⊆Z p ≥ codimX Z0 p+1 ≥ i (1.7) = lim Hx¯ Z (X Z0, ). ∼ p 0 x X⊕ Y ( − − F ∈ ∩ Z−→0 x¯ Here byx ¯ we mean the closure of x X regarded as a reduced closed subscheme of X. { } ⊆ p i For each x X we take this last expression as the deﬁnition of Hx(X, ); that is, ∈ F

i def i (1.8) Hx(X, ) = lim Hx¯ Z (X Z, ). F − − F −→Z(x¯ These groups are easily seen to be contravariant for flat morphisms in the following p sense: if f : X0 X is a flat morphism, then for each x X there is a map → ∈ i i H (X, ) H (X0, f ∗ ); x xi0 F → ⊕xi0 F 1 here the xi0 are the generic points of the irreducible components of f − (¯x) of codimension p; such points will exist by [GDb, Corollary 6.1.4]. The maps to the i 1 H (X0, f ∗ ) for x0 the generic point of an irreducible component of f − (¯x) of x0 F 2. THE LOCALIZATION SEQUENCE 59 codimension greater than p will all be zero, as the closed set Z in (1.8) can be 1 taken so that f − (Z) contains x0. The cohomology groups (1.8) are also covariant for finite flat morphisms: if p f : X0 X is a finite flat morphism, then for each x0 X0 there is a map → ∈ i i Hx (X0, f ∗ ) H (X, ) 0 F → f(x0) F induced by the trace map in étalecohomology; see [FK88, pp. 133-135] for the definition of the trace map. Note that codim f(x0) = p since f is finite flat. Summing up, we have the following proposition. Proposition 1.1. Let X be a scheme of finite Krull dimension, let Y be a closed subscheme of X and let be a torsion sheaf on X. Then there is a spectral sequence F pq p+q p+q (1.9) E1,Y (X, ) = Hx (X, ) HY (X, ). F x X⊕p Y F ⇒ F ∈ ∩ If f : X0 X is a flat morphism and Y 0 is a closed subscheme of X0 containing 1 → f − (Y ), then there is an induced morphism of spectral sequences pq pq E (X, ) E (X0, f ∗ ) r,Y F → r,Y 0 F and the map on E1-terms is the same as that coming from the contravariant functoriality of Hx∗. Furthermore, if for some p, q and r there are edge maps (see Section A.2) Epq (X, ) Hp+q(X, ) r,Y F → Y F pq p+q E (X0, f ∗ ) H (X0, f ∗ ), r,Y 0 F → Y 0 F then the diagram pq pq E (X, ) / E (X0, f ∗ ) r,Y F r,Y 0 F

p+q p+q H (X, ) / H (X0, f ∗ ) Y F Y 0 F commutes. The corresponding statements remain true for covariant functoriality with respect to finite flat morphisms, with the obvious modifications. Proof. The existence of the spectral sequence and the expression (1.9) were proven above. To see the functoriality, note first that f induces maps of the long exact localization sequences, and thus of the exact couples (1.3) for the pairs X,Y and X0,Y 0. (Note that flatness is needed here to insure that the relevant codimensions are compatible.) This yields the map of spectral sequences (see Section A.8), and the fact that for r = 1 this map is the same as that coming from the maps on i the Hx(X, )’s is immediate from the functoriality of the isomorphisms (1.7) The compatibilityF of the edge maps is proven in Proposition A.8.1. The same arguments work for the covariant case. 2. The localization sequence We continue with the notation of the previous section. Write U for the open pq subscheme X Y of X. We will write Er (X, ) for the spectral sequence previously pq − F denoted Er,X (X, ). Contravariant functoriality for flat morphisms and covariant functoriality for finiteF flat both yield natural maps Epq (X, ) Epq(X, ) r,Y F → r F 60 6. THE FLACH MAP arising from the identity map on X; these maps coincide. These maps are also compatible with either of the expressions (1.4) and (1.9). By contravariance for flat morphisms we also have a map Epq(X, ) Epq(U, ). r F → r F Lemma 2.1. Assume that U is dense in X. For each q, there is a short exact sequence of complexes

q q q (2.1) 0 E• (X, ) E• (X, ) E• (U, ) 0. → 1,Y F → 1 F → 1 F → This induces a long exact sequence

(2.2) Epq (X, ) Epq(X, ) Epq(U, ) Ep+1,q(X, ) · · · → 2,Y F → 2 F → 2 F → 2,Y F → · · · Furthermore, suppose that for some p, q there exist edge maps forming a square with the boundary maps of (2.2):

Epq(U, ) / Ep+1,q(X, ) 2 F 2,Y F

Hp+q(U, ) / Hp+q+1(X, ) F Y F Then this square commutes.

Proof. We have already seen the existence of the maps of spectral sequences in (2.1). Note that the differentials at this stage are vertical, so that for fixed q q we can consider E1• as a complex. The fact that the maps are maps of spectral sequences immediately implies that (2.1) is a maps of complexes. To construct (2.2) we therefore must show only that (2.1) is exact as a sequence of abelian groups. Since U is dense in X, U p = Xp U. The map ∩ Epq(X, ) Epq(U, ) 1 F → 1 F is the direct sum of the maps (2.3) Hp+q(X, ) Hp+q(U, ) x F → x F p p+q over all x U . In particular, for x / U the terms Hx (X, ) map to zero pq ∈ pq ∈ F pq in E1 (U, ); thus by (1.9) E1,Y (X, ) is in the kernel of the map E1 (X, ) pq F F F → E1 (U, ). FixF now one x U p. Plugging back into the definitions, in (2.3) we are considering the direct limit∈ over Z ( x¯ of the natural maps

p+q p+q (2.4) Hx¯ Z (X Z, ) Hx¯ U Z U (U Z U, ). − − F → ∩ − ∩ − ∩ F For any Z containing Y x¯ the map (2.4) is an isomorphism by excision; since such Z are coﬁnal in the∩ set of all Z, the direct limit of the maps (2.4) is an pq pq isomorphism. It is now clear that the kernel of the map E1 (X, ) E1 (U, ) is precisely Epq (X, ), so that (2.1) is exact as a sequence of abelianF → groups. F 1,Y F The exact sequence (2.1) yields (2.2) in the usual way. For the compatibility of the boundary maps with the edge maps, see [Fla95, Proposition 3]. 3. GROTHENDIECK’S PURITY CONJECTURE 61

3. Grothendieck’s purity conjecture We will need Grothendieck’s purity conjecture in order to get an additional expression for our coniveau spectral sequence. We begin by briefly recalling the statement of the conjecture as we will need it; see [Gro77, Exposé1, Section 3.1.4], [CT95, Section 3.2] or [?, Section 1] for more details. For a closed immersion i : Y, X and a sheaf on X, we write i! for the sheaf of sections of Y supported on Y ;→ see [FK88, ChapterF I, Section 10]. F Conjecture 3.1. Let X be a regular scheme and let i : Y X be a closed immersion of a regular scheme Y . Assume that i has codimension→c at every point. Let be a locally constant torsion sheaf on X of exponent invertible in X . Then thereF are functorial isomorphisms O 0 j = 2c; j ! ( R i ∼= 6 F i∗ ( c) j = 2c. F − In particular, j+2c j H (X, ) = H Y, i∗ ( c) Y F ∼ F − for all j. We will need the following results on the purity conjecture. Theorem 3.2. Let X be a regular scheme and let Y be an irreducible regular closed subscheme. Then the purity conjecture is known for the inclusion Y, X in any of the following circumstances: → (1) X and Y are both smooth over a base S; (2) X is a scheme of finite type over a perfect field; (3) X is a smooth scheme over a discrete valuation ring with perfect residue field and Y is a closed subscheme of the special fiber of X; (4) X is a separated scheme of finite type over a local or global field of positive characteristic and the sheaf has exponent divisible only by primes dim X + 2. F ≥

Proof. (1) is the usual purity theorem in étalecohomology; see [GAV73, Exposé16, Section 3] or [FK88, Chapter I, Theorem 10.1]. (2) follows from this and the fact [GDb, Proposition 17.15.1] that regular and smooth are the same for schemes of finite type over a perfect field. The case of (3) where Y is the special fiber of X is [Ras89, Lemma 2.1]; the general case follows easily from the long exact sequence of a pair and (2). (4) is proved in [?, Corollary 3.7], together with the cohomological dimension calculations of [GAV73, Exposé10, Theorem 4.3 and Theorem 5.2] and [Ser97, Corollary of Section II.4.2 and Proposition 12]. If Y is a closed subscheme of X, we will say that the pair X,Y satisfies relative purity at N if for all irreducible regular closed subschemes Z of Y (of pure codimension in X) and all locally constant N-torsion sheaves on X, the purity conjecture is satisfied for the inclusion of Z into X. We will sayF that X satisfies purity at N if the pair X,X satisfies relative purity at N. Now let X be a regular scheme of finite type over a field or a discrete valuation ring. Let Y be a closed subscheme. Assume that relative purity at N holds for the pair X,Y . In this situation we can further simplify the coniveau spectral sequence. Again, the main reference is [Gro68, Section 10.1] (we should note that 62 6. THE FLACH MAP in his treatment he seems to be assuming that the base field is perfect); see also [CTHK97, Section 1]. Let x Y be of codimension p in X, let be a locally constant N-torsion sheaf on X∈and consider F i i Hx(X, ) = lim Hx¯ Z (X Z, ). F − − F −→Z(x¯ Since X is assumed to be regular, X Z is certainly regular for each Z. Furthermore, x¯ is generically regular since the local− ring of the generic point is the field k(x). Together with [GDb, Corollary 6.12.6] this implies thatx ¯ is regular on a non-empty open subscheme. It follows that the regular open sets ofx ¯ are cofinal among all open sets ofx ¯, and thus that i i Hx(X, ) = lim Hx¯ Z (X Z, ). F − − F −→Z(x¯ x¯ Z regular − Purity tells us that for any such Z we have i i 2p Hx¯ Z (X Z, ) = H − x¯ Z, ( p) . − − F ∼ − F − Thus i i 2p H (X, ) = lim H − x¯ Z, ( p) . x F ∼ − F − −→Z(x¯ X Z regular − We can further restrict the direct system to affine open sets inx ¯, as they are a base for the topology onx ¯. In this situation étalecohomology commutes with the direct limit (see [GAV73, Exposé7, Section 5.8] or [Art62, Chapter 1, III.3]), so that i i 2p H (X, ) = H − lim (¯x Z), ( p) . x F ∼ − F − But this inverse limit is simply Spec k(x←−) (using here again the fact thatx ¯ is reduced), so we conclude finally that i i 2p H (X, ) = H − Spec k(x), ( p) . x F ∼ F − We summarize this in a proposition. Proposition 3.3. Let X be a regular scheme of finite type over a field or discrete valuation ring; let Y be a closed subscheme of X such that relative purity at N holds for the pair X,Y . Then the E1-term of the coniveau spectral sequence can be written as pq q p E1,Y (X, ) = H − Spec k(x), ( p) F ∼ x X⊕p Y F − ∈ ∩ and this identification respects the functorialities on both sides. Proof. The only new statement is the last one, and this follows from the functoriality assumptions in the purity conjecture. 4. The coniveau spectral sequence in K-theory One can redo the entire construction of the previous three sections using algebraic K-theory rather than étalecohomology. This construction is carried out in [Fla95, Sections 5.1 and 5.2]; for regular schemes it is equivalent to the construction in [Qui73, Section 7, Theorem 5.4]. We will also need the third equivalent construction given in [Gil81, pp. 239-240 and pp. 271-272]. For later reference we state what we will need as a proposition. Following [Gil81, Definition 2.13], if Y 4. THE CONIVEAU SPECTRAL SEQUENCE IN K-THEORY 63 is a closed subscheme of X, we define the relative K-groups Ki,Y (X) to be the homotopy fibers of K(X) K(X Y ). If X is a scheme of finite Krull dimension, we define the codimension→p Chow− group ApX to be the cokernel of the map

k(x)× Z x X⊕p 1 →x ⊕Xp ∈ − ∈ where the map

(4.1) k(x)× Z →x ⊕x¯1 0∈ sends a rational function f to its divisor in the sense of [Ful98, Section 1.3]; note that the definition there works perfectly well for schemes with are not finite type over a field. See, for example, [?, Chapter 1]. Proposition 4.1. Let X be a regular noetherian scheme of finite Krull dimension and let Y be a closed subscheme of X. Then there is a spectral sequence

pq (4.2) E1,Y (X) = K p qk(x) K p q,Y X. x X⊕p Y − − ⇒ − − ∈ ∩ This spectral sequence is contravariant for flat morphisms, covariant for finite flat morphisms, and these functorialities are compatible with edge maps. If U = X Y , then there is a localization sequence as in Lemma 2.1. Finally, for any p the spectral− sequence differential p 1, p p, p E1,Y− − (X) / E1,Y− (X)

k(x)× Z x Xp⊕ 1 Y x X⊕p Y ∈ − ∩ ∈ ∩ p, p identifies with the direct sum of the maps (4.1). In particular, E2 − (X) identifies with the codimension p Chow group ApX. Proof. The spectral sequence (4.2) is initially constructed in [Qui73, Sec- tion7, Theorem 5.4], where he also proves contravariant functoriality for flat morphisms. Covariant functoriality is proven in the same way, using [Qui73, Section 7, (2.7) and (2.8)]. The compatibility with edge maps is proven using Flach’s construction and the methods of the previous sections of this chapter. The localization sequence arises in the same way. Finally, the differential computation is in [Qui73, Proposition 5.14 and Remark 5.17], together with [Gra77]; note that Quillen’s statement is somewhat awkward, but in the proof it is apparent that he is proving precisely the statement above. pq p We will write a general element of E (X) as (αi, fi) where αi X Y and 1,Y P ∈ ∩ fi K p qk(αi). More generally, let α be a closed subscheme of Y such that each ∈ − − irreducible component αi has codimension p in X. If f is a section of the Zariski sheaf p qα defined on a dense open set, then we write (α, f) for the element K−mi− pq (αi, f ) of E (X); here fi is the restriction of f to αi and mi is the length of P i 1,Y the local ring of αi in α. The coniveau spectral sequences in K-theory and étalecohomology are connected by Chern class maps constructed by Gillet in [Gil81]; see also [Lev98, Chapter 3]. We summarize what we will need in another proposition. 64 6. THE FLACH MAP

Proposition 4.2. Let X be a regular noetherian scheme of finite Krull dimension, let Y be a closed subscheme of X and let be the sheaf Z/NZ for some N which is invertible on X. Then for any i and jFthere is a natural transformation of functors 2i j (4.3) Ki,Y X H − X, (i) . → Y F These combine to give a map of coniveau spectral sequences (4.4) Epq (X) Ep,q+2i X, (i) r,Y → r,Y F which is functorial in X and Y (both contravariantly for flat morphisms and co- variantly for finite flat morphisms). These maps also commute with the respective localization sequences. Lastly, assuming also that X is of finite type over a field or a discrete valuation ring and that the pair X,Y satisfies relative purity at N, the map

p, p p,p (4.5) E − (X) / E X, (p) r,Y r,Y F

Z / H0(Spec k(x), ) x X⊕p Y x X⊕p Y F ∈ ∩ ∈ ∩ is just the direct sum of the canonical maps Z H0(Spec k(x), ) for the constant sheaf . → F F Proof. See [Gil81, esp. Deﬁnition 2.22 and Lemma 2.23] and [Lev98, Chap- ter 3, Section 1.4] for the construction of (4.3). The corresponding maps (4.4) are given in [Gil81, pp. 239-240] and the localization behavior is in [Lev98, Chapter 3, Section 1.5]. (4.5) follows easily from Riemann-Roch without denominators as in [Lev98, Theorem 3.4.7], using the methods outlined in [Gil81, Theorem 3.9 and Remark 3.10] together with our purity hypotheses. (Note that [Gil81, Theorem 3.9] does not actually apply in this situation.)

5. Definition of the Flach map We are now in a position to define the Flach map. Let F be a field and let X be a smooth separated F -scheme of finite type and dimension n. Let N be an integer relatively prime to the characteristic of F and let be the constant sheaf Z/NZ on X. Fix also an integer m, 0 m n. To defineF the Flach map we need to assume that X satisfies purity at N≤. By≤ Theorem 3.2 this assumption holds if F is a perfect field or if F is a local or global field of positive characteristic and N is divisible only by primes n + 2. We will work with the second≥ stage of the coniveau spectral sequence for reasons which will become clear in a moment. We begin with the Chern class map (4.4) (with p = m, q = m 1, i = m + 1 and Y = X), which we denote by c(X): − − m, m 1 c(X) m,m+1 E − − (X) E X, (m + 1) . 2 −→ 2 F Since we assumed that purity holds for X at N by Proposition 3.3, we can write the E1-term of this second spectral sequence as pq q p E1 X, (m + 1) = H − Spec k(x), (m + 1 p) . F x ⊕Xp F − ∈ 5. DEFINITION OF THE FLACH MAP 65

But this certainly vanishes for q < p; it follows that there is an edge map (see Example A.2.1) at the second stage: d(X) Em,m+1 X, (m + 1) H2m+1 X, (m + 1) . 2 F −→ F Note that this edge map does not yet exist at the first stage. We now use the Leray spectral sequence [FK88, p. 28] for the morphism u : X Spec F : → (5.1) Hp Spec F,Rqu (m + 1) Hp+q X, (m + 1) . ∗F ⇒ F (5.1) yields an edge map (5.2) H2m+1 X, (m + 1) H0 F,R2m+1u (m + 1) . F → ∗F 2m+1 Define H (X, (m + 1))0 to be the kernel of (5.2); (5.1) yields a natural edge map F e(X) H2m+1 X, (m + 1) H1 F,R2mu (m + 1) F 0 −→ ∗F Often the 0-part fills up the entire étalecohomology group. 2m+1 Lemma 5.1. Assume that H (XFs , (m + 1)) has no GF -invariants. Then 2m+1 2m+1 F H (X, (m + 1))0 = H (X, (m + 1)). F F Proof. Recall that étalesheaves on Spec F can be identified with discrete GF -modules and that under this identification étalecohomology identifies with Galois cohomology. The sheaf R2m+1u (m+1) corresponds to the Galois module 2m+1 ∗F H (XF , (m + 1)); thus by the assumption of the lemma s F 0 2m+1 0 2m+1 H Spec F,R u (m + 1) = H F,H (XFs , (m + 1)) = 0. ∗F F This proves the lemma. Set m,m+1 1 2m+1 E X, (m + 1) = d(X)− H X, (m + 1) 2 F 0 F 0 m, m 1 1 1 2m+1 E2 − − (X)0, = c(X)− d(X)− H X, (m + 1) . F F 0 Definition 5.2. The Flach map m, m 1 1 2m σm : E2 − − (X)0, H Spec F,R u (m + 1) F → ∗F is defined to be to be e(X) d(X) c(X). ◦ ◦ We can also consider the Flach map as a map to Galois cohomology. The 2m 2m étalesheaf R u (m + 1) identifies with the GF -module H (XFs , (m + 1)); ∗F F denote this GF -module as V . Under these identifications, étalecohomology becomes Galois cohomology, so we can write the Flach map as m, m 1 1 σm : E2 − − (X)0, H (F,V ). F → The domain of σm is not a particularly complicated object. Indeed, the group m, m 1 E2 − − (X) is just the cohomology of the complex

K2k(x) k(x)× Z x X⊕m 1 →x ⊕Xm →x X ⊕m+1 ∈ − ∈ ∈ m, m 1 where the second map is the divisor map. Thus E2 − − (X) identiﬁes with a quotient of

(5.3) ker k(x)× Z ; x ⊕Xm →x X ⊕m+1 ∈ ∈ 66 6. THE FLACH MAP

m, m 1 this description makes it signiﬁcantly simpler to exhibit elements of E2 − − (X).

6. Functoriality and passage to the limit

We keep the hypotheses of the previous section. Let 0 be the constant sheaf F Z/N 0Z on X for some N 0 dividing N. Let π : 0 be the natural map. We have already assumed that X satisﬁes purity at FN, → so F there are two Flach maps m, m 1 1 σm : E2 − − (X)0, H (F,V ) F → m, m 1 1 σm0 : E2 − − (X)0, H (F,V 0) F 0 → 2m where V 0 = H (XF , 0(m + 1)). We claim that these two maps are compatible, s F in the sense that σm0 is the composition of σm with the natural map 1 1 H (F,V ) H (F,V 0) → coming from π. To check this compatibility we must check the commutativity of the diagram m, m 1 / m, m 1 E2 − − (X)0, E2 − − (X)0, F F 0

c(X) c0(X) m,m+1 π1 m,m+1 E X, (m + 1) / E X, 0(m + 1) 2 F 0 2 F 0

d(X) d0(X) 2m+1 π2 2m+1 H X, (m + 1) / H X, 0(m + 1) F 0 F 0

e(X) e0(X) 1 2m π3 1 2m H Spec F,R u (m + 1) / H Spec F,R u 0(m + 1) ∗F ∗F where the πi are the maps induced by π. (The fact that these maps send 0-parts to 0-parts is immediate.) This is quite easy: the commutativity of the first square follows from the functoriality of the Chern class maps; the commutativity of the second square is proven in Proposition A.8.1; and the commutativity of the third square is Proposition A.4.1. Fix now a prime l such that X satisfies purity at all powers of l. Considering Z/liZ as a Z/li+1Z-module, the compatibility above shows that the Flach maps are compatible for the constant sheaves associated to Z/liZ for all i. We will use m, m 1 this to define a Flach map for the sheaf Zl. Let E2 − − (X)0,Zl be the set of all m, m 1 m, m 1 elements of E2 − − (X) which lie in E2 − − (X)0,Z/liZ for some (and thus all sufficiently large) i. Set

2m def 2m i V = H XF , Zl(m + 1) = lim H XF , Z/l Z(m + 1) . s s ←−i Passing to the limit we obtain a Flach map m, m 1 1 σm : E − − (X)0,Z H (F,V ). 2 l → This is the version of the Flach map which we will almost always consider in later chapters. The next two results show that we can often consider this σm as origi- m, m 1 nating in E2 − − (X). 7. FUNCTORIALITY II 67

Lemma 6.1. Let X be a smooth and projective scheme over the ring of integers of a local field K. Let l be relatively prime to the residue characteristic of K; if K has positive characteristic then assume that l n + 2. Set X = XK . Then ≥ m, m 1 m, m 1 (6.1) E − − (X)0,Z Z Q = E − − (X) Z Q. 2 l ⊗ 2 ⊗ 2m+1 If H (XKs , Zl) is torsion-free, then m, m 1 m, m 1 (6.2) E2 − − (X)0,Zl = E2 − − (X). Proof. Let k denote the residue field of K. Smooth base change and the Weil 2m+1 conjectures [FK88, Chapter IV, Theorem 1.2] show that H (XKs , Ql(m + 1)) is unramified and that the eigenvalues of a geometric Frobenius on it are √#k. Thus this étalecohomology group has trivial GK -invariants. By Lemma 5.1 this implies that 2m+1 2m+1 H X, Zl(m + 1) Z Ql = H X, Zl(m + 1) Z Ql. 0 ⊗ l ⊗ l 2m+1 (6.1) now follows since any element of H (X, Zl(m + 1)) which is not GK - 2m+1 i invariant yields elements of H (X, Zl/l Z(m + 1)) which are not GK -invariant for all sufficiently large i. (6.2) is proven in the same way, since the torsion-free 2m+1 hypothesis implies that H (XKs , Zl(m+1)) itself already has no GK -invariants. Lemma 6.2. Let X be a smooth and projective scheme over a global field F . Let l be relatively prime to the characteristic of F , and assume that l n + 2 if F has positive characteristic. Then ≥ m, m 1 m, m 1 E − − (X)0,Z Z Q = E − − (X) Z Q. 2 l ⊗ 2 ⊗ 2m+1 If H (XFs , Zl) is torsion-free, then m, m 1 m, m 1 E2 − − (X)0,Zl = E2 − − (X). Proof. Standard arguments show that X extends to a smooth projective scheme over an open subscheme S of the spectrum of the ring of integers of F . Applying Lemma 6.1 to the completion of F at any closed point of S of residue characteristic different from l then proves the result. 7. Functoriality II

Let X and F be as before. Let F 0 be a field and let X0 be a smooth separated F 0-scheme of finite type and the same dimension n. Suppose that we are given an inclusion of fields F, F 0 and a flat morphism of schemes f : X0 X compatible → → with this inclusion. Suppose also that X0 satisfies purity at N, so that there is a m, m 1 1 2m Flach map σm0 : E2 − − (X0)0 H (F 0,V 0) where V 0 = H (XF0 , f ∗ (m + 1)). → s0 F We claim that σm0 is compatible with σm in the sense that there is a commutative diagram m, m 1 / m, m 1 E2 − − (X)0, E2 − − (X0)0,f F ∗F σ m σm0 1 1 H (F,V ) / H (F 0,V 0) Here the bottom map is the composition 1 1 1 H (F,V ) H (F 0,V ) H (F 0,V 0) → → 68 6. THE FLACH MAP induced by the map of Galois groups GF GF and the map on étalecohomology 0 → V V 0. →The proof of this is straightforward: we must check that the diagram

m, m 1 f1 / m, m 1 (7.1) E2 − − (X)0, E2 − − (X0)0,f F ∗F

c(X) c(X0) m,m+1 f2 m,m+1 E X, (m + 1) / E X0, f ∗ (m + 1) 2 F 0 2 F 0

d(X) d(X0) 2m+1 f3 2m+1 H X, (m + 1) / H X0, f ∗ (m + 1) F 0 F 0

e(X) e(X0) 1 2m f4 1 2m H Spec F,R u (m + 1) / H Spec F 0,R u0 f ∗ (m + 1) ∗F ∗ F is commutative. Here u0 : X0 Spec F 0 is the structure map and the fi are the obvious maps, which we discuss→ in more detail now. f1 is the map of coniveau spectral sequences induced by the flat morphism f : X0 X. f2 is induced by the same morphism, and this square commutes since Gillet’s→ construction is a natural transformation. f3 is the map on étalecohomology coming from the morphism X0 X; the second square commutes by Proposition 1.1. → For the fourth horizontal map, consider first the base change map 2m 2m (7.2) f 0∗R u R u0 f ∗ ∗F → ∗ F coming from the cartesian square

f X o X0

u u0

f 0 Spec F o Spec F 0 where f 0 is the obvious map. (7.2) induces a map 1 2m 1 2m H (Spec F 0, f 0∗R u ) H (Spec F 0,R u0 f ∗ ). ∗F → ∗ F Precomposing this with the map 1 2m 1 2m H (Spec F,R u ) H (Spec F 0, f 0∗R u ) ∗F → ∗F coming from the morphism f 0 defines f4. That this square commutes is a standard result on the Leray spectral sequence and edge maps; see Proposition A.5.1. This completes the proof of the functoriality of the Flach map for flat morphisms as above. There are two special cases of the above construction which are especially important: the first is when F = F 0, so that X0 X is just a flat morphism of → relative dimension 0 of F -schemes. The second is when X0 = XF 0 is just the base change of X and the morphism X0 X is the projection. → The last functoriality we will need is for finite flat morphisms. Let f : X0 X be a finite flat morphism of smooth separated F -schemes of dimension n and→ 7. FUNCTORIALITY II 69 suppose that all of the relevant Flach hypotheses are satisfied. Then there is a commutative diagram m, m 1 / m, m 1 E2 − − (X0)0, E2 − − (X)0,f F ∗F σ σm0 m 1 1 H (F,V 0) / H (F,V ) 2m where V 0 = H (XF0 , f ∗ (m + 1)) and the horizontal maps come from covariant functoriality for finite flatF morphisms. The proof of this is virtually the same as the proof above, given that all of our constructions were functorial both for flat and for finite flat morphisms.

CHAPTER 7

Local analysis of the Flach map

The usefulness of the Flach map in generating geometric Euler systems comes from a geometric description of the local ramiﬁcation of the image of the Flach map. The formulation and proof of this description is the focus of this chapter.

1. Overview Let S be a Dedekind scheme of positive dimension; that is, S is a normal noetherian scheme of dimension 1. (We will be concerned in this chapter with the local behavior of the Flach map, so there is no need to consider the case where S is the spectrum of a field.) We assume further that S is connected; since it is normal this implies that it is irreducible. Let F be the function field (i.e., the residue field at the generic point) of S. Let X be a smooth proper S-scheme of relative dimension n. Write X for the generic fibre X S Spec F of X. Fix an integer N relatively prime to the characteristic of F and let ×be the sheaf Z/NZ on X. Fix another integer m, 0 m n and F 2m ≤ ≤ let V denote H (XFs , (m + 1)), considered as a GF -module. If we assume that X satisfies purity at N,F then we have a Flach map

m, m 1 1 σm : E2 − − (X)0, H (F,V ) F → as in Section VI.5. Fix now a closed point v of S with residue characteristic prime to N; S,v is a discrete valuation ring. Let R denote its completion, with residue field kOand fraction field K. In this chapter we will give a description of the local behavior of the cohomology classes coming from σm at the closed point v. Note that by smooth base change and the local constancy of higher direct images [FK88, Chapter I, Theorem 8.9], V is unramified as a GK -module and there is an isomorphism 2m V = H Xk , (m + 1) . ∼ s F m, m 1 For (Z, f) E − − (X), let Z¯ denote the closure of Z in XR. Let ∈ 2 m, m 1 m divk : E − − (X) A Xk 2 → be the map sending (Z, f) to the divisor of f on Z¯; this divisor is supported entirely ¯ 1 on Zk (see (VI.5.3)) and has codimension m in Xk. Let Hs (K,V ) denote the 0 1 group H (k, H ( K ,V )) where K is the inertia group of K. Recall that there is a canonical isomorphismI I V ( 1)Gk = H1(K,V ) − ∼ s as in Lemma I.1.3. The goal of this chapter is to give a proof of the following theorem.

71 72 7. LOCAL ANALYSIS OF THE FLACH MAP

Theorem 1.1. Let S, X, and v be as above. Assume also that XK and Xk F satisfy purity at N, and that the pair XR, Xk satisﬁes relative purity at N. Then there is a commutative diagram

m, m 1 divk / m E2 − − (X)0, A Xk F

σm s 1 2m Gk H (F,V ) H Xk , (m) s F

' 1 1 H (K,V ) / Hs (K,V ) Here s is the cycle class map in ´etalecohomology, the unlabelled maps are the canonical restriction and singular restriction maps, and the unlabelled isomorphism is the isomorphism of Lemma I.1.3. The same diagram commutes if is a constant F sheaf of Zl-modules or Ql-vector spaces.

2. Local behavior I Let be a fixed constant N-torsion sheaf on X. The first step in the proof of TheoremF 1.1 is to connect the Flach map over F with the local Flach map over K. This is easy; indeed, by our assumption that XK satisfies purity at N, the Flach map m, m 1 1 2m (2.1) E2 − − (XK )0, H Spec K,R uv0 (m + 1) , F → ∗F is defined over XK ; here uv0 : XK Spec K is the structure map. We now relate (2.1) with the→ global Flach map. Specifically, we have the commutative diagram (VI.7.1): m, m 1 g1 / m, m 1 (2.2) E2 − − (X)0, E2 − − (XK )0, F F c(F ) c(K) m,m+1 g2 m,m+1 E X, (m + 1) / E XK , (m + 1) 2 F 0 2 F 0 d(F ) d(K) 2m+1 g3 2m+1 H X, (m + 1) / H XK , (m + 1) F 0 F 0 e(F ) e(K) 1 2m g4 1 2m H Spec F,R u (m + 1) / H Spec K,R uv0 (m + 1) ∗F ∗F Here we have labelled the maps in the definition of the Flach map by the base scheme, rather than the scheme itself; we will continue to do so for the remainder of this chapter.

3. Local behavior II The next step is to connect the local Flach map to a relative Flach-type map for the pair XR, Xk. Here we need the assumption that this pair satisﬁes relative purity at N. Note that XR is at least regular over R, since X is smooth over S. 3. LOCAL BEHAVIOR II 73

At this point it is crucial that we are working with the E2-terms of the coniveau spectral sequence. We will construct a commutative diagram

m, m 1 δ1 / m+1, m 1 (3.1) E2 − − (XK )0, E2,X − − (XR) F k c(K) c(R,k) m,m+1 δ2 m+1,m+1 E XK , (m + 1) / E XR, (m + 1) 2 F 0 2,Xk F d(K) d(R,k) 2m+1 δ3 2m+2 H XK , (m + 1) / H XR, (m + 1) F 0 Xk F e(K) e(R,k) 1 2m δ4 2 2m H Spec K,R uv0 (m + 1) / HSpec k Spec R, R uv (m + 1) ∗F ∗F where uv : XR Spec R is the structure map. → The left-hand vertical maps have already been defined. δ1 and δ2 are the localization maps of Lemma VI.2.1 and Proposition VI.4.1 for Xk , XR; note that → XK is dense in XR, so that these maps do exist. c(R, k) is Gillet’s Chern class map, and this square commutes by Proposition VI.4.2. d(R, k) is defined as an edge map in the same way as d(F ) and d(K), but its existence requires our purity hypothesis on XR. Given this assumption, the definition is almost the same as in the previous two cases: we can write pq q p E XR, (m + 1) = H − Spec k(x), (m + 1 p) 1,Xk p F x X⊕ Xk F − ∈ R∩ and this clearly vanishes for q < p. It follows from Example A.2.1 that the desired edge map d(R, k) exists. δ3 is the localization map in étalecohomology coming from the long exact sequence of the pair Xk , XR. This square commutes by Lemma VI.2.1. → e(R, k) is an edge map in the Leray spectral sequence with supports: p q p+q HSpec k Spec R, R uv (m + 1) HSpec X Spec XR, (m + 1) . ∗F ⇒ k F To show that it exists we must show that 0 2m+2 1 2m+1 HSpec k Spec R, R uv (m + 1) = HSpec k Spec R, R uv (m + 1) = 0. ∗F ∗F But this is immediate from purity as in Theorem VI.3.2(c) with X = Spec R and Y = Spec k (together with the fact that the higher direct images of the proper map uv are locally constant). Indeed, purity identifies these with cohomology groups ∗ over Spec k in dimensions 2 and 1, which automatically vanish. δ4 is again a localization map in étalecohomology,− − together with a base change map. That this square commutes follows from Proposition A.7.1. Specifically, take ! G = uv , F1 = Γ(Spec k, i ), F2 = Γ(Spec R, ), F3 = Γ(Spec K, ), where i : Spec k ∗ Spec R is the natural· map. The sequence· · → 0 F1( ) F2( ) F3( ) 0 → G → G → G → is left exact for any sheaf and since restriction maps are surjective for injective sheaves the sequence is exactG on injectives. Proposition A.7.1 now applies; the fact that the maps obtained agree with the usual boundary maps is a standard exercise in injective resolutions and is left to the reader. 74 7. LOCAL ANALYSIS OF THE FLACH MAP

4. Local behavior III We now use purity to connect the relative Flach map of the previous section to a “lower” Flach map over the residue ﬁeld k. We now need our assumption that Xk satisﬁes purity at N. With this assumption, there is a commutative diagram

m+1, m 1 p1 m, m (4.1) E − − (X ) o 2,Xk R E2 − (Xk) ' c(R,k) c(k)

p Em+1,m+1 X , (m + 1) o 2 Em,m X , (m) 2,Xk R 2 k F ' F d(R,k) d(k)

p H2m+2 X , (m + 1) o 3 H2m X , (m) Xk R k F ' F e(R,k) e(k) 2 2m p4 0 2m HSpec k Spec R, R uv (m + 1) o H Spec k, R uv00 (m) ∗F ' ∗F where uv00 : Xk Spec k is the structure map and all horizontal maps are isomorphisms. → Again, the left-hand vertical maps have all already been defined. The right- hand vertical maps are not difficult to define: c(k) is Gillet’s Chern class map for Xk, d(k) is defined as an edge map in the usual way (using purity for Xk) and e(k) is an edge map in the Leray spectral sequence for uv00. The horizontal maps are all incarnations of purity. Recall that the spectral pq sequence Er (Xk) is constructed from the filtration of coherent sheaves on Xk by codimension of support; see [Qui73, Section 7, Theorem 5.4]. Similarly, Epq (X ) r,Xk R is constructed from the filtration of coherent sheaves on XR, supported on Xk, by codimension of support. With these descriptions it is clear that these spectral sequences coincide, with a shift in the indices; that is, there is an isomorphism of spectral sequences

pq p+1,q 1 (4.2) E (Xk) E − (XR). r → r,Xk At the ﬁrst stage, this coincides with the obvious identiﬁcation of E1-terms: pq p+1,q 1 E1 (Xk) = K p qk(x) = K p qk(x) = E1,X − (XR). p − − p+1 − − k x ⊕X x X ⊕ Xk ∈ k ∈ R ∩

We let p1 be the bidegree (m, m) isomorphism in (4.2) with r = 2; at the ﬁrst stage this looks like −

m, m m+1, m 1 / E − − (X ) E1 − (Xk) 1,Xk R

K0k(x) K0k(x) m m+1 x ⊕X x X ⊕ Xk ∈ k ∈ R ∩

Next, recall that by relative purity for XR, Xk we have isomorphisms i i+2 H Xk, (m) H XR, (m + 1) Z F → Z F 5. THE DIVISOR MAP 75 for all irreducible regular closed subschemes Z of Xk. These compile to isomorphisms i p i+2 p+1 H Xk, (m) H XR, (m + 1) F → Xk F in the notation of Section VI.1. These identiﬁcations yield isomorphisms of the exact couples (VI.1.3) used to deﬁne the coniveau spectral sequence, with a shift in indices, and thus an isomorphism

pq p+1,q+1 (4.3) E Xk, (m) E XR, (m + 1) . r F → r,Xk F At the ﬁrst stage, this coincides with the previous purity identiﬁcation

pq 0 E1 Xk, (m) = H k(x), (m p) = F x ⊕Xp F − ∈ k 0 p+1,q+1 H k(x), (m p) = E XR, (m + 1) . p+1 1,Xk x X ⊕ Xk F − F ∈ R ∩

We let p2 be the bidegree (m, m) isomorphism of (4.3) with r = 2; at the ﬁrst stage it is just

m,m m+1,m+1 E Xk, (m) / E XR, (m + 1) 1 F 1,Xk F

H0(k(x), ) H0(k(x), ) m m+1 x ⊕X F x X ⊕ Xk F ∈ k ∈ R ∩ That the first square in (4.1) commutes is now immediate from the description of the Chern class maps given in (VI.4.5). Indeed, in both case these maps are induced by divisor maps which themselves coincide. p3 is just the usual purity isomorphism; the second square commutes by Propo- sition A.8.1 and the fact that p2 is defined in terms of purity. To define p4 we first must define an isomorphism of spectral sequences p q p+2 q (4.4) H Spec k, R uv00 (m) HSpec k Spec R, R uv (m + 1) . ∗F → ∗F This is most easily defined using the construction of the Leray spectral sequence and the description of purity given in [Gil81, pp. 205-207, esp. p. 207(vi)]. p4 is the second stage bidegree (0, 2m) part of (4.4). The desired commutativity is just the standard compatibility of Leray spectral sequences and edge maps, taking into account the shift in indices.

5. The divisor map The diagrams (2.2), (3.1) and (4.1) combine to put the Flach map in a commutative diagram

m, m 1 / m, m (5.1) E2 − − (X)0, E2 − (Xk) F

σm 1 2m 0 2m H Spec F,R u (m + 1) / H Spec k, R uv00 (m) ∗F ∗F We now need to evaluate the other three maps. 76 7. LOCAL ANALYSIS OF THE FLACH MAP

We begin with the top horizontal map

m, m 1 g1 m, m 1 δ1 (5.2) E2 − − (X)0, E2 − − (XK )0, F −→ F −→ m+1, m 1 p1 m, m E − − (XR) E − (Xk). 2,Xk ←− 2 Evaluating the four terms (using Proposition VI.4.1 and the computation of the K-groups of a ﬁeld) identiﬁes (5.2) with a subquotient of a diagram

(5.3) k(x)× k(x)× Z = Z. m m 99K m+1 m x ⊕X →x ⊕XK x X ⊕ Xk x ⊕Xk ∈ ∈ ∈ R ∩ ∈

Here the first and last maps exist at the first stage, but the middle one does not exist until the second stage. Consider a codimension m cycle x on X and a rational function f onx ¯. The K-schemex ¯ Spec F Spec K has a finite number of irreducible componentsx ¯i of codimension ×m [GDb, Corollary 4.5.10], and there are natural inclusions k(x) , → k(xi) of function fields. Under g1 in (5.3), the pair (x, f) maps to (xi, f), where P by f we mean the rational function on xi coming from the inclusion of function fields above. This description follows from the fact that these maps agree with the natural maps K1k(x) K1k(xi), which are as we just described. → ⊕ δ1 is a boundary map in the exact localization sequence. Recall that this boundary map is computed from the diagram

m+1, m 1 m+1, m 1 m+1, m 1 E − − (XR) / E − − (XR) / E − − (XK ) 1,Xk O 1 O 1 O

m, m 1 m, m 1 m, m 1 E − − (X ) / / 1,Xk R E1 − − (XR) E1 − − (XK )

by pulling back, pushing forward and pulling back. Pulling back a pair (xi, f) on XK to XR maps it to the pair (¯xi, f) on XR, wherex ¯i is the closure in XR of xi and f is regarded as a rational function onx ¯i. By Proposition VI.4.1, the differential at this stage is identified with the divisor map, so pushing this forward yields the divisor divXR (f) of the rational function f on XR. Since we are assuming that m, m 1 (x, f) lies in E2 − − (XK ) and thus that the divisor of f has no intersection with XK (see (VI.5.3)), this divisor is supported on Xk and thus pulls back. In summary, the image of (xi, f) under δ1 is nothing more than the divisor of f on XR, which is necessarily supported on Xk. p1 is just the canonical identification of terms given in Section 4; this reinter- prets the divisor above back on Xk. It follows, then, that the map

m, m 1 m, m E − − (X) E − (Xk) 2 → 2 of (5.2) sends a pair of a codimension m cycle x and a rational function f k(x)× ∈ to the part of the divisor of f which is supported on Xk. This is the map which we previously denoted divk. 6. THE CYCLE MAP 77

6. The cycle map We next compute the vertical map

m, m c(k) m,m d(k) E − (Xk) E Xk, (m) 2 −→ 2 F −→ 2m e(k) 0 2m H Xk, (m) H Spec k, R uv00 (m) . F −→ ∗F Recall that we can evaluate the ﬁrst two terms as subquotients of 0 (6.1) K0k(x) H (k(x), ), x ⊕Xm →x ⊕Xm F ∈ k ∈ k 0 and the map K0k(x) = Z M = H (k(x), ) (where M is the Z/NZ-module to which the constant sheaf→ is associated) isF just the natural map; see Propo- F m, m sition VI.4.2. That is, (6.1) sends an element (x, 1) of E2 − (Xk) to the “same” m,m element (x, 1) of E2 Xk, (m) . We next consider the edgeF map

m,m d(k) 2m (6.2) E Xk, (m) H Xk, (m) 2 F −→ F which we consider as the map induced from a first stage map 0 2m 2m (6.3) H (k(x), ) Hx Xk, (m) H Xk, (m) . x ⊕Xm F →x ⊕Xm F → F ∈ k ∈ k Recall that 2m 2m (6.4) Hx Xk, (m) = lim Hx¯ Z Xk Z, (m) . F − − F −→Z(x¯ 2m For each Z we have a well-defined fundamental class ofx ¯ Z in Hx¯ Z Xk − − − Z, (m) , and these classes are compatible with the maps of the direct system; see F [FK88, Chapter II, Corollary 2.3]. They therefore compile to give an element ξx of (6.4). Since for sufficiently large Z (specifically, large enough so thatx ¯ Z is regular) the purity map sends the element 1 of H0(k(x), ) to the fundamental− 2m F class ofx ¯ Z in Hx¯ Z (Xk Z, (m)), we see that under the first map of (6.3) the − − − F 2m element (x, 1) must map to ξx Hx (Xk, (m)). But ξx can already be realized as ∈ F 2m the fundamental class ofx ¯ at the initial Z = term Hx¯ (Xk, (m)) of the direct limit, and the map ∅ F 2m 2m H Xk, (m) H Xk, (m) x¯ F → F m, m of (6.3) is just the natural map. We conclude that (x, 1) E1 − (Xk, (m)) maps 2m ∈ F to the fundamental class ofx ¯ in H (Xk, (m)); the map (6.2) at the E2-level has the same description. F 2m 2m For e(k), we first identify R uv00 (m) with H (Xks , (m)). Under this identification, we are considering a map∗F F

2m 2m Gk (6.5) H Xk, (m) H Xk , (m) F → s F which is nothing more than the map induced from the map Xks Xk; see [Wei94, Section 5.8]. By transitivity of the fundamental class this sends→ the fundamental class ofx ¯ in Xk to the fundamental class ofx ¯ Spec k Spec ks in Xks ; it is necessarily Galois invariant since it comes from a cycle deﬁned× over k. Combining our descriptions of (6.1), (6.2) and (6.5), we see that the map

m, m 2m Gk Z E2 − (Xk) H (Xks , (m)) x ⊕Xm → F ∈ k 78 7. LOCAL ANALYSIS OF THE FLACH MAP sends an element with a 1 corresponding to x Xm and zero everywhere else to ∈ k the fundamental class ofx ¯ Spec k Spec ks; the general deﬁnition follows by linearity. m, m × m Identifying E2 − (Xk) with the Chow group A Xk as in Proposition VI.4.1, we will write this cycle map as

m 2m Gk s : A (Xk) H Xk , (m) . → s F 7. Relations with Galois cohomology The last map in (5.1) to identify is the bottom map

1 2m g4 1 2m δ4 H Spec F,R u (m + 1) H Spec K,R uv0 (m + 1) ∗F −→ ∗F −→ 2 2m p4 0 2m HSpec k Spec R, R uv (m + 1) H Spec k, R uv00 (m) . ∗F −→ ∗F We first evaluate some of the terms a bit more. R2mu (m+1) as an étalesheaf 2m ∗F on Spec F corresponds to the GF -module H (XFs , (m + 1)). Similarly, the 2m F2m étalesheaf R uv0 (m+1) on Spec K corresponds to H (XKs , (m+1)), which ∗F F is a GK -module. The smooth base change theorem [FK88, Chapter I, Section 8] shows that there is a natural isomorphism 2m 2m H XF , (m + 1) = H XK , (m + 1) s F ∼ s F as GK -modules. Note that these are both the Galois module previously denoted V . Under these identifications, the base change map g4 identifies with the usual restriction map H1(F,V ) H1(K,V ) in Galois cohomology. It remains to identify→ the sequence of maps

1 2m δ4 2 2m (7.1) H Spec K,R uv0 (m + 1) HSpec k Spec R, R uv (m + 1) ∗F −→ ∗F p4 0 2m H Spec k, R uv00 (m) . −→ ∗F The smooth base change theorem and the local constancy of higher direct images under proper maps ([FK88, Chapter I, Theorem 8.9]; it is here that it is crucial that X is proper over S) shows that V is unramified as a GK -module, and identifies 2m R uv00 (m) with V ( 1) as a Gk-module. Making these identifications we can rewrite∗F (7.1) as − 1 2 2m 0 (7.2) H (K,V ) HSpec k Spec R, R uv (m + 1) H k, V ( 1) . → ∗F → − 1 As in Section 1, the last term in (7.2) identifies with Hs (K,V ). Taking this into account, we see that we are trying to identify a map (7.3) H1(K,V ) H1(K,V ). → s Fortunately, Grothendieck showed that under these identifications, the map (7.3) which we are trying to evaluate really is nothing more than the natural singular restriction map; see [CT95, Section 3.3] and [Gro77, Exposé1, pp. 50-52]. This completes the proof of Theorem 1.1 for torsion sheaves.

8. Functoriality and passage to the limit

We continue with our running hypotheses. Let 0 be a second constant N- F torsion sheaf on X and suppose that we are given a map π : 0. Set V 0 = F → F 9. EXAMPLE : SCHEMES OVER GLOBAL FIELDS 79

2m H (XFs , 0(m + 1)). We now have two Flach maps ﬁtting into commutative diagrams F

m, m 1 divk / m E2 − − (X)0, A Xk F

σm s 1 2m Gk H (F,V ) H Xk , (m) s F

' 1 1 H (K,V ) / Hs (K,V ) and

m, m 1 divk / m E2 − − (X)0, A Xk F 0

σm s 1 2m Gk H (F,V 0) H Xk , 0(m) s F

' 1 1 H (K,V 0) / Hs (K,V 0)

These diagrams are connected by various obvious maps coming from π and we claim that the resulting three-dimensional diagram is commutative. This is not difficult; we checked that the Flach map is compatible with such maps in Section VI.6, and the rest of the commutativities are clear. Indeed, the only non-obvious one is the cycle map, which is proved in [FK88, Chapter II, Corollary 2.3]. Note that the somewhat daunting map divk doesn’t depend on or 0 at all. In particular, assuming that the purity hypotheses are all satisfiedF F and proceeding as in Section VI.6 we find that Theorem 1.1 holds for l-adic sheaves as well. Tensoring with Ql yields the result for sheaves of Ql-vector spaces, and completes the proof of the theorem.

9. Example : Schemes over global ﬁelds

Let F be a global field and let F denote its ring of integers. Let S be an open O subscheme of Spec F . In this section we restate Theorem 1.1 for schemes over S. Let X be a smoothO projective scheme of relative dimension n over S. Let X be the generic fiber of X. Let m be an integer 0 m n, and let l be a prime. If F is of characteristic 0 we allow l to be arbitrary,≤ while≤ if F has positive characteristic we require l to be relatively prime to the characteristic and greater than or equal 2m to n + 2. Set V = H (XF¯ , Zl(m + 1)). Theorem 9.1. Let F , S, X and l be as above. Let v be a place of F lying in S and of residue characteristic prime to l. Let K denote the completion of F at v 80 7. LOCAL ANALYSIS OF THE FLACH MAP and let k denote the residue field of v. Then there is a commutative diagram

m, m 1 divk m E − − (X) Ql / A Xk Ql 2 ⊗ ⊗

σm s 1 2m Gk H (F,V Ql) H Xk , Ql(m) ⊗ s

' 1 1 H (K,V Ql) / H (K,V Ql) ⊗ s ⊗ 2m+1 If H (XFs , Zl) is torsion-free, then there is also a commutative diagram

m, m 1 divk / m E2 − − (X) A Xk

σm s 1 2m Gk H (F,V ) H Xk , Zl(m) s

' 1 1 H (K,V ) / Hs (K,V ) Proof. Theorem VI.3.2 shows that all of the purity hypotheses are satisﬁed, so that the Flach maps exist. Theorem 1.1 now gives the diagrams above, with Lemma VI.6.2 identifying the 0-parts of the E2-terms.

10. Local behavior at places over l In this section we give a result which will allow us to control the behavior of Flach classes at places dividing l, at least in certain (somewhat restrictive) circumstances. It would be useful to have a stronger result. Note that despite the local nature of the statement of the proposition, the proof will require the global hypothesis. Let S be an open subscheme of the spectrum of the ring of integers of a number field F and let X be a smooth proper scheme of relative dimension n over S. Let l be a prime and let m be another integer 0 m n. Let X be the generic fiber of X. ≤ ≤ 2m Let T be a quotient of H (XF¯ , Zl(m + 1)) as a GF -module. Note that we can consider images of cycle classes of codimension m in T ( 1). Fix a place v dividing l (although the statement below will not depend on− which such v we pick) and let K be the completion of F at v, with valuation ring R. We let m, m 1 1 σ : E2 − − (X) H (K,T ) denote the composition of the Flach map σm with projection to T and→ restriction to K. m, m 1 ¯ Proposition 10.1. Let (Z, f) be an element of E2 − − (X) and let Z denote the closure of ZK in XR. If the fundamental class of ZKs in XKs maps to 1 0 in T ( 1), then σm(Z, f)v lies in H (K,T ) for the deRham local finite/singular − f structure at v. If further it is possible to realize U = XR Z¯ as the complement of a normal crossings divisor in a smooth proper variety over− R, then σ(Z, f) lies in 1 Hf (K,T ) for the crystalline finite/singular structure at v. 10. LOCAL BEHAVIOR AT PLACES OVER l 81

For deﬁnitions of the local conditions in the proposition, see Section I.4. In particular, the deﬁnitions imply that it is enough to prove the results after tensoring by Ql. Proof. For ease of notation, set i i H = H XK , Ql(m + 1) Z ZK i i H = H XK , Ql(m + 1) ¯ i i HZ = HZ XKs , Ql(m + 1) Ks i i H¯ = H XK , Ql(m + 1) . s We have the following commutative diagram of Flach and Jannsen (see [Jan90, Part II, Lemma 9.5] and [Fla92, pp. 323–324]):

ker H2m+1 (H2m+1)GK / ker H2m+1 (H¯ 2m+1)GK Z → →

e(XK ) ¯ 2m+1 ¯ 2m+1 GK 1 ¯ 2m ker HZ H H (K, H ) → VV VVVV δ VVVV VVVV VVVV* 1 ¯ 2m ¯ 2m H (K, H /HZ )

1 H (K,T Z Ql) ⊗ l where e(XK ) is the map of Section VI.5; δ is a boundary map in the Galois cohomology sequence of 2m 2m 2m 2m+1 2m+1 (10.1) 0 H¯ /H¯ H UK , Ql(m + 1) ker H¯ H¯ 0; → Z → s → Z → → 1 2m and the map to H (K,T Z Ql) exists since H¯ is generated by the (twisted) ⊗ l Z fundamental class of ZKs in XKs , which vanishes in T by hypothesis. Let d(XK ) and c(XK ) denote the maps of Section V.5. The element

(10.2) d(XK ) c(XK )(ZK , f) ◦ of H2m+1 maps to 0 in (H¯ 2m+1)GK since it can be lifted to a global element which 2m+1 GF factors through H (XF¯ , Ql(m+1)) = 0. It is also clear from the construction 2m+1 of the coniveau spectral sequence that (10.2) arises from an element of HZ . Let 2m+1 ¯ 2m+1 ¯ 2m+1 GK τ denote the image of this element of HZ in ker(HZ H ) . 1 ¯ 2m ¯ 2m→ Let σ0 denote the image of σm(Z, f)v in H (K, H /HZ ); it is just the image 1 ¯ 2m ¯ 2m of (10.2). Using the Yoneda extension interpretation of H (K, H /HZ ), the ¯ 2m ¯ 2m element σ0 corresponds to a GK -module extension of Ql by H /HZ . But by the above argument, σ0 is equal to δ(τ); the extension interpretation of δ thus implies that the extension corresponding to σ0 is obtained from (10.1) by pullback via the map ¯ 2m+1 ¯ 2m+1 Qlτ , ker(HZ H ). → 1 → The element σ(Z, f) Ql H (K,T Zl Ql) arises from this extension by ¯ 2m ¯ 2m ⊗ ∈ ⊗ pushout via H /HZ T Zl Ql. We conclude from (10.1) that the extension cor- → ⊗ 2m responding to σ(Z, f) Ql can be realized as a subquotient of H (UK , Ql(m+1)). ⊗ s 82 7. LOCAL ANALYSIS OF THE FLACH MAP

By [Fal89, Theorem 5.3 and Theorem 8.1], under the appropriate hypotheses 2m H (UKs , Ql(m + 1)) is deRham or even crystalline; since these properties are preserved under passage to subquotients, the characterization of deRham and crystalline structures in terms of extensions [BK90, (3.7)] completes the proof. In fact, recent results of Kisin [Kis] show that we could replace the deRham structure above with the potentially semistable structure. Since our applications will only involve potentially semistable representations (for which the deRham and potentially semistable structures coincide), we will not make any further use of this result. CHAPTER 8

Flach classes for correspondences

In the ﬁrst half of this chapter we study algebraic correspondences and the corresponding operations on algebraic K-theory and ´etalecohomology, culminating in the Leibniz relation of Theorem 6.1. We then apply this theory to set-up the methods for the production of cohesive Flach systems via the Flach map.

1. Algebraic correspondences In this chapter we will describe the additional algebraic structure on Flach classes associated to algebras of self-correspondences on varieties. With a view towards our intended applications we will work in a fairly restricted setting. It seems likely that many of the results of this chapter remain true more generally, but I have not attempted a proper formulation. Let X and Y be smooth proper varieties over a number field F . (For the first three sections we will only need that F is perfect, but for the remainder of the chapter we will be using Lemma VI.6.2 in an essential way.) All products in this chapter will be over Spec F or Spec F¯ unless otherwise noted; it should be clear from context which is meant. We assume further that both X and Y have the same dimension n. Definition 1.1. An irreducible correspondence from X to Y is a subscheme α , X Y such that the projections παX : α X and παY : α Y are both finite and→ faithfully× flat. A general correspondence→ from X to Y is a→ formal sum (with integer coefficients, or later with Zl-coefficients) of such irreducible correspondences. Note that an algebraic correspondence from X to Y necessarily has dimension n. Of course, the terminology “from X to Y ” is introduced purely for notational reasons. Note also that our definition is much less general than that in [Ful98, Chapter 16]. The difference lies in the fact that for us it will not be enough to work modulo rational equivalence. If α , X Y is an arbitrary closed subscheme such that the maps from each irreducible→ component× of α to X and Y are finite and faithfully flat, we define the associated correspondence as follows: Let α1, . . . , αr be the irreducible components of α. For each αi, let mi be the length of the local ring at the generic point of αi; the associated correspondence, which we will also denote α, is then miαi. We will use algebraic correspondences to define maps in K-theoryP and étaleco- homology. Let α be an irreducible correspondence from X to Y , with projections παX and παY . Given an étalesheaf on α, we define a map F i i i α : H (X, παX∗ ) H (α, ) H (Y, παY∗ ); ∗ F → F → F here the first map is the usual contravariant map on étalecohomology, and the second map is the trace map. Since παX and παY respect codimensions one sees immediately that we obtain maps of the exact couples (VI.1.3) used to define the

83 84 8. FLACH CLASSES FOR CORRESPONDENCES coniveau spectral sequence; this yields a map of spectral sequences which we also denote α : ∗ pq pq pq α : Er (X, παX∗ ) Er (α, ) Er (Y, παY∗ ). ∗ F → F → F Redoing the constructions in the opposite direction, we obtain maps i i α∗ : H (Y, π∗ ) H (X, π∗ ) αY F → αX F pq pq α∗ : E (Y, π∗ ) E (X, π∗ ). r αY F → r αX F We can also apply these constructions over F¯ to obtain maps which we again denote

i i α : H (XF¯ , πXα∗ F¯ ) H (YF¯ , παY∗ F¯ ) ∗ F → F i i α∗ : H (Y ¯ , π∗ ¯ ) H (X ¯ , π∗ ¯ ). F Y αFF → F αX FF Note that these last two maps commute with the action of GF since α is defined over F . They therefore can be used to induce maps on Galois cohomology. One checks immediately that these constructions are compatible with the natural map from étalecohomology over F to étalecohomology over F¯. We obtain analogous maps pq pq α : Er (X) Er (Y ) ∗ → pq pq α∗ : E (Y ) E (X) r → r in K-theory using the appropriate contravariant and covariant functoriality. For an p, p 1 explicit description in the case which we will need, the map α on E1 − − -terms ∗ (1.1) k(x)× k(a)× k(y)× x ⊕Xp →a ⊕αp →y ⊕Y p ∈ ∈ ∈ is as follows: an element (x, f) of the first direct sum maps to (a , π f), where i a∗iX 1 P the sum runs over ai π− (x); note that each of these has codimension p since ∈ αX παX is faithfully flat. The maps π∗ are the natural inclusions k(x) , k(ai). aiX → An element (a, f) of the second direct sum in (1.1) maps to (y, Nk(a)/k(y)f), where y = παY (a) and Nk(a)/k(y) is the norm mapping for the finite extension of fields p, p 1 k(y) , k(a). The map α∗ on E1 − − -terms has a similar description. We→ extend the definitions above to general correspondences (with integer coefficients) by linearity. Note also that if the sheaf is l-adic, then we can extend F the operations on étalecohomology to correspondences with Zl-coefficients.

2. Correspondences and operations on étalecohomology In this section we check that maps coming from correspondences are compatible with various maps in étalecohomology. We prove all results only for irreducible correspondences, but in each case they extend immediately to general correspondences by linearity. In order to discuss the theory with Zl-coefficients we will need the following definition. Definition 2.1. Let X be a variety over F . We say that X is cohomologically i torsion-free at l if the étalecohomology groups H (XF¯ , Zl) are torsion-free for all i. Note that the Künneththeorem shows that if X and Y are cohomologically torsion-free at l, then so is X Y . × 2. CORRESPONDENCES AND OPERATIONS ON étaleCOHOMOLOGY 85

2.1. Künnethprojections. Let X, Y be smooth proper varieties of dimension m over F and let X0, Y 0 be smooth proper varieties of dimension n over F . If α , X Y and β , X0 Y 0, are irreducible correspondences, one checks immediately→ × (using [GDb→, Proposition× 4.2.4]) that α β can be viewed as a (not × necessarily irreducible) correspondence from X X0 to Y Y 0. This construction generalizes in the obvious way to general correspondences× ×α and β. Suppose also that all of these varieties are cohomologically torsion-free at l. In this situation we have natural Künnethprojections fitting into a commutative diagram

i+j i j (2.1) H X ¯ X0 , Zl(a + b) / H X ¯ , Zl(a) Z H X0 , Zl(b) F × F¯ F ⊗ l F¯ (α β) α β × ∗ ∗× ∗ i+J i j H Y ¯ Y 0 , Zl(a + b) / H Y ¯ , Zl(a) Z H Y 0 , Zl(b) F × F¯ F ⊗ l F¯ for any i, j 0 and any integers a, b. (See [FK88, Chapter 1, Corollay 8.17] for information≥ on the Künneththeorem.) The commutativity of (2.1) follows immediately from the compatibility of Künnethprojections with maps coming from finite, flat morphisms; indeed, its inverse comes from cup product and maps on cohomology induced by various projections, and these are all appropriately functorial. The corresponding diagram for (α β)∗ commutes for the same reason. Of course, this × commutativity is true for far more general étalesheaves than twists of Zl; we state it this way purely for notational reasons. If the varieties are not all cohomologically torsion-free, we can still define Künnethprojections and a diagram analogous to (2.1) provided that we work with Ql-coefficients rather than Zl-coefficients. 2.2. Poincaréduality. Let X and Y be smooth proper varieties of dimension n over F and let α be an irreducible correspondence from X to Y . Let i 2n i ϕX : H (X, Zl) Z H − (X, Zl) Zl( n) ⊗ l → − i 2n i ϕY : H (Y, Zl) Z H − (Y, Zl) Zl( n) ⊗ l → − be the Poincarépairings for some i 0; see [FK88, Chapter 2, Section 1]. These pairings are compatible in the sense≥ that

(2.2) ϕX (h, α∗h0) = ϕY (α h, h0) ∗ i 2n i for h H (X, Zl) and h0 H − (Y, Zl); and ∈ ∈ (2.3) ϕX (α∗h, h0) = ϕY (h, α h0) ∗ i 2n i for h H (Y, Zl) and h0 H − (X, Zl). (2.2)∈ follows immediately∈ from the commutative diagram

i 2n i 2n H X, Zl(a) Z H − X, Zl(b) / H X, Zl(a + b) ⊗ l O O

π∗ πX πX X ∗ ∗ i 2n i 2n H α, Zl(a) Z H − α, Zl(b) / H α, Zl(a + b) ⊗ l O

πY π∗ πY ∗ Y ∗ i 2n i 2n H Y, Zl(a) Z H − Y, Zl(b) / H Y, Zl(a + b) ⊗ l 86 8. FLACH CLASSES FOR CORRESPONDENCES

(where the horizontal maps are cup product) and the fact that πX and πY induce the canonical isomorphisms on the top cohomology groups. The∗ proof of∗ (2.3) is similar.

2.3. The Flach map. Let X be as above. Note that the purity hypothesis required to deﬁne the Flach map on Xare automatic since F is perfect. For any irreducible correspondence α , X Y there are commutative diagrams → × m, m 1 α m, m 1 ∗ / E2 − − (X)0,Zl E2 − − (Y )0,Zl

σm σm 1 2m α 1 2m H F,H (X ¯ , Zl(m + 1)) ∗ / H F,H (Y ¯ , Zl(m + 1)) F F

m, m 1 α∗ / m, m 1 E2 − − (Y )0,Zl E2 − − (X)0,Zl

σm σm 1 2m α∗ 1 2m H F,H (Y ¯ , Zl(m + 1)) / H F,H (X ¯ , Zl(m + 1)) F F The commutativity of these diagrams follows immediately from the compatibility of the Flach map with pullback under flat morphisms and trace maps under finite, flat morphisms; see Section VI.7.

3. Composition of correspondences Let X,Y,Z be smooth proper varieties of dimension n over F . Let α , X Y and β , Y Z be irreducible correspondences. Under certain circumstances→ × we will deﬁne→ the× composition β α as a correspondence from X to Z. Begin by considering the◦ scheme-theoretic intersection Γ = (α Z) (X β) , X Y Z. × ∩ × → × × Let Γ1,..., Γr be the irreducible components of Γ. Each has dimension at least n; we will see in a moment that each in fact has dimension exactly n.

Lemma 3.1. Each irreducible component Γi is generically reduced. Proof. Let A and B be smooth open subsets of irreducible components of α Z and X β respectively. Further shrink A so that the projection A Y is× smooth; we× can do this by [GDb, Corollaries 6.12.5 and 17.15.2], using→ the fact that F is perfect. To prove the lemma it will suﬃce to show that A and B intersect transversally at all geometric points; see for example [Ful98, Section 8.2, esp. Remark 8.2] and [GDb, Deﬁnition 17.13.3 and Proposition 17.13.8]. Let c be a geometric point of A B. Since X Y Z is smooth of dimension ∩ × × 3n, the tangent space Tc(X Y Z) has dimension 3n over F¯ and has a canonical × × direct sum decomposition as TcX TcY TcZ. The tangent spaces TcA and TcB are both 2n-dimensional, since A and⊕ B are⊕ smooth of dimension 2n at c. Clearly by our construction we have canonical injections TcX, TcB and TcZ, TcA. → → For A and B to intersect non-transversally at c means precisely that TcA TcB ∩ has dimension greater than n. In particular, if this is the case then TcA must have non-trivial intersection with TcX. Since already TcZ injects into TcA, it follows that the projection TcA TcY is not surjective. Since the map A Y was → → 3. COMPOSITION OF CORRESPONDENCES 87 assumed to be smooth at c this contradicts [GDb, Theorem 17.11.1] and completes the proof.

Let γ , X Z be the scheme-theoretic image of Γ under the projection → × πXZ : X Y Z X Z and let γi be the scheme-theoretic image of Γi. Each γi is irreducible× × and→ generically× reduced by Lemma 3.1 and [GDc, Proposition 9.5.9].

Lemma 3.2. For each i, the projections γi X and γi Z are ﬁnite and surjective. → →

Proof. We first show that γ X is quasi-finite and surjective. Since everything in sight is finite type over a→ perfect field, in both cases it is enough to work on the level of geometric points; see [GDb, Proposition 9.3.2, Corollaries 10.4.8 and 13.1.4], although what we are using is really much easier. Let x X(F¯) be an arbitrary geometric point. Since the map α X is finite and surjective,∈ there is a → finite non-empty set of points (x, y1),..., (x, yd) in the fiber over x. Since β Y → is also finite and surjective, for each yi there is a finite non-empty set of points

(yi, zi1),..., (yi, ziei ) in the fiber over yi. Thus the fiber over x in Γ is precisely the finite non-empty set of points (x, yi, zij) and the fiber over x in γ consists of the points (x, zij). Thus γ X is quasi-finite and surjective. This also shows that Γ X and Γ γ are quasi-finite→ and surjective; in particular Γ has dimension n → → since X does. Since each Γi has dimension at least n, it follows that they all have dimension exactly n. By base change we also see that each Γi γi is quasi-finite and surjective. In → particular, each γi has dimension exactly n. γ is a closed subscheme of X Z and thus is proper over X. Since quasi-finite and proper imply finite [GDa, Proposition× 4.4.2], we conclude that the projection γ X is finite and surjective. → Now consider the projection γi X of an irreducible component of γ. This is → the composition of the closed immersion γi γ with the finite map γ X, and thus is finite. In particular, it is also proper,→ so the image is a closed subset→ of X. Since X is irreducible, if this image were not all of X, then it would have smaller dimension; since γi X is a finite map of schemes of the same dimension, this → is impossible. Thus γi X is surjective. (See also [GDb, Proposition 5.4.1(ii)].) → The proof for γi Z is identical. → Given all of this, we define the composition β α only under the additional assumption: ◦

The projections γi X and γi Z are ﬂat for all i; • → → With these hypotheses, we deﬁne γ = β α as ◦ m γ X i i

where mi = [k(Γi): k(γi)]. This makes sense since by Lemma 3.2 and the assumption above the maps γi X,Z are finite and faithfully flat. (The fact that the γi are generically reduced→ means that we need not introduce any multiplicities back on Γ.) If α1, . . . , αr and β1, . . . , βs are correspondences such that each composition βj αi is defined, we define the composition of αi and βj in the obvious way. ◦ P P 88 8. FLACH CLASSES FOR CORRESPONDENCES

4. Marked varieties Fix integers n, k, w such that k = nw. For any F -scheme X, deﬁne

k w X = Ω⊗ ; L ∧ X/F X is always to be considered as a Zariski sheaf, not an étalesheaf. The construc- L tion of X is functorial, in the sense that if there is a map f : X Y over Spec F , L → then there is an induced map f ∗ Y X of Zariski sheaves on X; this is immediate from the functoriality of sheavesL → of L differentials. If X is a smooth variety of dimension n over F , then X is an invertible sheaf by our choice of k and w. L Definition 4.1. A marking ωX on a smooth F -scheme X of dimension n is a non-zero rational section of X . That is, a marking is an equivalence class of pairs L of a dense open set U X and a non-zero section ω X (U). ⊆ ∈ L We should note that which sheaf we use here is not particularly important; one could replace X by any other functorial Zariski sheaf which is invertible on the smooth locus ofL F -schemes of dimension n. Now let X and Y be smooth proper varieties of dimension n over Spec F . Let ωX and ωY be markings on X and Y and let α , X Y be an irreducible correspondence from X to Y . We will use the markings→ on×X and Y to define a rational function fα = fα(ωX , ωY ) on α; we will always assume that ωX and ωY are fixed for the discussion and suppress them from the notation. The definition of fα is as follows: let UX and UY be open sets on which ωX and ωY are defined, respectively. Let V be an open subset of α contained in the 1 1 intersection of πX− (UX ), πY− (UY ) and the smooth locus of α; further shrink V so that α is free (necessarily of rank 1) over V . Since F is perfect, such V exist by L [GDb, Corollaries 6.12.5 and 17.15.2]. πX is flat and thus open, so πX (V ) and πY (V ) are open subsets of UX and UY respectively. Evaluating the map of sheaves πX∗ X α at V and composing with appropriate restriction maps we obtain a mapL → L

(4.1) X (UX ) α(V ). L → L We denote by πX∗ ωX the image of ωX under (4.1), viewed as a rational section of α. We deﬁne πY∗ ωY similarly. The rational function fα k(α)× is now simply theL ratio ∈

πX∗ ωX (4.2) k(α)×; πY∗ ωY ∈ this makes sense as α V is free of rank 1 over V , and it is clear that (4.2) is L | O independent of the choices of UX , UY and V . fα is non-zero and “not inﬁnite” since ωX , ωY are non-zero and πX , πY are ﬁnite and surjective. If α = miαi is a general correspondence, we use the markings on X and Y to associateP to α the rational function f on α given by f mi on α . α αi i n, n 1 We view the pair (α, fα) as an element of the spectral sequence E − − (X Y ); 1 × if α = m α is not irreducible then we view it as the element (α , f mi ) of i i i αi n, n 1P P E − − (X Y ) in the usual way. 1 × Definition 4.2. Let α be an algebraic correspondence from X to Y . We will say that α is admissible for the given markings ωX , ωY on X and Y if the Weil divisor of fα is trivial on α. 5. DIVISORS AND COMPOSITIONS 89

Here by the Weil divisor of fα we mean the sum of the Weil divisors on the irreducible component; in particular, we allow these to be non-trivial so long as they cancel each other out. A similar effect can occur if α is irreducible but singular. If α is an admissible correspondence, then by (VI.5.3) (α, fα) defines an element n, n 1 of E − − (X Y ). By Lemma VI.6.2 we can define a Flach class (depending also 2 × on ωX and ωY ) 1 2n (4.3) σX,Y (α) = σn(α, fα) H F,H (X ¯ Y ¯ , Ql(n + 1)) . ∈ F × F 2n+1 If H (X ¯ Y ¯ , Zl) is torsion-free, then we can even realize this class as F × F 1 2n (4.4) σX,Y (α) = σn(α, fα) H F,H (X ¯ Y ¯ , Zl(n + 1)) . ∈ F × F Note that in order to make this construction it does not seem to be enough to know α up to rational equivalence; this is why we are forced to use the somewhat restricted definition of correspondence which we are using.

5. Divisors and compositions

Let X,Y,Z be smooth proper varieties of dimension n over F . Let ωX , ωY , ωZ be markings on X,Y,Z (for some fixed k, w as in Section 4) and let α , X Y and β , Y Z be irreducible correspondences. Suppose also that the composition→ × → × γ = β α is defined as a correspondence from X to Z; let γ0 be an irreducible ◦ component. The markings determine rational functions fα, fβ, fγ0 on α, β, γ0 respectively. We want to relate the admissibility condition on α and β to that on γ0. For this result we will need to use pullbacks of divisors by finite, surjective maps. That is, given a finite, surjective map π : X Y and a codimension 1 cycle → Z on Y , we define π∗Z to be the cycle class (in the sense of [Ful98, Section 1.3]) 1 of π− Z = X Y Z. One checks easily from the fact that π is integral that every × 1 irreducible component of π− Z has codimension 1 in X. There is not a particularly good theory of such pullbacks (for example, they may not respect rational equivalence), but they do satisfy the following two properties which will be sufficient for our purposes: First, the composition π π∗ of the pullback with the proper pushforward is injective on the free abelian group∗ of codimension 1 cycles; indeed, it sends any cycle Z to a non-zero multiple of itself, from which this injectivity follows immediately. Second, if π0 : Y Y 0 is a finite, flat morphism, → then the finite surjective pullback (π0π)∗ is the same as the composition π∗π0∗; here π0∗ is the usual intersection theoretic pullback.

Lemma 5.1. With the above notation, suppose also that divα fα = divβ fβ = 0.

Then divγ0 fγ0 = 0.

Proof. Let Γ0 be the irreducible component of Γ = (α Z) (X β) mapping × ∩ × to γ0. Note that the projection πΓ X :Γ0 X factors through the map παX : 0 → α X. Similarly, the projection πΓ Y factors through both παY and πβY , and the → 0 projection πΓ0Z factors through πβZ . The statement that divα fα = 0 is precisely the statement that divα παX∗ ωX = divα παY∗ ωY . By compatibility of ﬁnite, ﬂat pullback with divisors, this is the same as the equality

(5.1) παX∗ divX ωX = παY∗ divY ωY . 90 8. FLACH CLASSES FOR CORRESPONDENCES

Pulling back (5.1) by the ﬁnite, surjective morphism πΓ0α yields π div ω = π div ω . Γ∗0X X X Γ∗0Y Y Y Using the same sort of argument for β, we conclude that π div ω = π div ω . Γ∗0X X X Γ∗0Z Z Z

Applying the proper pushforward πΓ0γ0 to this and using the functoriality of finite, surjective pullbacks with flat pullbacks,∗ we find that

(5.2) πΓ0γ0 πΓ∗ γ πγ∗ X divX ωX = πΓ0γ0 πΓ∗ γ πγ∗ Z divZ ωZ . ∗ 0 0 0 ∗ 0 0 0

Since πΓ0γ0 πΓ∗ γ is injective, we conclude from (5.2) that ∗ 0 0 π div ω = π div ω . γ∗0X X X γ∗0Z Z Z Compatibility of flat pullbacks with divisors now yields the desired equality. 6. The Leibniz relation We keep the notation of the previous section. Further assume that α and β are admissible; Lemma 5.1 insures that γ = β α is as well. Assuming that X, Y and Z are cohomologically torsion-free at l, we◦ can define Flach classes 1 2n σX,Y (α) H F,H (X ¯ Y ¯ , Zl(n + 1)) ;(6.1) ∈ F × F 1 2n σY,Z (β) H F,H (Y ¯ Z ¯ , Zl(n + 1)) ; ∈ F × F 1 2n σX,Z (γ) H F,H (X ¯ Z ¯ , Zl(n + 1)) ∈ F × F as in (4.4); even if the groups are not torsion-free, we can still define these classes after tensoring with Ql as in (4.3). These classes are related by the following beautiful formula of Mazur and Beilinson. We will first need some notation. Let ∆Z , Z Z be the diagonal viewed → × as an algebraic correspondence from Z to Z; both ∆ and ∆∗ are the identity ∗ map on K-theory and étalecohomology. View α ∆Z , X Y Z Z as a correspondence from X Z to Y Z; one checks immediately× → that× × it satisfies× the × × required hypotheses. Similarly, view ∆X β , X X Y Z as a correspondence from X Y to X Z. Recall that we can also× use→ maps× coming× × from correspondences to yield× maps on× Galois cohomology. We will consider the induced maps 1 2n 1 2n (α ∆Z )∗ : H F,H (Y ¯ Z ¯ , Zl(n + 1)) H F,H (X ¯ Z ¯ , Zl(n + 1)) × F × F → F × F 1 2n 1 2n (∆X β)∗ : H F,H (X ¯ Y ¯ , Zl(n + 1)) H F,H (X ¯ Z ¯ , Zl(n + 1)) . × F × F → F × F Theorem 6.1. Let α be a correspondence from X to Y and let β be a correspondence from Y to Z. Assume that γ = β α is defined as a correspondence from X to Z and that α and β are admissible for◦ our fixed choice of markings. If all of the integral Flach classes (6.1) are defined then

(6.2) σX,Z (γ) = (α ∆Z )∗σY,Z (β) + (∆X β) σX,Y (α). × × ∗ If the integral Flach classes are not deﬁned, then this formula still holds after tensoring with Ql as in (4.3). Proof. By linearity we can assume that α and β are irreducible. We ﬁrst prove the formula on the level of algebraic cycles and K-theory. That is, we wish n, n 1 to show that in E − − (X Z) we have the equality 1 × (6.3) (γ, fγ ) = (α ∆Z )∗(β, fβ) + (∆X β) (α, fα). × × ∗ 6. THE LEIBNIZ RELATION 91

Consider first (α ∆Z )∗(β, fβ). The “cycle” part of this is obtained as follows: one pulls back and pushes× forward β , Y Z in the diagram → × α ∆Z × K ss KKK sss KK ss KKK sy ss K% Y ZX Z × × Let β0 be the image of β under the map id ∆ : Y Z Y Z Z. Pulling back × × → × × β to α ∆Z is the same as forming the scheme-theoretic intersection × (6.4) (X β0) (α ∆Z ) , X Y Z Z. × ∩ × → × × × The projection from here to X Z factors through X Y Z; here the image of (6.4) is just the intersection of ×X β and α Z. In particular,× × by our definition of composition of correspondences× the final image× of β in X Z is nothing other than β α = γ. × ◦ Since fβ is πβY∗ ωY /πβZ∗ ωZ , tracing through the maps we see that the corresponding rational function on an irreducible component γi of γ is N π ω k(Γi)/k(γi) Γ∗iY Y π ωmi γ∗iZ Z where Γi is the irreducible component of Γ surjecting onto γi and mi = [k(Γi): k(γi)]. That is, writing γ = miγi as a sum of irreducible correspondences, we have P N π ω k(Γi)/k(γi) Γ∗iY Y (6.5) (α ∆Z )∗(β, fβ) = γi, ! . X π ωmi × γ∗iZ Z Similarly, we have π ωmi γ∗iX X (6.6) (∆X β) (α, fα) = γi, ! . ∗ X N π ω × k(Γi)/k(γi) γ∗iY Y n, n 1 Adding (6.5) and (6.6) in E − − (X Z) yields 1 × m π ω i γ∗iX X (α ∆Z )∗(β, fβ) + (∆X β) (α, fα) = γi, ! ! ∗ X π ω × × γ∗iZ Z

which is precisely the element (γ, fγ ). n, n 1 Since α, β and γ are all admissible, the equality (6.3) in E1 − − (X Z) yields n, n 1 × the same equality in E2 − − (X Z). The fact that (6.2) holds in étalecohomology now follows immediately from the× compatibility of the Flach map with maps coming from correspondences as in Section 2. Assume now that X, Y and Z are all cohomologically torsion-free at l. In this situation we have Künnethprojections on étalecohomology as in Section 2. Let

σX,Y0 (α) denote the image of σX,Y (α) under the map 1 2n 1 n n H F,H (X ¯ Y ¯ , Zl(n + 1)) H F,H (X ¯ , Zl) Z H (Y ¯ , Zl)(n + 1) F × F → F ⊗ l F induced by the Künnethprojection; we define σY,Z0 (β) and σX,Z0 (γ) similarly. By the compatibility of correspondences with Künnethprojections, we see that (6.2) 92 8. FLACH CLASSES FOR CORRESPONDENCES now takes the form

(6.7) σX,Z0 (γ) = (α∗ 1)σY,Z0 (β) + (1 β )σX,Y0 (α). ⊗ ⊗ ∗

We are again using α∗ and β to induce maps on Galois cohomology: ∗ n n α∗ : H (Y ¯ , Zl) H (X ¯ , Zl) F → F n n β : H (YF¯ , Zl) H (ZF¯ , Zl). ∗ → As usual, we can obtain analogous results after tensoring by Ql even if the varieties are not all cohomologically torsion-free.

7. Algebras of correspondences Let X be a smooth proper variety of dimension n over F . By an algebra of correspondences on X we will mean a set of correspondences from X to X which forms a (possibly infinitely generated andA non-commutative) Z-algebra with composition of correspondences as multiplication. In particular, it is assumed that every composition of elements of is defined. We assume that ∆X lies in ; it serves as a multiplicative identityA element. A We say that a marking ωX is admissible for 0 if every α 0 is admissible A ∈ A for ωX . Note that to check that an algebra is admissible for a given marking, by Lemma 5.1 it suffices to check on a set of algebraA generators of . A Now let 0 be an algebra of correspondences on X and let = 0 Z Zl for some fixedA prime l; is a (possibly infinitely generated and non-commutative)A A ⊗ A m Zl-algebra. For any fixed m, admits two maps to EndZ H (X ¯ , Zl), one given A l F by α α and one given by α α∗. 7→ ∗ 7→ Let ωX be an admissible marking for 0. Assume also that X is cohomologically A torsion-free at l. We write the map σX,X of (4.3) as σ; we consider it as a map

n, n 1 1 2n σ : E − − (X X) Z Zl H F,H (X ¯ X ¯ , Zl(n + 1)) A → 2 × ⊗ → F × F sending α to σX,X (α, fα). We now apply the K¨unnethanalysis at the end of the n previous section. Speciﬁcally, let V = H (XF¯ , Zl) and let

1 τ : H F,V Z V (n + 1) A → ⊗ l denote the composition of σ with the map

1 2n H F,H (X ¯ X ¯ , Zl(n + 1)) F × F → 1 n n H F,H (X ¯ , Zl) Z H (X ¯ , Zl)(n + 1) F ⊗ l F coming from the K¨unnethprojection. The Leibniz relation (6.7) takes the form

(7.1) τ(βα) = (α∗ 1)τ(β) + (1 β )τ(α). ⊗ ⊗ ∗

As always we can obtain the same formula over Ql without the cohomologically torsion-free hypothesis. For the remainder of the chapter we will assume that X is cohomologically torsion-free; however all results remain true over Ql even without this hypothesis. We will not comment on this further. 8. DERIVATIONS IN THE SELF-ADJOINT CASE 93

8. Derivations in the self-adjoint case We keep the hypotheses of the previous section: X is a smooth proper variety of dimension n over F and is a Zl-algebra of self-correspondences on X with an A admissible marking ωX . We assume that X is cohomologically torsion-free at l. Set n V = H (XF¯ , Zl); we have a map

1 τ : H F,V Z V (n + 1) . A → ⊗ l

The maps α α and α α∗ yield two maps EndZl V . Let B and → ∗ → A → ∗ B∗ denote their images; they are finite, flat Zl-algebras since EndZl V is. For this section we make the following assumptions: is commutative; •A is self-adjoint in the sense that the two maps EndZ V coincide; •A A → l None of these assumptions will actually be used in this section, but if they are not satisfied then the constructions here are not appropriate. We will discuss the elimination of the self-adjoint hypothesis in later sections. For now, we write B for the image of in EndZl V ; acts on V in a canonical way via B. In this situation,A the functionalA equation (7.1) for the map τ can be unambigu- ously written as τ(βα) = (α 1)τ(β) + (1 β)τ(α). ⊗ ⊗ That is, τ is a bilateral derivation in the sense of Section A.6. We wish to pass from the bilateral derivation τ to bilateral derivations and derivations to the Galois cohomology of certain quotients of V Zl V (n + 1). In the self-adjoint case, this is straightforward. Let m be a maximal⊗ ideal of B and let A denote the completion of B at m. A is a finite, flat, local Zl-algebra and is canonically a direct summand of B. H = V B A is therefore canonically a direct ⊗ summand of V ; let i : H, V and j : V H denote the corresponding maps. We define a bilateral→ derivation

1 : H F,H Z H(n + 1) D A → ⊗ l as the composition of τ with the map on cohomology induced by j j. We deﬁne a map ⊗ 1 ∂ : H F,H A H(n + 1) A → ⊗ as the composition of with the map on cohomology induced by the natural surjection D

H Z H(n + 1) H A H(n + 1). ⊗ l ⊗ Since satisﬁes D (βα) = (α 1) (β) + (1 β) (α), D ⊗ D ⊗ D 1 and the A Z A action on H (F,H A H(n+1)) factors through the diagonal map ⊗ l ⊗ A Z A A, we see that ∂ satisﬁes ⊗ l → ∂(βα) = α∂(β) + β∂(α); that is, ∂ is a derivation. 94 8. FLACH CLASSES FOR CORRESPONDENCES

9. Local diagrams in the self-adjoint case In the applications of our constructions it is often more convenient the cohomology of EndA H(1) than H A H(n + 1). In this section we explain how to make the transition; it is also useful⊗ for computational purposes. For the remainder of this chapter, for any Zl-module M we denote by M † its integral Pontrjagin dual HomZ (M, Zl). If ϕ : M Z N Zl is any pairing, we l ⊗ l → write ϕr : N M † for the induced map. Central to→ the transition are various pairings induced by Poincar´eduality. The basic Poincar´epairing is a Galois equivariant, perfect pairing

ϕ : V Z V (n) Zl. ⊗ l → Since is self-adjoint, ϕ satisﬁes (see Section 2) ϕ(bv, v0) = ϕ(v, bv0) for all b B, A ∈ v V and v0 V (n); that is, ϕ is B-hermitian. ∈ Let m be∈ a maximal ideal of B as before, and deﬁne a pairing

ψ : H Z H(n) Zl ⊗ l → by ψ(h, h0) = ϕ(ih, ih0). ψ is an A-hermitian, Galois equivariant perfect pairing. (The fact that ψ is perfect is an easy computation using properties of localization.) We have a commutative diagram

id ϕr ⊗ (9.1) V Z V (n) / V Z V † / EndZl V ⊗ l ⊗ l j j j i f jfi ⊗ † 7→ ⊗ id ψr ⊗ H Z H(n) / H Z H† / EndZl H ⊗ l ⊗ l (To show that (9.1) commutes requires the fact that ϕ(ijh, iv) = ϕ(ijh, v) for all h H and v V (n); this follows from the fact that both ϕ and ψ are perfect.) All∈ of the maps∈ of (9.1) are Galois equivariant and B-linear. We now introduce the sort of maximal ideals of B which we can use to make the desired translation. Definition 9.1. A maximal ideal m of B is said to be dualizing if

Bm is reduced; • Vm is free of rank 2 over Bm. • By Lemma B.4.1 these conditions imply that A = Bm is a Gorenstein Zl- algebra. Fix a Gorenstein trace tr : A Zl; by Lemma B.3.1 this choice induces → an isomorphism H† ∼= HomA(H,A). Furthermore, by Lemma B.4.2 there exists a unique A-linear, Galois equivariant perfect pairing ψ0 : H A H(n) A such that ψ factors as ⊗ →

ψ0 tr H Z H(n) H A H(n) A Zl. ⊗ l −→ ⊗ −→ −→ We use these trace identiﬁcations to extend (9.1) to

id ψr ⊗ (9.2) H Z H(n) / H Z H† / EndZl H ⊗ l ⊗ l

id ψ0 ⊗ r H A H(n) / H A HomA(H,A) / EndA H ⊗ ⊗ 10. DERIVATIONS IN THE GENERAL CASE 95

Recall that the map EndZ H EndA H has an especially simple description on l → the submodule EndA H of EndZl H; see Lemma B.3.3. In any event, we can deﬁne the desired derivation 1 ∂0 : H F, EndA H(1) A → as the composition of ∂ with the isomorphism from H(n + 1) A H to EndA H(1) coming from the bottom row of (9.2). Note that this isomorphism⊗ depends on the choice of Gorenstein trace tr, and thus is canonical only up to an element in A×.

10. Derivations in the general case In this section we carry out the construction of the previous two sections without the self-adjoint hypothesis. Otherwise we continue with the hypotheses of

Section 8. We again define B and B∗ as the images of in EndZl V . V has a ∗ A canonical module structure over B and B∗, and the Poincarépairing ∗ ϕ : V Z V (n) Zl ⊗ l → now satisfies

ϕ(α v, v0) = ϕ(v, α∗v0); ∗ ϕ(α∗v, v0) = ϕ(v, α v0). ∗ We will need to modify ϕ to obtain a B∗-hermitian pairing. We note in passing that the constructions of these sections can be carried out n with V replaced by a direct summand of H (XF¯ , Zl) which is stable under both actions of 0 and which is self-dual under Poincar´eduality. We will not comment further onA this. Definition 10.1. An untwisting of V (with respect to ) is a triple (w, B,˜ ξ) A of an isomorphism of abelian groups w : V V satisfying w(α v) = α∗w(v) and → ˜ ∗ w(α∗v) = α w(v); a free B∗-module of rank 1 B with a B∗-linear action of GF ; and a chosen∗ generator ξ of B˜ such that the map

ξ w : V B˜ B V ⊗ → ⊗ ∗ is Galois equivariant. Note that the notion of untwisting is actually independent of the choice of generator ξ. We include ξ in the notation for simplicity, although our final constructions will not depend on it. Fix an untwisting (w, B,˜ ξ) and set V˜ = B˜ B V . We define a pairing ⊗ ∗ ϕ0 : V Z V˜ (n) Zl ⊗ l → 1 by ϕ0(v, ξ v0) = ϕ(v, w− v0). ϕ0 is B∗-hermitian and Galois equivariant by the definition of⊗ an untwisting. Define 1 0 : H F,V Z V˜ (n + 1) D A → ⊗ l to be the composition of τ with the map on cohomology induced by id ξ w. We ⊗ ⊗ claim that 0 can be regarded as a bilateral derivation. To check this, let α, β D ∈ A and γ GF be any elements, and write ∈ τ(α)(γ) = t0 ti X i ⊗ τ(β)(γ) = u0 ui X i ⊗ 96 8. FLACH CLASSES FOR CORRESPONDENCES be the evaluation of the cocycles at γ. We compute

0(βα)(γ) = (id w)τ(βα)(γ) D ⊗ = (id w) (α∗ 1)τ(β) + (1 β )τ(α) ⊗ ⊗ ⊗ ∗ = (id w) α∗ui0 ui + ti0 β ti ⊗ X ⊗ X ⊗ ∗

= α∗ui0 wui + ti0 wβ ti X ⊗ X ⊗ ∗ = α∗u0 wui + t0 β∗wti X i ⊗ X i ⊗ = (α∗ 1) 0(β) + (1 β∗) 0(α). ⊗ D ⊗ D

Thus 0 is indeed a bilateral derivation when V and V˜ are given -module struc- D A tures via B∗. ˜ ˜ Now choose a maximal ideal m of B∗. Let A = Bm∗ and set A = B B∗ A; we ˜ ˜ ˜ ⊗ will also write ξ for the image of ξ in A. Set H = V B∗ A and H = A A H. We have natural maps i : H, V and j : V H. We deﬁne⊗ a pairing ⊗ →

ψ : H Z H˜ (n) Zl ⊗ l → by

1 ψ(h, ξ h0) = ϕ0(ih, ξ ih0) = ϕ(ih, w− ih0). ⊗ ⊗ ψ is A-hermitian and Galois equivariant.

Definition 10.2. A maximal ideal m of B∗ is said to be dualizing if

Bm is reduced; • Vm is free of rank 2 over Bm. • Fix a dualizing maximal ideal m. By Lemma B.4.1 A = Bm is Gorenstein. Let tr : A Zl be a choice of Gorenstein trace and let →

ψ0 : H A H˜ (n) A ⊗ → be the A-linear, Galois equivariant perfect pairing induced by ψ. We deﬁne the -bilateral derivation A 1 : H F,H Z H˜ (n + 1) D A → ⊗ l to be the composition of 0 with the map induced by j j. We deﬁne the - derivation D ⊗ A

1 ∂ : H F,H A H˜ (n + 1) A → ⊗ to be the composition of with the map on cohomology induced by the surjection D H Z H˜ (n + 1) H A H˜ (n + 1). We regard H and H˜ as -modules via the map ⊗ l ⊗ A B∗ A. A → → 12. DERIVATIONS MODULO η 97

We can use the following diagram to pass from our constructions above to the Galois cohomology of EndA H(1):

id ϕr ⊗ (10.1) V Z V (n) / V Z V † / EndZl V ⊗ l ⊗ l id ξ w id id ⊗ ⊗ id ϕ0 ˜ ⊗ r / / V Z V (n) V Z V † EndZl V ⊗ l ⊗ l

j j j i† f jfi ⊗ ⊗ 7→ id ψr ˜ ⊗ / / H Z H(n) H Z H† EndZl H ⊗ l ⊗ l

id ψ0 ˜ ⊗ r H A H(n) / H A H† / EndA H ⊗ ⊗ All of these maps are Galois equivariant. The maps are B∗-linear except for id ϕr ⊗ and id ξ w, both of which interchange the action of B and B∗. ⊗ ⊗ ∗ 11. Untwistings and cycle classes It will be useful to understand the behavior of certain cycle classes under the top row of (10.1). Let f : X X be a morphism and let Γf be the graph of f in X X: that is, it is the scheme-theoretic→ image of the morphism × id f : X X X. × → × By [FK88, pp. 155–156] the image of the cycle class 2n s(Γf ) H X ¯ X ¯ , Zl(n) ∈ F × F under the maps

2n id ϕr (11.1) H X ¯ X ¯ , Zl(n) V Z V (n) ⊗ V Z V † EndZ V F × F ⊗ l −→ ⊗ l → l is nothing other than the endomorphism f ∗ of V . # We denote by Γf the scheme-theoretic image of f id : X X X. × → × # Again by [FK88, pp. 155–156] the image of s(Γf ) under (11.1) is the Poincar´ead- adj joint f ∗ of f ∗. It is characterized by the equality adj ϕ(f ∗ v, v0) = ϕ(v, f ∗v0) for all v, v0.

12. Derivations modulo η We return now to the notation and hypotheses of Section 10; in particular we assume that we have an untwisting w and a dualizing maximal ideal m. Fix also a Gorenstein trace tr : A Zl. We have a bilateral derivation → 1 : H F,H Z H˜ (n + 1) D A → ⊗ l and a derivation 1 ∂ : H F,H A H˜ (n + 1) A → ⊗ determined by this collection of data. 98 8. FLACH CLASSES FOR CORRESPONDENCES

Let I be the kernel of the surjection A. By Lemma B.6.1, and ∂ induce -module homomorphisms A → D A 2 1 ˜ : I/I H F,H Z H˜ (n + 1) D → ⊗ l δ ˜ 2 1 ∂ : I/I H F,H A H˜ (n + 1) . → ⊗ By Lemma B.6.2, our choice of Gorenstein trace yields an A-linear Galois equivariant isomorphism (H Z H)δ = H A H ⊗ l ∼ ⊗ ﬁtting into a commutative diagram (12.1) (H Z H)δ / H Z H ⊗ l ⊗ l

' η H A H / H A H ⊗ ⊗ Here η is the congruence element for tr. If we assume that every Jordan-Holder factor of H˜ (n + 1) A H (as a GF -module) has no GF -invariants, then combining ⊗ this with Lemma B.6.3 we can view ˜ as a map D 2 1 0 : I/I H F,H A H˜ (n + 1) , D → ⊗ ˜ which by (12.1) satisﬁes η 0 = ∂. The following proposition is an immediate consequence. D

Proposition 12.1. Let W = H A H˜ (n + 1). There exists an A-derivation Θ: A H1(F, W/ηW ) fitting into a commutative⊗ diagram → 0 / I / / A / 0 A ˜ ∂ Θ D η H1(F,W ) / H1(F,W ) / H1(F, W/ηW ) Proof. The bottom exact sequence is part of the long exact sequence in cohomology associated to the short exact sequence η 0 H˜ A H H˜ A H H˜ A H/η 0. → ⊗ −→ ⊗ −→ ⊗ → Note that the first map is injective since η is a non-zero divisor by Lemma B.2.2 and the definition of dualizing. The commutativity of the first square is the relationship η ˜ = ∂˜, and the map Θ is the induced map on cokernels. D Of course, we can use the identifications of (10.1) to regard Θ as a derivation 1 Θ: A H F, EndA H/ηH(1) . → CHAPTER 9

Construction of geometric Euler systems

In this chapter we combine the results of Chapters VII and VIII to give geometric conditions for the existence of geometric Euler systems.

1. Divisorial liftings of cycles Let F be a global field and let S be an open subscheme of the spectrum of the ring of integers of F . Let X be a smooth proper S-scheme of relative dimension n and let X be the generic fiber of X. For v a closed point of S, we will often need to consider liftings of cycles on the special fiber Xkv up to X. The relevant notion of lifting is the following.

Definition 1.1. Let Z be a codimension m cycle on Xkv . We say that a ﬁnite set (Zi, fi) of pairs of codimension m cycles Zi on X and rational functions fi { } on Zi is a divisorial lifting of Z if

divZ f = Z; P i here Zi is the closure of Zi in X and Z is considered as a vertical cycle on X.

The first step in constructing a divisorial lifting of a cycle Z is to find a cycle Z0 on X which has Z as an irreducible component over kv. This can be done by using a complete intersection containing Z as an irreducible component; any complete intersection can easily be lifted to X. It is much harder to find a rational function with trivial divisor on Z0 which separates out Z over kv. (Although if Z itself is a complete intersection one can use the methods of Lemma 1.2 for this.)

Note that by definition a divisorial lifting (Zi, fi) of Z in Xkv has no divisor on any fibers of X S other than the fiber overP v. In particular, it has no divisor → m, m 1 on X, so that (Zi, fi) defines an element of E2 − − (X) by (VI.5.3). Thus a P m, m 1 divisorial lifting of Z yields an element (Zi, fi) of E − − (X) such that P 2 0 w = v; divw (Zi, fi) = ( 6 P Z w = v.

Here divw is the map denoted divkw in Section VII.1. The simplest example of divisorial liftings are given by the following lemma. Lemma 1.2. Let Z be a codimension m cycle on X with closure Z on X. Let v be a closed point of S and let p be the corresponding prime of F . Let Zv be the O special ﬁber of Z at v. Then hZv admits a divisorial lifting to X, where h is the order of p in the ideal class group of F,S. O Proof. By deﬁnition of the ideal class group, the ideal ph is principal; let π be a generator. Then π is a regular function on S, non-vanishing away from v and vanishing to order h at v. The pair (Z, π) is thus a divisorial lifting of hZv.

99 100 9. CONSTRUCTION OF GEOMETRIC EULER SYSTEMS

Of course, it is unreasonable to expect all cycles on special fibers of X to admit liftings as in Lemma 1.2; this is why we introduced the more general notion of divisorial liftings. Divisorial liftings are designed to give useful elements for Theorem VII.1.1. We will also need to consider the more subtle conditions at certain bad places. Assume 2m for this that F is a number field. Fix a prime l and set V = H (XF¯ , Zl(m + 1)). Let T be a torsion-free l-adic GF -module over Zl equipped with a map V T . We will consider the composition of the Flach map with the projection V →T : → m, m 1 1 σ : E − − (X)0,Z H (F,T ). 2 l → Definition 1.3. Let K be the completion of F at a place v above l. An m, m 1 element (Zi, fi) of E2 − − (X)0,Zl is said to be cohomologically deRham (resp. P 1 cohomologically crystalline) at v for T if σ( (Zi, fi)) lies in Hf (K,T ) for the deRham (resp. crystalline) local finite/singularP structure on T . We have the following sufficient conditions for a pair (Z, f) to be cohomologically deRham or crystalline. Recall that we have a cycle class map s : AmX V ( 1) T ( 1). → − → − Lemma 1.4. Let K be the completion of F at a place v above l and let Zi { } be codimension m cycles on X. If s(Zi) vanishes for each i, then each element of m, m 1 E2 − − (X)0,Zl of the form (Zi, fi) (for any rational functions fi on the Zi) is cohomologically deRham. P Let R denote the ring of integers of K and let Zi denote the closure of Zi in XR. If it is further possible to realize each XR Zi as the complement of a normal crossings divisor in a smooth proper variety over− R (for example, by embedded resolution of singularities of the Zi), then each (Zi, fi) is cohomologically crystalline as well. P Proof. This follows immediately from Proposition VII.10.1 and the definitions of cohomologically deRham and crystalline.

2. Construction of partial Euler systems In this section we describe the geometric data required to use the Flach map to construct partial geometric Euler systems. Let F , S and X be as before; we once again allow F to have positive characteristic. Fix an integer m and a prime 2m l and let V denote H (XFs , Zl(m + 1)). (If F is a function field we assume that l n + 2; we will need this in order to invoke Theorem VII.9.1.) Fix a Zl-algebra ≥ A of scalars and let T be a torsion-free l-adic GF -module over A equipped with a map V T such that the image of V has finite index in T . V itself need not have any structure→ of A-module; we do however assume that it is torsion-free, so that by Lemma VI.6.2 we have a Flach map m, m 1 1 σm : E − − (X) H (F,V ). 2 → We let m, m 1 1 σ : E − − (X) H (F,T ) 2 → denote the composition of σm with the map on cohomology induced by V T . → V is unramified at all places of S Σl by smooth base change. Since the image of V has finite index in T and T is torsion-free,− it follows that T is also unramified 3. PARTIAL EULER SYSTEMS ON PRODUCTS 101 at all places of S Σl. We let V and T have the unramified finite/singular structure at all of these places.− (We will worry about the structures at the other places later.) We wish to consider the cycle class map G m 2m kv Gk Gk (2.1) s : A Xk H X¯ , Zl(m) = V ( 1) v T ( 1) v . v → kv ∼ − → − Definition 2.1. Let v be a closed point of S Σl and let η be an element of A. − We will say that a collection of codimension m cycles Z1,...,Zr on Xkv generate G T with depth η if the A-submodule of T ( 1) kv generated by the s(Zi) contains G − ηT ( 1) kv . − Lemma 2.2. Let Z1,...,Zr be cycles on Xkv which generate T with depth η. Assume that each of the Zi admit divisorial liftings to X. Then there is an A- 1 submodule C of H (F,T ) such that Cw,s = 0 for w S Σl distinct from v and 1 ∈ − such that Cv,s has depth η in Hs (Fv,T ).

Proof. Let (Zij, fj) be a divisorial lifting of Zi and define C to be the A- 1P submodule of H (F,T ) generated by the σ( (Zij, fj)) for all i, j. We will check that this C satisfies the conditions of the lemma.P Let w be a closed point of S Σl. If w = v, then the divisor of (Zij, fj) − 6 P vanishes on Xkw by the definition of a divisorial lifting; thus by Theorem VII.9.1 1 1 σ( (Zij, fj)) vanishes in H (Fw,V ). It therefore vanishes in H (Fw,T ) as well, P s s which shows that Cw,s = 0. 1 Applying Theorem VII.9.1 at v shows that Cv,s Hs (Fv,T ) is generated by 1 ⊆ Gk the s(Zi), where we have identified Hs (Fv,T ) and T ( 1) v . Since the s(Zi) G − are assumed to fill up ηT ( 1) kv , we see that Cv,s does indeed have depth η in 1 − Hs (Fv,T ), as claimed. We will consider three different choices of finite/singular structure on T . Let w (resp. d, resp. c) denote the finite/singular structure on T which is weak away S S S from S, unramified at S Σl and weak (resp. deRham, resp. crystalline) at Σl S. − ∩ Theorem 2.3. Let be a set of closed points of S Σl. Assume that for each L − v there is a set of codimension m cycles of Xkv which generate T with depth η ∈and L which admit divisorial liftings to X. Then there is a partial Euler system v C v of depth η for T with the structure w. If further the divisorial liftings are all{ cohomologically} ∈L deRham (resp. cohomologicallyS crystalline) then this is a partial Euler system for the structure d (resp. c) as well. S S Proof. This is immediate from Lemma 2.2 and the definitions of cohomologically deRham and crystalline. One can combine Theorem 2.3 with Corollary III.3.2 to obtain annihilation results for the Selmer groups of T ∗; we do not give a precise statement as it becomes notationally quite unpleasant.

3. Partial Euler systems on products In this section we give the simplest method for the construction of geometric Euler systems for l-adic GF -modules of endomorphisms. Let F , S and X be as before. Fix a prime l such that X is cohomologically torsion-free at l (and such that 2m l n+2 if F is a function field) and some m n. Set V = H (XFs , Zl(m+1)). Let d≥denote the rank of V as a Z -module. Let≤T be the l-adic G -module End0 V (1) l F Zl over Zl. Let w, d and c denote the finite/singular structures on T analogous to S S S 102 9. CONSTRUCTION OF GEOMETRIC EULER SYSTEMS those in the previous section; as in Chapter IV, control of the Selmer groups of T ∗ has implications for the deformation theory of V . As in Section VIII.9, our assumption on the cohomology of X yields a canonical map 2n (3.1) H XF XF , Zl(n + 1) V Z V (n + 1) = EndZ V (1) T s × s → ⊗ l ∼ l via the Künnethprojection and Poincaréduality. We will produce an Euler system for T from the geometry of XFs XFs . We first need to specify at which places to form our Euler system. ×

Definition 3.1. A d d matrix τ over Fl is said to be of general type if: × The characteristic polynomial and the minimal polynomial of τ coincide; • dimF M MnFl Mτ = τM = d. • l { ∈ | } One checks easily that τ is of general type if τ has distinct eigenvalues. th We write Γv,i for the graph in Xk Xk of the i power of Frobenius on Xk ; v × v v Γv,0 is nothing other than the diagonal correspondence. Lemma 3.2. Let τ be a matrix of general type. Let v be a place of F such that Fr(v) acts on V/lV Fd as a conjugate of τ. Then the cycles Γ ,..., Γ ∼= l v,1 v,d 1 generate T via (3.1). −

Proof. Fix an identiﬁcation of EndFl V/lV with MnFl such that Fr(v) cor- Gkv responds to τ. Then (EndFl V/lV ) identiﬁes with the set of elements of MnFl 2 d 1 which commute with τ. Since τ is of general type, the matrices id, τ, τ , . . . , τ − generate (as an Fl-vector space) the subspace of MnFl of matrices which commute Gkv with τ. We conclude that (EndFl V/lV ) is generated (as Fl-vector space) by 2 d 1 id, Fr(v), Fr(v) ,..., Fr(v) − . d 1 The identity matrix corresponds to the scalars, so Fr(v),..., Fr(v) − generate 0 G d 1 (End V/lV ) kv . By Nakayama’s lemma we conclude that Fr(v),..., Fr(v) Fl − 0 G generate (End V ) kv , which is equivalent to the statement of the lemma since Zl the cycle class of Γ in End0 V is just Fr(v)i by Section VIII.11 and [FK88, v,i Zl Chapter II, Section 4].

Of course, if τ is not in the image of GF EndZl V , then there are no places as in Lemma 3.2. →

Theorem 3.3. Let τ be a d d matrix over Fl of general type and let denote × L the set of places of S Σl with Frobenius conjugate to τ on V/lV . Suppose that − there is an integer η such that for each v , the cycles ηΓv,1, . . . , ηΓv,d 1 admit ∈ L v − divisorial liftings to X. Then there is a partial Euler system C v for T of depth { } ∈L η with the structure w. If these liftings are also cohomologically deRham (resp. S cohomologically crystalline) then the Euler system is for d (resp. c) as well. S S Proof. This follows immediately from Lemma 3.2 and Theorem 2.3. 4. Construction of Flach systems in the self-adjoint case In this section we will refine the results of the previous section, via the methods of Sections VIII.8 and VIII.9, to produce Flach systems. Let F be a number field with at least one real embedding and such that Fv is absolutely unramified for every v Σl. Let S be an open subscheme of the spectrum of the ring of integers of F . Let∈ X be a smooth proper S-scheme of relative dimension n with generic fiber X. 4. CONSTRUCTION OF FLACH SYSTEMS IN THE SELF-ADJOINT CASE 103

We assume that n is odd. (This assumption is necessary to insure that complex conjugation will act as a non-scalar; to consider the case of even dimension one needs to use the methods of the non-self-adjoint case.) Fix a choice τ of complex conjugation for F . Fix a prime l such that S contains the set Σl of places of F above l. Set n V = H (XF¯ , Zl). We assume that X is cohomologically torsion-free at l. Let be a commutative l-adic algebra of correspondences on X. We assume for A this section that is self-adjoint, and we let B denote the image of in EndZ V . A A l Assume also that we have a dualizing maximal ideal m of B; set A = Bm and 0 H = V B A. Set T = EndA H(1). We consider H and T as l-adic GF -modules over A. ⊗ Let k denote the residue field of A. Since m is dualizing, A is Gorenstein and H is free of rank 2 over A. We fix a Gorenstein trace tr : A Zl; let η A be the associated congruence element. → ∈ We need to check that H is a Galois representation of Taylor-Wiles type; we also need to check the conditions of Section IV.4 required to discuss the existence of a Flach system. Note that H is unramified away from the set Σ consisting of Σl and the places of F not in S. H is also crystalline at every place of v by [Fal89] and [FM87]. The pairing required in the definition is simply the pairing ψ of Section n VIII.9. It follows that the determinant of H is ε− . Since n is odd, this is indeed an odd character. We assume also the following conditions:

(1) For every v Σ Σl, H A k is minimally ramified at v and the minimally ramified structure∈ − at v agrees⊗ with the weak structure; (2) H A k and T A k are absolutely irreducible over k; 1⊗ ⊗ (3) H (F (T ∗[a])/F, T ∗[a]) = 0 for every ideal a of finite index in A; (4) A is generated by the Hecke operators Tv for v / Σl; ∈ (5) H is crystalline of weight k > l for each v Σl for every v Σl. ∈ ∈ Recall that Tv is defined as the trace of Fr(v) acting on H. Note that by Lemma I.5.2 the first assumption is satisfied in the case of ordinary representations. Let denote S the finite/singular structure on T which is minimally ramified away from Σl and crystalline at Σl. # For a place v S Σl, let Γv denote the graph of Fr on Xk . Let Γ denote its ∈ − v v transpose. Let = τ denote the set of non-archimedean places of F which have L L Frobenius conjugate to τ on H A k. ⊗ Lemma 4.1. Fix a place v and let a, b be integers such that l does not # ∈ L divide a b. Then aΓv + bΓv generates T with depth η (via the cycle class map (3.1)). −

Gk 0 Gk Proof. Recall that by Lemma IV.3.2, T ( 1) v = (EndA H) v is a free rank − 1 0 one A-module which is generated by the matrix 0 1 . − # We must compute the image of the cycle class of aΓv + bΓv under the map 2n H X¯ X¯ , Zl(n) V Z V (n) = EndZ V EndZ H EndA H, kv × kv ⊗ l ∼ l l → of (VIII.9.1) and (VIII.9.2). The discussion of Section VIII.11 shows that the image of s(Γv) in EndZ V is the morphism Fr∗ of V ; here Fr : X¯ X¯ is the base l kv → kv change of the Frobenius morphism of Xkv . By [FK88, Chapter II, Section 4], Fr∗ is nothing other than the geometric Frobenius automorphism Fr(v) of V . By

(VIII.9.1) this maps to the Frobenius automorphism of H in EndZl H. Since Fr(v) 104 9. CONSTRUCTION OF GEOMETRIC EULER SYSTEMS is A-linear, (VIII.9.2) and Lemma B.3.3 show that this ﬁnally maps to η times the Frobenius morphism in EndA H:

(4.1) s(Γv) η Fr(v) EndA H. 7→ ∈ # adj Again by Section VIII.11, Γv maps to the Poincar´eadjoint Fr(v) of Frobenius on V . Viewing the Poincar´epairing ϕ as a Zl-linear, Galois equivariant pairing V Z V Zl( n), we can compute this as follows: ⊗ l → − 1 ϕ v, Fr(v)v0 = Fr(v)ϕ Fr(v)− v, v0 n 1 = ε(v)− ϕ Fr(v)− v, v0 n 1 = ϕ ε(v)− Fr(v)− v, v0 . adj n 1 Thus Fr(v) = ε(v)− Fr(v)− . The same analysis as for (4.1) now shows that:

# n 1 (4.2) s(Γ ) ηε(v)− Fr(v)− EndA H. v 7→ ∈ By Lemma IV.3.1 we can choose a basis of H with respect to which Fr(v) is given by a matrix α 0 . Since H has determinant ε n, we have αβ = ε(v) n. 0 β − − Thus Fr(v)adj is given by the matrix β 0 . Thus a Fr(v) + b Fr(v)# is just 0 α aα + bβ 0 . 0 bα + aβ This projects to 1 1 0 (4.3) (a b)(α β) 2 − − 0 1 − 0 in EndA H. # We conclude by (4.1) and (4.2) that the image of the cycle class of aΓv + bΓv 0 in EndA H is η times (4.3). As in the proof of Lemma IV.3.2, α β is a unit in A. Since we assumed that l does not divide a b, this is indeed− of depth η in 0 Gk − (EndA H) v , as required.

Theorem 4.2. Assume that for every v there are integers av, bv such that ∈ L # l does not divide av bv and such that the cycle avΓv + bvΓv admits a divisorial lifting to X X. Assume− also that these divisorial liftings are cohomologically crystalline. Then× T admits a Flach system of depth η for the structure . S v 1 Proof. Let c H (F,T ) denote the image of the divisorial lifting of avΓv + # ∈ bvΓv under the Flach map

n, n 1 1 (4.4) σ : E − − (X) H (F,T ). 2 → v We will show that c v is a Flach system of depth η. To do this we must check { } ∈Lv 1 v 1 that the A-submodule C of H (F,T ) generated by c maps to 0 in Hs (Fw,T ) for w = v and has strict depth η at v. The conditions for w / Σ and v are dealt with 6 1 ∈ as in the proof of Lemma 2.2. For w Σ Σl, Hs (Fw,T ) = 0 by assumption, so v ∈ − the local condition for C is automatic. Finally, the conditions for w Σl are part ∈ of the hypotheses. 5. CONSTRUCTION OF FLACH SYSTEMS IN THE GENERAL CASE 105

5. Construction of Flach systems in the general case We now give the analogue of Theorem 4.2 without the self-adjoint hypothesis. Let F , S and X be as before. We no longer assume that the dimension n is odd. We again fix a prime l such that S contains Σl and such that X is cohomologically n torsion-free at l. Set V = H (XF¯ , Zl). We could also allow V to be a direct n summand of H (XF¯ , Zl) as discussed in Section VIII.10. Let be a commutative l-adic algebra of correspondences on X. Let B and A ∗ B∗ denote the images of in EndZl V as in Section VIII.10. Assume also that we ˜ A have an untwisting (w, B, ξ). Let m be a dualizing maximal ideal of B∗; set A = Bm∗ and H = V B A. Since m is dualizing, A is Gorenstein; let tr be a fixed choice of ⊗ ∗ Gorenstein trace with associated congruence element η. Let χ : GF A× denote the determinant character of H. We assume that χ is odd. → We again check that H is of Taylor-Wiles type and satisfies the assumptions of Section IV.4. As before, H is unramified away from the set Σ consisting of Σl and the places of F not in S, and H is crystalline at every place of Σl. We assume also the conditions 1,2,3,4,5 on H given in Section 4. All other hypotheses are satisfied as before. Let denote the finite/singular structure on T which is S minimally ramified away from Σl and crystalline at Σl. We will need one last piece of data. Definition 5.1. Let v be a place of S. By a diamond operator for v we mean n an automorphism v of X such that j v ∗ i = χ(v)ε(v) as an automorphism of H. h i h i

Let v be a place in S Σl. Let Γv denote the graph of Fr(v) on Xk . Assume − v that there exists a diamond operator v for v and let Γ0 denote the image of h i v Fr(v) v : X X X. × h i → ×

Let = τ denote the set of places of F which have Frobenius conjugate to τ on L L H A k. ⊗ Lemma 5.2. Fix a place v and let a, b be integers such that l does not ∈ L divide a b. Then aΓv + bΓ0 generates T with depth η. − v Proof. This proof is quite close to that of Lemma 4.1. The only difference is the computation of the cycle classes. Applying the analysis of Section VIII.12, we α 0 see that the cycle class of Γv in EndA H is still just 0 β , where we have chosen adj a basis for H as before. The cycle class of Γv0 in EndZl V is v ∗ Fr(v) . The n 1 h i Poincaréadjoint of Fr(v) is still ε(v)− Fr(v)− . We now see from the definition of adj 1 diamond operators that v ∗ Fr(v) is χ(v) Fr(v)− . Since χ(v) is the determinant h i of Fr(v) on H, from here the analysis is exactly as in Lemma 4.1.

Theorem 5.3. Suppose that for every v there are integers av, bv such that ∈ L l does not divide av bv and such that the cycle avΓv + bvΓv0 admits a divisorial lifting to X X. Assume− also that these divisorial liftings are cohomologically crystalline).× Then T admits a Flach system of depth η for the structure . S Proof. This is proven in the same way as Theorem 4.2, using Lemma 5.2 instead of Lemma 4.1. 106 9. CONSTRUCTION OF GEOMETRIC EULER SYSTEMS

6. Construction of cohesive Flach systems It is quite easy to extend the methods of the previous sections to construct cohesive Flach systems. We continue with the hypotheses of the previous section; we will no longer treat the self-adjoint case separately. (Of course, the self-adjoint case is a special case of the general case via the untwisting (id,B∗, 1). Note also that in the self-adjoint case diamond operators are given simply by the identity map.) If ω is a marking on the curve X (that is, a rational section of some invertible exterior power of a sheaf of differentials on X), then we write fα for the induced rational function on a correspondence α as in Section VIII.4. Note that if (α, fα) is a divisorial lifting (of anything) only if α is admissible for the marking ω. Theorem 6.1. Let ω be an admissible marking on X. Assume that for all v / Σl there is a correspondence Tv such that (Tv, fTv ) is a divisorial lifting of ∈ # ∈ A avΓv + bvΓ ; here av, bv are integers such that l does not divide av bv. Assume v − also that Tv∗ yields the Hecke operator Tv in A. Further assume that the (Tv, fTv ) are cohomologically crystalline. Then T admits a cohesive Flach system of depth η for the structure . If the differences av bv are a constant independent of v, then the cohesive FlachS system is of Eichler-Shimura− type of weight twice this constant. v Proof. The classes c are defined to be σ(Tv, fTv ), with σ the general case of (4.4). The local analysis is as in the previous constructions; note that the fact that v 1 c maps to 0 in Hs (Fw, T/ηT ) for all v, w is immediate from the fact that the map

EndZ H EndA H l → is multiplication by η on A-linear maps. The derivation Θ : A H1(F, T/ηT ) is that constructed in Proposition VIII.12.1. This completes the construction→ of the cohesive Flach system. The fact that the cohesive Flach system is of Eichler-Shimura type if the differences are constant follows immediately from the definition of Eichler-Shimura type and the fact that Ver(v) (as defined in Section IV.6) agrees with the cycle class of Γv0 as computed in the proof of Lemma 5.2. Part 3

Examples

CHAPTER 10

The modular curve X0(N)

In this chapter and the next we construct an explicit cohesive Flach system of Eichler-Shimura type for representations associated to weight 2 newforms with trivial character.

1. The geometry of X0(N)

We begin by recalling the basic geometry of the modular curve X0(N). We will work logically somewhat out of order, as we will actually deﬁne X0(N) in terms of X1(N) in the next chapter. We give most references to the summary [DI95, Sections 8 and 9], which in turn contains references to the standard sources [DR73] and [KM85]. See also [Gro90, Sections 2 and 3] and [MW84, Chapter 2, Sections 3-5].

1 1.1. The model X0(N). Let E/S be a generalized elliptic curve over a Z[ N ]- scheme S. We define a Γ0(N)-structure on E/S to be a finite flat subgroup scheme C with all geometric fibers cyclic of order N; we further require that C meet every irreducible component of fibers of E/S which are Néronpolygons. In particular, we see that a Néron d-gon can only have a Γ0(N)-structure if d divides N. We consider two Γ0(N)-structures (E/S, C) and (E0/S, C0) to be isomorphic if there is an S-isomorphism E ' E0 taking C to C0. 1 −→ X0(N) is a Z[ N ]-scheme which coarsely represents the Γ0(N)-moduli problem; see [DI95, Sections 9.2 and 9.3]. X0(N) is a smooth, proper, geometrically con- 1 nected Z[ N ]-scheme of relative dimension 1; this will all follow from our description of X0(N) in terms of X1(N) in the next chapter, together with [KM85, Theorem 7.1.3]. In fact, X0(N) admits a proper, regular model over Z; see [DI95, Section 8.3].

1.2. The degeneracy maps. For all N dividing M, there is a natural degeneracy map

jM,N : X0(M) X0(N); → 1 here we are taking the model of X0(M) over Z[ N ] obtained from the proper regular model over Z. jM,N is deﬁned on the moduli level by sending the Γ0(M)-structure (E/S, C) to the Γ0(N)-structure (E/S, CM ), where CN is the unique subgroup scheme of C of order N. We will also need an alternate degeneracy map in the case that p is a prime not dividing N:

j0 : X0(Np) X0(N). Np,N → On moduli, jNp,N0 sends (E/S, C) to the pair ((E/Cp)/S, C/Cp) where Cp is the unique subgroup of C of order p. Both maps jNp,N and jNp,N0 are ´etaleover 1 Spec Z[ Np ] away from the cusps (which we will deﬁne in the next section); see

109 110 10. THE MODULAR CURVE X0(N)

[Gro90, Section 3]. One should keep in mind that the moduli deﬁnitions above become more complicated (including contractions of irreducible components) on N´eronpolygons.

1.3. The cusps. X0(N) has a certain finite set of distinguished horizontal closed subschemes called the cusps; in terms of the moduli problem they correspond to Néronpolygons. For our purposes it will suffice to describe the cusps over an arbitrary algebraically closed field k of characteristic prime to N. (In fact, our description is valid over algebraically closed fields of any characteristic so long as we use models of X0(N) over Z and we use cyclic in the sense of [KM85, Chapter 1, Section 4].) We will say that a cusp of X0(N)k is of type d if the corresponding Néronpolygon is a d-gon; as we observed above, d must be a divisor of N. To begin we allow N to be arbitrary. Fix an integer d dividing N and let d = Gm Z/dZ denote the Néron d-gon over k. We will classify Γ0(N)-structures E × th on d; these are the type d cusps of X0(N)k. Fix a primitive N root of unity ζ in k. E The N-torsion on d is µN Z/dZ. We will call an element of d[N] primary if it has exact order NE and projects× to 1 Z/dZ. Note that by definitionE every ∈ Γ0(N)-structure on d is generated by a primary element. Thus to determine the E type d cusps it suffices to classify such subgroups up to automorphisms of d. One a E sees immediately that the primary elements of d[N] are of the form ζ 1 for a N E N × relatively prime to d . In particular, there are dφ( d ) primary elements; here φ is the Euler totient function. a b Two primary elements ζ 1 and ζ 1 generate the same subgroup of d[N] × × E precisely when a b (mod d). We denote this subgroup by Sd,a(N); here the second subscript is≡ understood to run through those congruence classes in Z/dZ N which contain representatives relatively prime to d . One finds that there are dφ N φ(d) d φ(N) such subgroups. These subgroups may still be related by automorphisms of d and thus give rise E to the same Γ0(N)-structure. By [DR73, Chapter I], the automorphism group of d is isomorphic to Z/2Z n µd. The Z/2Z acts by “inversion” and thus preserves E all of the subgroups Sd,a(N). On the other hand, ξ µd acts on a primary element ζa 1 by ζa 1 ζaξ 1. It follows that ∈ × × 7→ × d,Sd,a(N) = d,Sd,b(N) E ∼ E precisely when a b (mod g) with g = gcd(d, N ). We will write the correspond- ≡ d ing cusp of X0(N)k as Cd,a(N) with a Z/gZ; in fact, since the only additional ∈ N condition is that a is relatively prime to d , we see that a runs precisely through (Z/gZ)×. In particular, there are φ(g) cusps of type d. The absolute ramification degree of Cd,a(N) over the unique cusp of X0(1)k is equal to the number of d subgroups of d contained in Cd,a(N); this is just . E g We will also need to understand the behavior of the cusps under the maps jNp,N and jNp,N0 for p not dividing N. We now restrict to the case when N is squarefree. In this case X0(N)k has a unique cusp of each type d for dividing N; we denote it by Cd(N). Similarly, X0(Np)k has a unique cusp Cd(Np) for each d 1. THE GEOMETRY OF X0(N) 111 dividing Np. One now computes easily the image of each Cd(Np) under jNp,N and jNp,N0 ; one finds that 1 j− Cd(N) = Cd(Np),Cdp(Np) with ramification degrees 1 and p, respectively:

(1.1) Cd(Np) Cdp(Np) JJ s JJ 1 p ss JJ ss JJ ss J ss Cd(N)

The behavior under jNp,N0 is the same except that the cusp of type d is exchanged with the cusp of type dp:

(1.2) Cd(Np) Cdp(Np) JJ s JJ p 1 ss JJ ss JJ ss J ss Cd(N) 1.4. The Hecke correspondences. Fix a prime p not dividing N. We deﬁne th the p Hecke correspondence Tp on X0(N) to be the scheme-theoretic image of the map

jNp,N jNp,N0 : X0(Np) X0(N) Spec Z[ 1 ] X0(N). × → × N Tp is birational to X0(Np) away from characteristic p and has pure codimension 1 1 in X0(N) Spec Z[ ] X0(N). It is possible to view Tp,Fp as an algebraic self- × N correspondence on X0(N)Fp (we will explain how in our discussion of the Hecke algebra T0(N) below) and we have the Eichler-Shimura relation # (1.3) Tp,Fp = Γp + Γp , # where Γp is the graph of the Frobenius morphism on X0(N)Fp and Γp is its trans- # pose; here we regard Γp and Γp as algebraic self-correspondences on X0(N)Fp in the obvious way. See [Gro90, p. 454] and [DI95, Section 8.4]. 1.5. The Atkin correspondences. Fix a prime p dividing N. We define a Γ0(N; p)-structure on a generalized elliptic curve E/S to be a pair (C,C0) of finite flat subgroup schemes of E/S of order N and p respectively such that C C0 = ∩ 0. We further require that C + C0 meets all irreducible components of fibers of E/S which are Néronpolygons. One sees easily that Γ0(N; p)-structures exist on N Néron d-gons only for d = pd0 with d0 a divisor of p . We have the obvious notion of an isomorphism of Γ0(N; p)-structures. The Γ0(N; p)-moduli problem is coarsely 1 represented by a proper, regular Z[ N ]-scheme X0(N; p); see [MW84, Chapter 2, Section 5.5].

There are two natural degeneracy maps jN;p,p and jN0 ;p,p from X0(N; p) to X0(N). The ﬁrst sends the triple (E/S, C, C0) to the pair (E/S, C) and the second th sends it to the pair ((E/C0)/S, (C+C0)/C0). We deﬁne the p Atkin correspondence Tp to be the scheme-theoretic image of the map

jN;p,p jN0 ;p,p : X0(N; p) X0(N) Spec Z[ 1 ] X0(N). × → × N Tp is birational to X0(N; p) away from characteristic p and has pure codimension 1 in X0(N) Spec Z[ 1 ] X0(N). Often in the literature our Tp is denoted Up. × N 112 10. THE MODULAR CURVE X0(N)

We will need to understand how the cusps of X0(N; p)k sit over the cusps of X0(N)k for k an algebraically closed field of characteristic not dividing N. For simplicity we only give an explicit description in the case that N is squarefree. The analysis of these structures is similar to (although slightly more complicated than) the analysis of Section 1.3. Let ( dp,C,C0) be a Γ0(N; p)-structure. We define the type (i, j) of the cor- E responding cusps of X0(N; p)k as follows: we let i (resp. j) equal the order of the image of the map C Z/pdZ (resp. C0 Z/pdZ). → → One finds that the cusps of X0(N; p)k are precisely indexed by these types: the N allowable types are (dp, 1), (dp, p) and (d, p) for d dividing p . These are related to the cusps of X0(N) under jN;p,N by:

(1.4) Cd,p(N; p) Cdp,1(N; p) Cdp,p(N; p) p 1 ooo p 1 − oo ooo ooo Cd(N) Cdp(N) and under jN0 ;p,N by

(1.5) Cdp,1(N; p) Cd,p(N; p) Cdp,p(N; p) p 1 ppp p 1 − pp ppp ppp Cd(N) Cdp(N)

1.6. The Hecke algebra T0(N). The modular curve X0(N) is the generic fiber X0(N) Spec Z[ 1 ] Spec Q of X0(N); it is a smooth projective curve over Q. × N Write Tp for the Hecke correspondence Tp,Q on X0(N). One can also define Hecke correspondences Tn for all n (using an appropriate moduli interpretation; see [Roh97, Chapter 2, Sections 1-3]). Note that the Tn are divisors on X0(N), and therefore are local complete intersections; in particular, they are Cohen-Macaulay. The two projections Tn X0(N) are visibly quasi-finite and proper, so by [GDb, Proposition 15.4.2] and [→GDa, Proposition 4.4.2] they are both finite flat. Checking on the level of geometric points shows that they are both finite, faithfully flat of the same degree. In particular, we see that the Hecke correspondences Tn really are algebraic correspondences in the sense of Definition VIII.1.1. (This argument works over fields of any characteristic relatively prime to N and therefore completes our description of the Eichler-Shimura relation in (1.3).) The Hecke correspondences satisfy the relations

Tmn = TmTn for m, n relatively prime;

T n = T n 1 T pT n 2 for p not dividing N; p p − p p − n − Tpn = Tp for p dividing N; see [Lan76, Chapter 7, Theorem 2.1]. It follows that the Tp (for all p) generate a commutative algebra of correspondences T0(N). In fact, T0(N) can be generated without Tl for any ﬁxed prime l so long as we assume that l does not divide N; see [DDT97, Lemma 4.1]. 2. THE MODULAR UNIT ∆ 113

1.7. Poincaréduality. Fix a prime l 7 and not dividing N and let V = 1 ≥ H (X0(N)Q¯ , Zl); we wish to fit V into the general framework of Section VIII.8. It follows from the formulas in [MW84, p. 236] and the fact that the diamond automorphisms act trivially on V that the Hecke operators Tp for p not dividing N are self-adjoint acting on V . However, here we must make the unfortunate assumption that all Tp are self-adjoint acting on V . This is certainly false in general; we make this assumption so that we can proceed in the simpler self-adjoint case. We will remove it in the next chapter. Given this, the Poincarépairing ϕ : V Z V (1) ⊗ l → Zl is a T0(N)-hermitian, skew-symmetric, Galois equivariant perfect pairing.

1.8. Non-Eisenstein maximal ideals. Let B denote the image of T0(N) in

EndZl V ; by self-adjointness this is independent of which map we take. We wish to find a dualizing maximal ideal m of B. There is a very well developed theory of such ideals; see [DDT97, Chapter 4] and [Til97] for an exposition. The first step is to find a maximal ideal m of B associated to a newform and such that the Galois representation V B k (where A = Bm and k = A/mA as always) is absolutely irreducible. This occurs⊗ precisely when m is not Eisenstein. (Recall that m is said to be Eisenstein if Tp p + 1 (mod m) for all primes p congruent to 1 modulo N; see [DDT97, Section≡ 4.3, esp. Lemma 4.12].) Since l does not divide N,[Til97, Theorem 3.4] and the proofs of its corollaries show that a non-Eisenstein m containing l is dualizing in our sense. In particular, A is Gorenstein. Fix for the remainder of the chapter one such maximal ideal m and a corresponding choice of Gorenstein trace tr : A Zl. Let η A denote the corresponding congruence element. → ∈ 0 1.9. The finite/singular structure. Let T = EndA H(1) considered as a GQ-module. We give T the finite/singular structure as in Section IX.4: that is, it is unramified away from Nl, minimally ramified at N and crystalline at l. We will assume below that N is squarefree so that H is ordinary at primes dividing N; thus the minimally ramified structure will coincide with the weak structure. We will return to this below.

2. The modular unit ∆ We assume from now on that N is squarefree. In order to construct a cohesive Flach system for T as in Chapter IX.6, we ﬁrst must ﬁnd an admissible marking for the algebra T0(N). Following Flach we choose ∆, the unique normalized rational cusp form of weight 12 and level 1. In fact, we will see that this is essentially the only choice, at least if we constrain the divisor to the cusps. (We prefer not to even contemplate non-cuspidal divisors.)

Lemma 2.1. The marking ∆ is admissible for the Hecke algebra T0(N). Fur- thermore, the only cusp forms of any weight (and level dividing N) which yield admissible markings for T0(N) are rational multiples of powers of ∆. Proof. Since one can check if a divisor vanishes after base extension, for this proof all of the varieties and divisors we consider will be over Q¯ ; we will omit this 6 from our notation. On the modular curve X(1), ∆ lies in ΩX⊗(1) and has divisor 6 C1(1). It follows easily that on X0(N), ∆ lies in Ω⊗ and has divisor X0(N) (2.1) dC (N). X d d N | 114 10. THE MODULAR CURVE X0(N)

(Recall that since N is squarefree its cusps correspond bijectively to the divisors of N.) We must show that each Tp is admissible for the marking ∆. Suppose ﬁrst that p is a prime not dividing N. We have a diagram

X0(Np) ?? ?? ?? ? j ?? j0 Tp ?? oo OOO ?? π1 oo OO π2 ? ooo OOO ?? ooo OOO ?? ow oo OO' ? X0(N) o X0(N) X0(N) / X0(N) × To show that Tp is admissible for ∆, we must show that the rational function

π1∗∆ fp = k(Tp)× π2∗∆ ∈ has trivial divisor. We will compute ﬁrst the divisor of the rational function

j∗∆ (2.2) k X0(Np) × j0∗∆ ∈ on X0(Np); since the map X0(Np) Tp is birational, we can identify (2.2) with → fp. (2.1) and (1.1) show that the divisor of j∗∆ on X0(Np) is (2.3) dC (Np) + dpC (Np). X d dp d N | In the same way, we see from (1.2) that j0∗∆ has divisor (2.4) dpC (Np) + dC (Np). X d dp d N | Thus on X0(Np) the rational function fp has divisor

(2.5) d(1 p) Cd(Np) Cdp(Np) . X − − d N | Since under j j0 : X0(Np) Tn both Cd(Np) and Cdp(Np) map to Cd(N) × → × Cd(N), we see from (2.5) that fp has trivial divisor on Tp. Thus Tp is divisorless for ∆ for all p not dividing N. The same is true for p dividing N. We can write the divisor (2.1) on X0(N) as dC (N) + dpC (N). X d dp d N | p

It follows from (1.4) that on X0(N; p), j∗∆ has divisor

dpCd,p(N; p) + dpCdp,1(N; p) + dp(p 1)Cdp,p(N; p). X − d N | p

By (1.5), j0∗∆ has divisor

dpCdp,1(N; p) + dpCd,p(N; p) + dp(p 1)Cdp,p(N; p). X − d N | p 3. THE DIVISOR OF fp IN POSITIVE CHARACTERISTIC 115

Therefore the rational function fp has trivial divisor on X0(N; p); it follows that Tp is admissible for the marking ∆ for p dividing N as well. This leaves the uniqueness assertion. In fact, one checks immediately as above that any cusp form is admissible for the sub-algebra of T0(N) generated by the Tp for p not dividing N. The diﬃculty arises on the Tp for p dividing N. Let g be a rational cusp form of level N and weight k, with divisor n C (N) X d d d N | on X0(N). Fix a prime p dividing N. One checks that the condition that fg be divisorless for Tp is that

(2.6) npd = pnd N where d runs over the divisors of p . (This essentially means that g must come N from level p .) The conditions (2.6) for all primes p dividing N show that the nd are given by nd = dn1.

Thus the divisor of g is a multiple of the divisor of ∆. Since X0(N) is a nonsingular curve and ∆ is the modular form of weight 1 and minimal positive weight, this implies that g is a multiple of a power of ∆, as claimed.

3. The divisor of fp in positive characteristic In this section we prove the following key lemma.

Lemma 3.1. For all p not dividing N, the pair (Tp, fp) is a divisorial lifting of # 6(Γ Γp). p − Proof. Since fp has no divisor on Tp in characteristic 0, we know that the divisor of fp on Tp has no horizontal component. We must show that it also has no vertical component in characteristics different from p (and not dividing N) and that it has the appropriate divisor in characteristic p. Assume first that r is a prime not dividing Np. In this case the calculation is exactly the same as the calculation in characteristic 0. Indeed, the degeneracy maps jNp,N : X0(Np)¯ X0(N)¯ Fr → Fr j0 : X0(Np)¯ X0(N)¯ Np,N Fr → Fr are étaleaway from the cusps, so the divisors of j∗∆ and j0∗∆ on X0(Np)F¯ r are the usual cuspidal divisors (2.3) and (2.4). As before it follows that the divisor of fp on Tp,F¯ r is trivial.

We now consider the case r = p. Here we compute directly on Tp,F¯ p . By the # Eichler-Shimura relation (1.3), we can write Tp,F¯ p = Γp + Γp . Since we know that the divisor of fp is of codimension 0 in Tp,F¯ p , it suﬃces to compute it generically # ¯ on the irreducible components Γp and Γp . Recall that Γp is the base change to Fp of the scheme-theoretic image of

id Fr : X0(N)F X0(N)F X0(N)F . × p → p × p Thus π1 Γ is an isomorphism, so the divisor of π∗∆ has no support on Γp. However, | p 1 π2 Γ is purely inseparable, so by [Sil86, Chapter 2, Proposition 4.2(c)] the divisor | p of π2∗∆ picks up 6 factors of Γp. (The 6 comes from the fact that ∆ is an element 116 10. THE MODULAR CURVE X0(N)

6 # of Ω⊗ .) In the same way, the divisor of π ∆ # is 6Γ , while the divisor of X0(N) 1∗ Γp p | # π2∗∆ # is trivial. Combining these results, we find that fp has divisor 6(Γp Γp) |Γp − in characteristic p. 4. The cohesive Flach system Given our analysis to this point, the proof of the following theorem is not difficult. Recall that we have assumed that N is squarefree and that l 7 is a 1 ≥ prime not dividing N. B is the image of T0(N) in EndZl H (X0(N)Q¯ , Zl) where we have assumed that all Tp are self-adjoint. We have A = Bm and 1 H = H X0(N) ¯ , Zl Q m for m a non-Eisenstein maximal ideal of B associated to a newform. and we have fixed a Gorenstein trace tr : A Zl with congruence element η. → Theorem 4.1. Let H be a modular Galois representation as above and set 0 T = End H(1). Assume that T A k is absolutely irreducible and that A ⊗ 1 H (Q(T ∗[a])/Q,T ∗[a]) = 0 for all ideals a of finite index in A. Then T admits a cohesive Flach system of Eichler-Shimura type of depth η and weight 12 for the structure c. − S Proof. This is an immediate consequence of Theorem IX.6.1 and Lemma 3.1 once we check that the hypotheses of Theorem IX.6.1 are satisfied. X0(N) is cohomologically torsion-free at l since it is a curve. The fact that Tp∗ is the trace of Frp on H is [DDT97, Theorem 3.1] (taking into account that the representation there is the dual of ours and Frobenius there is arithmetic). This leaves the four numbered conditions of Section IX.4. By [DDT97, Lemma 3.27], H is minimally ramified and ordinary; Lemma I.5.2 now gives condition 1 since l 7. That H A k is absolutely irreducible follows from the fact that m is non-Eisenstein,≥ as discussed⊗ in Section 1.8; the rest of condition 2 is one of our hypotheses. Condition 3 is as well, and condition 4 is shown in [DDT97, Lemma 4.1]. Finally, the crystalline conditions come from Proposition VII.10.1 on checking that the cycle class of Tp 0 vanishes in EndA H; that is, that it is a scalar. This is clear from the computations of Lemma IX.4.1 and the compatibility of the cycle class map with specialization as in [GBI71, Appendix to Exposé10]. (The singularities of the Hecke correspondences are all ordinary double points, which can indeed be resolved over Zl.) This completes the proof. Note that the cohesive Flach system of Theorem 4.1 is canonically determined up to the choice of Gorenstein trace tr. Note also that the depth η is canonically determined as the congruence element of tr. CHAPTER 11

The modular curve X1(N)

In this chapter we construct cohesive Flach systems of Eichler-Shimura type for Galois representations associated to modular forms of weight 2 and arbitrary character. We assume for this chapter that N is a squarefree integer greater than or equal to 7.

1. The geometry of X1(N)

1 1.1. The model X1(N). Let E/S be a generalized elliptic curve over a Z[ N ]- scheme S. We define a Γ1(N)-structure on E/S to be a section P : Z/NZ , E of exact order N on fibers; we further require that the subgroup generated by →P meet every irreducible component of fibers which are Néronpolygons. (As before this implies that only Néron d-gons for d dividing N can support Γ1(N)-structures.) We consider two Γ1(N)-structures (E/S, P ) and (E0/S, P 0) to be isomorphic if there is an isomorphism E ' E0 taking P to P 0. −→ 1 The functor from the category of Z[ N ]-schemes to sets sending a scheme S to the set of isomorphism classes of Γ1(N)-structures on generalized elliptic curves over S is representable (only coarsely representable for N 4) by a scheme X1(N). X1(N) 1 ≤ is a proper, smooth, geometrically connected Z[ N ]-scheme of relative dimension 1. X1(N) admits a proper, regular model over Z as well; see [DI95, Sections 8.2, 8.3, 9.2, 9.3].

1.2. The degeneracy maps. For all N dividing M, there is a natural degeneracy map

jM,N : X1(M) X1(N); → 1 here we are using a model for X1(M) deﬁned over Z[ N ]. This map is deﬁned on the moduli level by sending the Γ1(M)-structure (E/S, P ) to the Γ1(N)-structure M (E/S, N P ).

1.3. The diamond automorphisms. For d (Z/NZ)× there is an automorphism of the above moduli problem sending a pair∈ (E/S, P ) to the pair (E/S, dP ); the corresponding automorphism of X1(N) is called a diamond automorphism and is written d . Note that 1 acts trivially on X1(N), and that (Z/NZ)×/ 1 acts h i h− i ± freely on X1(N). X0(N) is deﬁned to be the quotient of X1(N) by (Z/NZ)×/ 1; the natural quotient map ±

πN : X1(N) X0(N) → realizes X1(N) as a ﬁnite Galois covering of X0(N), of degree φ(N)/2.

117 118 11. THE MODULAR CURVE X1(N)

1.4. The cusps. The closed subschemes of X1(N) corresponding to families containing Néronpolygons are called the cusps. The map πN is a finite Galois covering unramified at the cusps, so over any algebraically closed field k there are exactly φ(N)/2 cusps of X1(N) sitting over each cusp of X0(N).

1.5. The Hecke correspondences. Fix a prime p not dividing N. Let E/S 1 be a generalized elliptic curve over a Z[ N ]-scheme S. We define a Γ1(N; p)-structure on E/S to be a pair of a Γ1(N)-structure P and a finite flat subgroup scheme C of E with all geometric fibers of rank p; we require that the group generated by P and C meet every irreducible component of Néronpolygon fibers. The Γ1(N; p)- moduli problem is representable by a proper, regular, geometrically irreducible 1 1 Z[ N ]-scheme X1(N; p) of relative dimension 1; it becomes smooth over Z[ Np ]. (See [DI95, Sections 8.3 and 9.3].) X1(N; p) admits two natural degeneracy maps

jN;p,N : X1(N; p) X1(N) → j0 : X1(N; p) X1(N). N;p,N → jN;p,N sends the triple (E/S, P, C) to the pair (E/S, P ), and jN0 ;p,N sends it to ((E/C)/S, P ). These maps are both generically ´etaleaway from characteristic p. th We deﬁne the p Hecke correspondence Tp to be the scheme-theoretic image of the map

jN;p,N jN0 ;p,N : X1(N; p) X1(N) Spec Z[ 1 ] X1(N). × → × N Tp is birational to X1(N; p) away from characteristic p, and has pure codimension 1 in X1(N) Spec Z[ 1 ] X1(N). × N We can give a precise description of the closed subscheme Tp,Fp of the proper smooth variety X1(N)F Spec F X1(N)F . We will see later that Tp,F can be p × p p p considered as an algebraic self-correspondence on X1(N)Fp ; the Eichler-Shimura relation states that

Tp,Fp = Γp + Γp0 where Γp is the graph of the Frobenius morphism on X1(N)Fp and Γp0 is its modiﬁed transpose given as the image of

Fr p : X1(N)F X1(N)F X1(N)F . × h i p → p × p See [Gro90, p. 454] and [DI95, Section 8.4].

1.6. The Atkin correspondences. Fix a prime p dividing N. Let E/S be 1 a generalized elliptic curve over a Z[ N ]-scheme S. We define a Γ1(N; p)-structure on E/S to be a pair of a Γ1(N)-structure P and a finite flat subgroup scheme C of E with all geometric fibers of rank p; we require that the group generated by P and C meet every irreducible component of Néronpolygon fibers and we further require that C has trivial intersection with the group generated by P . The Γ1(N; p)- 1 moduli problem is representable by a proper, regular Z[ N ]-scheme X1(N; p) with geometrically irreducible fibers and of relative dimension 1; it becomes smooth over 1 Z[ Np ]. See [Gro90, p. 454]. X1(N; p) admits two natural degeneracy maps jN;p,N and jN0 ;p,N to X1(N): the first sends the triple (E/S, P, C) to the pair (E/S, P ) and the second sends (E/S, P, C) to ((E/C)/S, P ); here P denotes the induced section of E/C, which 1. THE GEOMETRY OF X1(N) 119 still had order N since C is not contained in the group generated by P . We define th the p Atkin correspondence Tp to be the scheme-theoretic image of the map

jN;p,N jN0 ;p,N : X1(N; p) X1(N) Spec Z[ 1 ] X1(N). × → × N

As before, Tp is birational to X1(N; p) away from characteristic p and has pure codimension 1 in X1(N) Spec Z[ 1 ] X1(N). See [MW84, Section 5.5] for more × N details.

There is a natural projection πN0 : X1(N; p) X0(N; p); in fact, it is simply the quotient map by the diamond automorphisms→ (acting on the point of order N) and is a Galois covering of degree φ(N)/2. It is unramiﬁed at the cusps, so it allows us to understand the cusps on X1(N; p) in terms of the cusps of X0(N; p).

1.7. The Hecke algebra T1(N). The modular curve X1(N) is the generic fiber X1(N) Spec Z[ 1 ] Spec Q of X1(N); it is a smooth projective curve over Q. We × N regard the diamond automorphisms d as algebraic correspondences on X1(N) via h i their graphs. Write Tp for the Hecke correspondence Tp,Q on X1(N). One can also define Hecke correspondences Tn for all n as with X0(N). The same argument as in the X0(N) case shows that these are all algebraic correspondences in our sense. Note also that the compositions d Tn and Tn d are trivially defined since d is an automorphism. h i ◦ ◦ h i h i The Hecke correspondences satisfy the relations

Tmn = TmTn for m, n relatively prime;

T n = T n 1 T p p T n 2 for p not dividing N; p p − p p − n − h i Tpn = Tp for p dividing N; see [Lan76, Chapter 7, Theorem 2.1]. The Tp and diamond automorphisms generate a commutative algebra of correspondences which we denote T1(N). We can omit any Tl from the set of generators of T1(N) so long as l does not divide N by [DDT97, Lemma 4.1].

1.8. The involution wζ . We will need one “exotic” involution of X1(N)Q¯ . th 1 Fix a primitive N root of unity ζ and for every elliptic curve E/S over a Z[ N ]- scheme S, let e : E[N] E[N] µN be the scheme-theoretic Weil pairing. wζ is the × → automorphism of X1(N)Z[1/N,ζ] sending a pair (E/S, P ) to the pair ((E/C)/S, Q), where C is the subgroup of E generated by P and Q is the unique point of E[N] such that e(Q, P ) = ζ; see [MW84, Section 5.2] for more details. Consider now the corresponding involution of X1(N)Q¯ , which we also write as wζ . wζ is self-adjoint in the sense that wζ∗ = wζ acting on cohomology. wζ ∗ interacts with elements of T1(N) via the relation α∗wζ∗ = wζ∗α . Since wζ is an ∗ involution, this implies also that α wζ∗ = wζ∗α∗; see [Til97]. We also have a simple interaction∗ with the Galois action on ´etalecohomology: 1 for v H (X1(N) ¯ , Zl) and σ GQ, we have (considering w∗ as an involution on ∈ Q ∈ ζ cohomology) 1 σ(w∗v) = (σw∗)(σv) = σ ∗ − w∗(σv). ζ ζ h i ζ + Here we are using the natural identiﬁcation of Gal(Q(ζN ) /Q) with (Z/NZ)×/ 1 ± to regard as a character of GQ. See [MT73, Section 2] for details. h·i 120 11. THE MODULAR CURVE X1(N)

1.9. Poincaréduality. Fix a prime l 7 and not dividing N and let V be 1 ≥ the étalecohomology group H (X1(N)Q¯ , Zl). The algebra T1(N) is not self-adjoint with respect to the Poincarépairing ϕ : V Z V (1) Zl. Since wζ is self-adjoint, ⊗ l → ϕ does satisfy ϕ(wζ∗t, t0) = ϕ(t, wζ∗t0).

Let B and B∗ be the images of T1(N) in EndZl V , as usual. We deﬁne an ∗ ˜ untwisting of T1(N) as follows: let B be a free B∗-module of rank 1 (with a chosen generator ξ) with a B∗-linear action of GQ given by

σξ = σ ∗ ξ. h i We claim that the map

ξ w∗ : V B˜ B V ⊗ ζ → ⊗ ∗ sending v to ξ w∗v is Galois equivariant. Indeed, ⊗ ζ σ ξ w∗(v) = σξ σ w∗(v) ⊗ ζ ⊗ ζ 1 = σ ∗ ξ σ ∗ − w∗(σv) h i ⊗ h i ζ = ξ w∗(σv) ⊗ ζ = (ξ w∗)(σv). ⊗ ζ ˜ Thus the triple (wζ∗, B, ξ) is an untwisting of V . 1.10. Non-Eisenstein maximal ideals. The theory of maximal ideals in T1(N) is very similar to that of T0(N) in Section X.1.8; see [DDT97, Chapter 4] and [Til97, Theorem 3.4]. Let m be a non-Eisenstein maximal ideal of B∗ associated ˜ ˜ to a newform at level N. Set A = B∗ , k = A/mA, H = V B A and A = B B A. m ⊗ ∗ ⊗ ∗ H A k is absolutely irreducible as a GQ-module, and as usual we know that A is ⊗ Gorenstein. Fix a choice of Gorenstein trace tr : A Zl with congruence element 0 → η. We set T = EndA H and we endow T with the finite/singular structure which is unramified away from Nl, minimally ramified at N and crystalline at l.

2. Admissible markings We once again will use ∆ as our admissible marking.

Lemma 2.1. The marking ∆ is admissible for the Hecke algebra T1(N).

Proof. Since the maps X1(N; p) X0(N; p) are finite Galois coverings un- → ramified at the cusps, the verification that ∆ is divisorless for all Tp follows immediately from Lemma X.2.1. It remains to check that ∆ is divisorless for the diamond automorphisms d . The induced rational function on the correspondence h i d (which is isomorphic to X1(N)) is simply ∆ d /∆; since ∆ has trivial character,h i this is the constant 1. Thus d is divisorless◦ h fori ∆, and we conclude that all h i of T1(N) is divisorless for ∆.

Let fp denote the rational function on Tp induced by ∆. The calculation of the divisor of fp in positive characteristic is entirely similar to that of Lemma X.3.1, taking into account the modification in the Eichler-Shimura relation and the fact that p is étale.One finds the following result; here Γp is the graph of Frobenius h i and Γp0 is its modified transpose as in Section 1.5. Lemma 2.2. For all p not dividing N, the pair (Tp, fp) is a divisorial lifting of 6(Γ0 Γp). p − 3. THE COHESIVE FLACH SYSTEM 121

3. The cohesive Flach system Recall that we have assumed that N 5 is squarefree and that l 7 is a prime ≥ 1 ≥ not dividing N. B∗ is the image of T1(N) in EndZl H (X1(N)Q¯ , Zl). We have A = Bm∗ and 1 H = H X0(N) ¯ , Zl Q m for m a non-Eisenstein maximal ideal of B∗ associated to a newform, and we have fixed a Gorenstein trace tr : A Zl with congruence element η. → Theorem 3.1. Let H be a modular Galois representation as above and set 0 T = End H(1). Assume that T A k is absolutely irreducible and that A ⊗ 1 H Q(T ∗[a])/Q,T ∗[a] = 0 for all ideals a of finite index in A. Then T admits a cohesive Flach system of Eichler-Shimura type of depth η and weight 12. − Proof. The fact that the diamond automorphisms are diamond operators in the sense of Definition IX.5.1 follows from [DDT97, Theorem 3.1]. The rest of the proof is virtually identical to the proof of Theorem X.4.1. The one complication is the check that the minimally ramified structure agrees with the weak structure. For this one uses [Car86, ThéorèmeA] to see that H is either ordinary or else a direct sum of an unramified character and a tamely ramified character. The first case is dealt with via Lemma I.5.2, while the second case is a straightforward computation.

CHAPTER 12

Kuga-Sato varieties

In this chapter we extend the methods of the previous two chapters to construct cohesive Flach systems of Eichler-Shimura type for modular Galois representations of higher weight. There are many possible approaches to this. We choose the least elegant but simplest geometrically: we realize these representations in the cohomology of an “open” Kuga-Sato variety. This requires some modiﬁcations of the results of Chapters VII and VIII, but has the advantage of not involving any resolution of singularities.

1. The geometry of Kuga-Sato varieties

1.1. The universal elliptic curve. Recall that X1(N) represents the Γ1(N)- 1 moduli problem for Z[ N ]-schemes. We will be interested only in the complement 1 1(N) of the cusps in X1(N); 1(N) is a smooth separated Z[ ]-scheme, but it Y Y N is not proper. 1(N) represents the Γ1(N)-moduli problem for elliptic curves over 1 Y Z[ N ]; in particular, there exists a universal elliptic curve (with a point of exact order N) 1(N) over 1(N). E Y 1.2. Open Kuga-Sato schemes. For k 0, we let k(N) denote the k-fold ≥ E1 ﬁber product of 1(N) over 1(N): E Y k k 1 (N) = 1(N) 1(N) 1(N) 1(N) 1(N) . E Ez ×Y E }|× · · · ×Y E { k 1 1 (N) Z[ N ] is smooth and separated of relative dimension k + 1, but is not E → k k proper. We call 1 (N) the open Kuga-Sato scheme of weight k + 2. 1 (N) has a canonical compactiﬁcationE in each characteristic (see [Del71, LemmaE 5.4] and [Con, Theorem 4.3.1.1]) but (with one exception) we will not need this.

1.3. The Hecke algebra. For d (Z/NZ)× there is an automorphism d of ∈ h i 1(N) sitting over the automorphism d of 1(N); see [Con, Section 4.2.7] where E h i Y k it is denoted Id. This in turn yields an automorphism of 1 (N) which we denote by (k) E d . Similarly, the Hecke correspondences Tp (for all p not dividing N) on 1(N) h i (k) k Y yield (by base change) Hecke correspondences Tp on 1 (N). (One can also obtain these operators by a generalization of the methods ofE Section XI.1.5; see [Sch90, Section 4].) k k We define E1 (N) to be the generic fiber of 1 (N); it is a smooth separated (k) E (k) variety of dimension k + 1 over Q. Let Tp denote the Hecke correspondence Tp,Q k (k) k on E1 (N). We note that the projections Tp,Q E1 (N) are finite flat of the same (k) → degree (by base change), although Tp,Q is not technically an algebraic correspon- k (k) dence in the sense of Chapter VIII as E1 (N) is not proper over Q. However, Tp

123 124 12. KUGA-SATO VARIETIES still yields maps in K-theory and the notion of composition as in Section VIII.3 is still valid in this setting. In particular, it follows from the relations of [Lan76, (k) (k) Chapter 7, Theorem 2.1] that the Tp and d generate a commutative algebra of k h i (k) correspondences for the open variety E1 (N); we denote this algebra by T1(N) . (Note that we have not yet deﬁned any maps on ´etalecohomology associated to (k) these correspondences.) As always we can omit any Tl from the generators of (k) T1(N) so long as l does not divide N. Let p be a prime not dividing N. For such a p we have the usual Eichler-

Shimura relation for Tp,Fp , regarded as an algebraic correspondence on the smooth k k variety (N)F Spec F (N)F : E1 p × p E1 p

(1.1) Tp,Fp = Γp + Γp0 k where Γp is the graph of the Frobenius morphism on 1 (N)Fp and Γp0 is its modified transpose given as the image of E k k k Fr p : (N)F (N)F Spec F (N)F . × h i E1 p → E1 p × p E1 p See [Con, Theorem 5.3.3.1] for details. 1.4. Untwistings. Let ζ denote a fixed primitive N th root of unity. By [Con, Section 4.2.7], the involution wζ of 1(N)Z[1/N,ζ] lifts to an involution wζ (called ϕζ Y (k) k in [Con]) of 1(N) ; this in turn yields an involution w of (N) . E Z[1/N,ζ] ζ E1 Z[1/N,ζ] 1.5. Galois representations. The production of Galois representations from V and maximal ideals of B∗ is due to Deligne; see [Del71]. We follow the presentation of [Con] via the integral theory of [FJ95]. Fix a prime l max 7, k + 1 not dividing N. We consider first the image ≥ { } ˜ 1 k 1 V = H Y1(N)Q¯ , Sym R f Zl ∗ of the map 1 k 1 1 k 1 Hc Y1(N)Q¯ , Sym R f Zl H Y1(N)Q¯ , Sym R f Zl ∗ → ∗ of compactly supported cohomology into ordinary cohomology. Here f : E1(N) ¯ Q → Y1(N)Q¯ is the structure map. By [Con, Section 5.4], [Sch90, Proposition 4.1.1] and [Car94] (which discusses the integral structure), V is a subquotient of the ˜ k+1 k étalecohomology group H (E1 (N)Q¯ , Zl). We must assume that: k E1 (N) is cohomologically torsion-free at l; • k+1 k V is a direct summand of H˜ (E (N) ¯ , Zl) as a Zl-module. • 1 Q Both conditions hold for almost all l. Given these assumptions, we claim that we can apply the methods of Chapters k k VII and VIII for the open variety E (N) Spec Q E (N) to the direct summand 1 × 1 (via the above assumptions and the Künnethformula) EndZl V inside of 2k+2 k k (1.2) V0 = H E (N) ¯ ¯ E (N) ¯ , Zl . 1 Q ×Spec Q 1 Q To show this we must consider where proper hypotheses were used in these chapters. Properness was used in three places in Chapter VII. In Section VII.3 it was used to guarantee the local constancy of a higher direct image for an application of purity; however, [Ras89, Lemma 2.1] applies even without this, so that here properness was not required. The critical application of properness was in Section VII.7 where it was used to relate the cohomology of the generic fiber with the cohomology of 2. ADMISSIBLE MARKINGS 125 the special fiber. By [Con, Theorem 5.2.8.1], our V is compatible with such base change, and one sees immediately that this compatibility is compatible with the k+1 k inclusion of V into H (E1 (N)Q¯ , Zl). This is sufficient to extend Theorem VII.1.1 to apply to EndZl V inside of (1.2); that is, there is a commutative diagram (1.3) div k+1, k 2 k k F p k+1 k k E − − E (N) E (N) / A (N)F (N)F 2 1 × 1 E1 p × E1 p

σk+1 s 1 2k+2 k k GFp H (Q,V0) H (N)¯ (N)¯ , Zl(k + 1) E1 Fp × E1 Fp

' 1 GFp H (Q, EndZl V ) (EndZl )

1 / 1 H (Qp, EndZl V ) Hs (Qp, EndZl V ) for each p not dividing Nl. The last application of properness in Chapter VII occurred in Section 10 where it was assumed of the ambient variety. However, the deRham cases of these results remain valid without this properness, and in our particular case the crystalline k case remains valid by the explicit compactification of E1 (N) given by Kuga-Sato varieties. In Chapter VIII properness was used frequently, but always to insure that appropriate operations in étalecohomology (covariant functoriality, Poincarédu- ality and Künnethprojections) operate entirely on ordinary cohomology (that is, without introducing compact cohomology). The results of [Con, Section 5.2.1.1], [Sch90, Proposition 4.1.1] and especially [FJ95, Theorem 2.1] show that these op- (k) erations for the Hecke algebra T1(N) can be regarded as operating entirely on V . We conclude that the results of Chapter VIII, and thus the results of Chapter IX, remain valid for the Flach map k+1, k 2 k k 1 E − − E (N) Spec Q E (N) H (Q, EndZ V ) 2 1 × 1 → l of (1.3). (k) We let B and B∗ denote the images of T1(N) in EndZl V . We define an ∗ (k) untwisting of V via wζ exactly as in Section XI.1.9, using [FJ95] to check that it is an untwisting. Let m be a non-Eisenstein maximal ideal of B∗ associated to a newform; by [FJ95, Theorem 2.1 and Theorem 3.38], such an m is dualizing. In particular, H = Vm is free of rank 2 over the Gorenstein ring A = B∗ and H A k m ⊗ is absolutely irreducible. We fix a choice of Gorenstein trace tr : A Zl and let 0 → η A be the associated congruence element. We set T = EndA H(1) and give it the∈ finite/singular structure which is unramified away from Nl, minimally ramified at N and crystalline at l.

2. Admissible markings k Note that by pullback the modular form ∆ yields a diﬀerential form on E1 (N). (k) We use ∆ as a marking for the Hecke algebra T1(N) . It is actually possible in 126 12. KUGA-SATO VARIETIES this setting to use any modular unit as a marking, but we will not pursue this here.

(k) Lemma 2.1. The marking ∆ is admissible for the Hecke algebra T1(N) .

(k) Proof. Recall that Tp is obtained by pullback from the Hecke correspondence Tp on Y1(N). By the definition of the rational function induced by a marking, (k) (k) we see that the rational function fp on Tp induced by ∆ is precisely the pullback of the rational function fp on Tp induced by ∆. Since by Lemma XI.2.1 we know (k) that fp has trivial divisor on Tp, we see that Tp is divisorless for ∆. The proof for the diamond automorphisms is entirely similar. (k) (k) k k Lemma 2.2. For all p not dividing N, the pair (Tp , fp ) on E1 (N) E1 (N) k k × is a divisorial lifting of the cycle 6(Γ0 Γp) on (N)F Spec F (N)F . p − E1 p × p E1 p Proof. The proof of this is identical to the proof of Lemma XI.3.1 using the Eichler-Shimura relation (1.1) in this context and regarding ∆ as a differential form on k(N); see also [Con, Theorem 5.3.3.1]. E1 3. The cohesive Flach system Recall that we have assumed that N 5 is squarefree and that l max 7, k+1 is a prime not dividing N. We have also≥ assumed that ≥ { } k E1 (N) is cohomologically torsion-free at l; • k+1 k V is a direct summand of H˜ (E (N) ¯ , Zl) as a Zl-module. • 1 Q (k) B∗ is the image of T1(N) in EndZl V for ˜ 1 k 1 V = H Y1(N)Q¯ , Sym R f Zl ∗ k+1 k regarded as a direct summand of H (E1 (N)Q¯ , Zl). We have A = Bm∗ and H = Vm for m a non-Eisenstein maximal ideal of B∗ associated to a newform, and we have fixed a Gorenstein trace tr : A Zl with congruence element η. → Theorem 3.1. Let H be a modular Galois representation as above and set 0 T = End H(1). Assume that T A k is absolutely irreducible and that A ⊗ 1 H Q(T ∗[a])/Q,T ∗[a] = 0 for all ideals a of finite index in A. Then T admits a cohesive Flach system of Eichler-Shimura type of depth η and weight 12. − Proof. The proof of this is virtually identical to the proof of Theorem XI.3.1. In particular, the crystalline condition again follows from Proposition VII.10.1 since the cycle class of Tp is easily seen to be scalar. 4. Applications As observed in [Fla92], results like Theorem 3.1 can also be used to show that certain deformation problems are unobstructed. Theorem 4.1. Let H and T be as above; in particular, we assume that l does not divide N. Let S denote the set of places of Q dividing Nl together with the archimedean place. Assume that:

T A k is absolutely irreducible; • 1⊗ H Q(T ∗[a])/Q,T ∗[a] = 0 for all ideal a of ﬁnite index in A; • 4. APPLICATIONS 127

η = A; • 0 0 H Qp, End (H A k)(1) = 0 for all p dividing Nl. • k ⊗ Then the deformation problem (with ﬁxed determinant) for the residual representation ρ : GQ Autk(H A k) S → ⊗ is cohomologically unobstructed. In particular, the associated universal deformation ring is a power series ring in two variables over W (k). Proof. This is derived from Theorem 3.1 as in [Fla92, Section 3]; see also [Wes00, Sections 3–5]. We are forced to impose the determinant condition as the Shafarevich-Tate group of the determinant of H A k is diﬃcult to control if H ⊗ does not have cyclotomic determinant.

Appendix

APPENDIX A

Edge maps of spectral sequences

In this appendix we prove various compatibility results for edge maps of spectral sequences which are used in the text.

1. Notation for filtered complexes We will use the notation of [Wei94, Chapter 5]; for clarity we review what we will need. We work only in the level of generality which we will need for the applications. Note that we can afford to be a bit carefree, as we already know that everything we are looking at is well-defined. We will also ignore all categorical issues without further comment. Let C• be a cochain complex supported in non-negative degree. We will write d for the differential on C•. We assume that C• has a filtration F •C•: n+1 n 1 0 F C• F C• F C• F C• = C•. · · · ⊆ ⊆ ⊆ · · · ⊆ ⊆ We assume that this filtration is canonically bounded: recall that this means that F n+1Cn = 0 for each n, so that we have filtrations 0 = F n+1Cn F nCn F 1Cn F 0Cn = Cn. ⊆ ⊆ · · · ⊆ ⊆ We associate a spectral sequence to this data as follows. For each p, q, r 0, set ≥ Apq = c F pCp+q d(c) F p+rCp+q+1 . r ∈ | ∈ Note that from this definition the differential yields a map pq p+r,q r+1 (1.1) d : A A − . r → r pq p p+q p+1 p+q p p+q pq Set E0 = F C /F C and let ηpq : F C E0 be the quotient map. Define → pq pq Zr = ηpq(Ar ) pq p r+1,q+r 2 Br = ηpq d(Ar−1 − ) − pq pq pq pq for r 0. Set Z = rZr and B = rBr . We have inclusions ≥ ∞ ∩ ∞ ∪ pq pq pq pq pq pq pq 0 = B0 B1 B Z Z1 Z0 = E0 . ⊆ ⊆ · · · ⊆ ∞ ⊆ ∞ ⊆ · · · ⊆ ⊆ Set pq pq pq Er = Zr /Br ; one checks immediately that this agrees with our previous definition for r = 0. The maps (1.1) above now yield maps pq p+r,q r+1 d : E E − r → r and one checks that this yields a spectral sequence pq p+q (1.2) E H (C•). 0 ⇒ 131 132 A. EDGE MAPS OF SPECTRAL SEQUENCES

Recall that convergence of (1.2) means that for each p, q there is an isomorphism (1.3) Epq = F pHp+q/F p+1Hp+q. ∞ ∼ One checks immediately that (1.3) is induced by the inclusion (1.4) Apq , F pCp+q ∞ → pq pq where A is defined to be rAr . ∞ ∩ 2. Edge maps pq p+q For this section let Ea H be any first quadrant spectral sequence. Fix z pq ⇒ pq pq p, q 0, r a. Define Er to be the elements of Er which survive to E ; in the ≥ ≥ pq pq ∞ case of a filtered complex, this is just Z /Br . We obtain the following sequence of maps: ∞ pq z pq pq p p+q p+1 p+q p+q p+1 p+q p+q (2.1) Er - Er E F H /F H , H /F H H . ← ∞ ↔ → pq z pq pq We will say that this spectral sequence has an x-axis edge map at Er if Er = Er and F p+1Hp+q = 0. (Often we will omit the “x-axis” if it is clear from context.) Under these hypotheses (2.1) becomes pq pq p p+q p+q Er E F H , H ; ∞ ↔ → that is, we obtain a map pq p+q Er H , → pq which explains the terminology. Note that if there is an x-axis edge map at Er pq then there is an x-axis edge map at E for all r0 r. r0 ≥ Example 2.1. Suppose that for some p, q, r the spectral sequence satisfies p+1,q 1 p+2,q 2 p+q,0 (2.2) E − = E − = = E = 0 r r ··· r and p+r,q r+1 p+r+1,q r p+r+2,q r 1 p+q+1,0 (2.3) E − = E − = E − − = = E = 0. r r r ··· r (2.2) implies that the corresponding E terms vanish; this in turn implies that ∞ F p+1Hp+q/F p+2Hp+q = = F p+qHp+q/F p+q+1Hp+q = 0; ··· p+1 p+q pq z pq thus F H = 0. (2.3) implies that Er = Er since every later differential from the (p, q) entry maps to 0. We conclude that under these conditions there is pq an x-axis edge map at Er . b pq There is a similar theory for y-axis edge maps: define Er to be the quotient pq of Er by all boundaries which ever map to it; in the case of a filtered complex, it pq pq is Zr /B . There is a sequence of maps ∞ p+q p p+q p p+q p+1 p+q pq b pq pq (2.4) H -F H F H /F H E , Er Er . ← ↔ ∞ → pq b pq pq We will say that a spectral sequence has a y-axis edge map at Er if Er = Er and F pHp+q = Hp+q. Under these conditions (2.4) reduces to p+q p p+q p+1 p+q pq pq H F H /F H E , Er , ↔ ∞ → so we obtain a map Hp+q Epq. → r Again, if there is a y-axis edge map at Epq, then there is a y-axis edge map at Epq r r0 for all r0 r. ≥ 3. EDGE MAPS IN SPECTRAL SEQUENCES OF FILTERED COMPLEXES 133

Example 2.2. Suppose that for some p, q, r we have p 1,q+1 p 2,q+2 0,p+q E − = E − = = E = 0 r r ··· r and p r,q+r 1 p r 1,q+r p r 2,q+r+1 0,p+q+1 Er − − = Er − − = Er − − = = Er = 0. ··· pq Then as in Example 2.1 one shows that there is a y-axis edge map at Er . In the following we will work exclusively with x-axis edge maps, but the proofs all adapt immediately to the y-axis case as well.

3. Edge maps in spectral sequences of ﬁltered complexes

Let C1• and C2• be ﬁltered complexes as before. Suppose that we are given a map C• C• compatible with the ﬁltrations. This induces a map 1 → 2 pq pq E (C•) E (C•) r 1 → r 2 of spectral sequences and a map p+q p+q H (C•) H (C•) 1 → 2 of cohomology groups. Proposition 3.1. Suppose that for some p, q, r there are edge maps pq p+q E (C•) H (C•); r 1 → 1 pq p+q E (C•) H (C•). r 2 → 2 Then the diagram pq pq Er (C1•) / Er (C2•)

p+q p+q H (C1•) / H (C2•) commutes. Proof. Consider the expanded diagram

pq pq (3.1) Er (C1•) / Er (C2•)

pq pq E (C1•) / E (C2•) ∞ ∞

p p+q p+1 p+q p p+q p+1 p+q F H (C1•)/F H (C1•) / F H (C2•)/F H (C2•)

p+q p+q H (C1•) / H (C2•) All maps in (3.1) exist by our assumption that the edge maps exist. The first and last squares commute by the definitions of morphisms of spectral sequences and filtered complexes. This leaves the middle square: as we observed in (1.4), the maps pq p p+q p+1 p+q E (Ci•) F H (Ci•)/F H (Ci•) ∞ → 134 A. EDGE MAPS OF SPECTRAL SEQUENCES are induced by the inclusions Apq , F pCp+q, and now the commutativity of the ∞ → middle square is clear as well.

4. Edge maps in Grothendieck spectral sequences I Let , and be abelian categories such that and have enough injectives. Let G : A B andC F : be left exact functors.A SupposeB that G maps injective objectsA to →F -acyclic B objects.B → C Under these hypotheses for each object A of one obtains a Grothendieck spectral sequence A

(4.1) Epq = (RpF )(RqG)(A) Rp+q(FG)(A). 2 ⇒

(4.1) is constructed as follows: one begins with an injective resolution A I• of A. → The complex G(I•) admits a Cartan-Eilenberg resolution J ••. The Grothendieck spectral sequence (4.1) is the spectral sequence associated to the ﬁltration by rows of the total complex of the double complex F (J ••); see [Wei94, Section 5.8] for details. pq pq Now let A1 and A2 be two objects of and write Er (A1) and Er (A2) for the corresponding Grothendieck spectral sequences.A Suppose that there is a morphism pq pq A1 A2 in . We now construct a corresponding morphism E (A1) E (A2) → A r → r of spectral sequences. Begin with injective resolutions A1 I• and A2 I•. Stan- → 1 → 2 dard properties of injective resolutions show that the map A1 A2 lifts to a map → of complexes I• I•. This in turn yields a map of the complexes G(I•) G(I•), 1 → 2 1 → 2 and this lifts to a map J1•• J2•• of any corresponding Cartan-Eilenberg resolutions. Taking F of these complexes→ and passing to the associated total complex yields a map

(4.2) Tot F (J ••) Tot F (J ••) 1 → 2 compatible with ﬁltrations by rows or columns. (4.2) in turn yields a map of the associated Grothendieck spectral sequences. Proposition 3.1 can be restated in this situation as follows.

Proposition 4.1. Let A1 A2 be a morphism in . This morphism induces pq pq → A a morphism Er (A1) Er (A2) of spectral sequences. Suppose further that for some p, q, r there are edge→ maps

pq p+q E (A1) R (FG)(A1); r →

pq p+q E (A2) R (FG)(A2). r → Then the diagram

pq pq Er (A1) / Er (A2)

p+q p+q R (FG)(A1) / R (FG)(A2) commutes. 5. EDGE MAPS IN GROTHENDIECK SPECTRAL SEQUENCES II 135

5. Edge maps in Grothendieck spectral sequences II Suppose that one is given categories and functors forming a commutative diagram G1 / F1 / 1 1 ? A B C α β F2 G2 2 / 2 A B Suppose that the Fi,Gi are left exact, that Gi maps injectives to Fi-acyclics, and that α and β are exact. Given an object A of 1, these hypotheses allow us to pq A pq form Grothendieck spectral sequences Er,1 for A and Er,2 for αA. Proposition 5.1. Under the above hypotheses there is a natural map of spectral pq pq sequences Er,1 Er,2 deﬁned at the r = 0 stage. For r = 2 it agrees with the natural map → p q p q R F1R G1(A) R F2R G2(αA). → Furthermore, if for some p, q, r there exist edge maps pq p+q E R (F1G1)(A); r,1 → pq p+q E R (F2G2)(αA); r,2 → then the diagram

pq pq Er,1 / Er,2

p+q p+q R (F1G1)(A) / R (F2G2)(αA) commutes, where the bottom map is the natural map.

Proof. Begin with an injective resolution A I•. Let J •• be a Cartan- → 1 1 Eilenberg resolution of the complex G1(I1•). The Grothendieck spectral sequence for F1 and G1 is the spectral sequence associated to the filtration by rows of the total complex of F1(J1••). The same construction for αA yields an injective resolution I2• and a Cartan- Eilenberg resolution J2•• of G2(I2•). Since α is exact, αI1• is still exact; thus it is a resolution of αA. This implies that there exists a map of complexes αI• I•. This 1 → 2 yields a map G2αI• G2I•. Since G2α = βG1, we obtain a map βG1I• G2I•. 1 → 2 1 → 2 β is exact, so βJ1•• is a resolution of βG1I1• and thus maps to the Cartan-Eilenberg resolution J2••. Applying F2 yields a map F2βJ •• F2J ••. Since F2β = F1, this is a map 1 → 2 (5.1) F1J •• F2J •• 1 → 2 of double complexes. (5.1) induces a map of filtered complexes, and the proposition pq pq now follows from Proposition 3.1 so long as we check that the maps E2,1 E2,2 p+q p+q → and R (F1G1)(A) R (F2G2)(αA) are the natural maps. →pq pq We begin with E2,1 E2,2. These maps are induced from (5.1) after first taking horizontal cohomology→ and then vertical cohomology. By the definition of

a Cartan-Eilenberg resolution, horizontal cohomology of Ji•• yields resolutions (of 136 A. EDGE MAPS OF SPECTRAL SEQUENCES

q q complexes) H (GiI•) Ji •. By the definition of derived functors, we can identify these with resolutions→ q q R G1(A) J • → 1 q q R G2(αA) J •. → 2 Since the map βJ1•• J2•• sits over a map βG1I1• G2I2•, we now see immediately pq → pq → that the map E1,1 E1,2 (which is what we obtain after horizontal cohomology) → q q pq sits over the natural maps βR G1(A) R G2(αA). The E1,i’s form injective resolutions of these and the cohomology→ computes the right derived functors of pq pq the Fi. Since the map E1,1 E1,2 sits over these natural maps we see now that pq pq → p q p q E2,1 E2,2 coincides with the natural maps R F1R G1(A) R F2R G2(αA), as claimed.→ → For the naturality of the other maps, it is immediate from the definitions that p+q p+q the map R F1(G1I1•) R F2(G2I2•) induced by the spectral sequence map pq pq → Er,1 Er,2 is the natural map. We must check that the corresponding map p+q → p+q I pq R (F1G1)(A) R (F2G2)(αA) obtained from the collapsing of Er,i is the natural map. Recall→ that to compute in this spectral sequence we first take vertical cohomology and then take horizontal cohomology. But since Gi takes injectives q to Fi-acyclics, after taking vertical cohomology (which computes R Fi(GiI•)), we 0 are left with a single row R Fi(GiI•) = FiGi(I•). One now sees as before that p+q p+q horizontal cohomology yields the usual map R (F1G1)(A) R (F2G2)(αA). →

6. Boundary maps of exact sequences of ﬁltered complexes

Let C• be a filtered complex as in Section 1. Note that in the first stage of the spectral sequence constructed from C• the differentials are all horizontal. That q is, for fixed q, E1• can be considered as a complex as well, and its cohomology is q nothing other than E2• . Now let 0 C• C• C• 0 → 1 → 2 → 3 → be an exact sequence of filtered complexes. Suppose in addition that for some q the induced sequence q q q 0 E• (C•) E• (C•) E• (C•) 0 → 1 1 → 1 2 → 1 3 → is also exact. We obtain long exact sequences of cohomology in both cases: n 1 n n n n+1 (6.1) H − (C•) H (C•) H (C•) H (C•) H (C•) · · · → 3 → 1 → 2 → 3 → 1 → · · · p 1,q pq pq pq p+1,q (6.2) E − (C•) E (C•) E (C•) E (C•) E (C•) · · · → 2 3 → 2 1 → 2 2 → 2 3 → 2 1 → · · · In particular, we have boundary maps p+q p+q+1 H (C•) H (C•) 3 → 1 pq p+1,q E (C•) E (C•) 2 3 → 2 1 for each p. Proposition 6.1. Suppose that for some p, q 0 there exist edge maps ≥ pq p+q E (C•) H (C•); 2 3 → 3 p+1,q p+q+1 E (C•) H (C•). 2 1 → 1 7. BOUNDARY MAPS OF GROTHENDIECK SPECTRAL SEQUENCES 137

Then the diagram pq / p+1,q (6.3) E2 (C3•) E2 (C1•)

p+q p+q+1 H (C3•) / H (C1•) commutes. Proof. Consider the diagram (in the notation of Section 1)

(6.4) Ap+1,q(C ) / Ap+1,q(C ) / Ap+1,q(C ) 1 O 1• 1 O 2• 1 O 3•

pq pq pq A1 (C1•) / A1 (C2•) / A1 (C3•) On the one hand, (6.4) surjects onto the diagram

Ep+1,q(C ) / Ep+1,q(C ) / Ep+1,q(C ) 1 O 1• 1 O 2• 1 O 3•

pq pq pq E1 (C1•) / E1 (C2•) / E1 (C3•) from which the boundary maps of (6.2) are computed. We can therefore also compute these boundary maps after lifting to the diagram (6.4). (Recall that pq pq the boundary map is computed by lifting from E1 (C3•) to E1 (C2•), mapping to p+1,q p+1,q E1 (C2•) and pulling back to E1 (C1•).) On the other hand, (6.4) naturally injects into the diagram Cp+q+1(C ) / Cp+q+1(C ) / Cp+q+1(C ) O 1• O 2• O 3•

p+q p+q p+q C (C1•) / C (C2•) / C (C3•) from which one computes the boundary maps of (6.1). The edge maps (when they exist) are induced by these injections. Since boundary maps are computed by the same procedure in both cases, it is now clear (6.3) commutes.

7. Boundary maps of Grothendieck spectral sequences We return now to the set-up of Section 4. We now assume that we have three left exact functors F1,F2,F3 : B → C and that G : takes injectives to Fi-acyclic objects for each i. Suppose further that forA any → injective B object I of , the sequence B (7.1) 0 F1(I) F2(I) F3(I) 0 → → → → is exact. Let us now go through the construction of the Grothendieck spectral sequence again. Begin with an object A of . One ﬁrst forms an injective resolution A I• A → of A. Next, one takes a Cartan-Eilenberg resolution J •• of the complex G(I•). 138 A. EDGE MAPS OF SPECTRAL SEQUENCES

Applying each Fi to this, we obtain three double complexes Fi(J ••) in . In fact, we obtain an exact sequence C

0 F1(J ••) F2(J ••) F3(J ••) 0 → → → → of double complexes by the exactness of (7.1). This in turn yields an exact sequence

0 Tot F1(J ••) Tot F2(J ••) Tot F3(J ••) 0 → → → → of the total complexes, compatible with filtrations by rows and columns. Consider now the filtrations of the Fi(J ••) by rows. At the first stage of the associated spectral sequence one takes the horizontal cohomology. Since J •• is a Cartan-Eilenberg resolution, we can first form the cohomology of the complex J •• and then apply Fi. By the definition of a Cartan-Eilenberg resolution, the horizontal cohomology of J •• yields injective resolutions of the cohomology complex H•(G(I•)). The first stage of the spectral sequence is obtained by applying Fi to this double complex; since the complex still consists of injectives, we again obtain an exact sequence of double complexes

0 E•• E•• E•• 0 → 1,1 → 1,2 → 1,3 → where the spectral sequence E1••,i is the Grothendieck spectral sequence for the p+1,q p,q composition of Fi and G. In particular, we get boundary maps E2 E2 as in (6.2). Given all of this, ﬁxing q and translating Proposition 6.1 into the→ language of Grothendieck spectral sequences yields the following result.

Proposition 7.1. Let F1,F2,F3,G be functors as above. Suppose that for some p, q, r and A there exist edge maps p q p+q R F3R G(A) R (F3G)(A) → and p+1 q p+q+1 R F1R G(A) R (F1G)(A). → Then the diagram

p q p+1 q R F3R G(A) / R F1R G(A)

p+q p+q+1 R (F3G)(A) / R (F1G)(A) of edge maps and boundary maps commutes.

8. Edge maps of exact couples We conclude this appendix by considering the spectral sequence of an exact couple. Since [Wei94, Section 5.9] does not consider the cohomological case, we ﬁrst set our notation. We will work in a somewhat restricted setting, as this is all which we will need for the applications. We being by recalling the construction of the derived couple. Our terminology is adapted to our situation and is not standard. Consider an exact diagram 1 E i / D ` D @@ ~ @@ ~~ k @@ ~~j @ ~~ E 8. EDGE MAPS OF EXACT COUPLES 139

We define the (second) derived couple 2 to be E i(D) i / i(D) Nf NNN ppp NNN ppp k NN pp (2) NN px pp j ker(jk)/ im(jk) where j(2)(i(d)) = j(d). One checks easily that these maps are all well defined and that 2 is an exact couple; see [Wei94, Definition 5.9.1]. The rth derived couple r ofE 1 is defined to be the exact couple obtained from 1 after r 1 iterations of thisE construction.E E − pq pq Now suppose further that D and E are bigraded: D = pqD , E = pqE . Assume also that the maps i, j and k have bidegrees ( ⊕1, 1), (0, 0) and⊕ (1, 0) r th −pq respectively. Let be the r derived couple, and let Er be the (p, q)-graded E (r) pq piece of the E-term in this couple. Letting j k be the differential for Er , one (r) pq sees immediately that j k has bidegree (r, r + 1), so that Er is a spectral sequence. − We now make some more simplifying assumptions. Assume that E is concen- pq pq pq p 1,q+1 trated in the first quadrant; that D = 0 for q < p; and that i : D D − pq→ is an isomorphism for p 0. In this situation one shows easily that Er converges n 0,n ≤ p n r p,n p to H = D , with filtration F H = i (D − ) for any r p (here using the fact ≥ pq pq that i is eventually an isomorphism). For any r large enough so that Er = E , the isomorphism ∞

p n p+1 n r p,n p r+1 p,n p pq pq (8.1) F H /F H = i (D − )/i (D − ) = Er = E ∼ ∞ is induced by j(r). Suppose now that we have a morphism (of degree (0, 0)) 1 2 of bigraded exact couples. It is immediate from the construction aboveE that→ Ethis induces a morphism pq pq E ( 1) E ( 2) r E → r E of the corresponding spectral sequences. In the proposition below we consider only x-axis edge maps; the y-axis case is somewhat diﬀerent. Proposition 8.1. Suppose that for some p, q, r there exist x-axis edge maps

pq p+q E ( 1) H ( 1); r E → E

pq p+q E ( 2) H ( 2). r E → E Then the diagram

pq pq E ( 1) / E ( 2) r E r E

p+q p+q H ( 1) / H ( 2) E E commutes, where the bottom map is induced from the map D0,n D0,n. 1 → 2 140 A. EDGE MAPS OF SPECTRAL SEQUENCES

Proof. Consider the diagram pq pq E ( 1) / E ( 2) r E r E

pq pq E ( 1) / E ( 2) ∞ E ∞ E

p p+q p+1 p+q p p+q p+1 p+q F H ( 1)/F H ( 1) / F H ( 2)/F H ( 2) E E E E

p+q p+q H ( 1) / H ( 2) E E Recalling the deﬁnitions of each object, the only non-obvious commutativity is the middle square, and this is also clear since by (8.1) the maps can be taken to be (r) induced by j in both cases. APPENDIX B

Gorenstein linear algebra

In this appendix we will prove the results on linear algebra over Gorenstein rings which we will need in the text. We also include a basic discussion of bilateral derivations and a few results on torsion Zl-modules.

1. Deﬁnitions

We will restrict ourselves to the case of ﬁnite, ﬂat, local Zl-algebras.

Definition 1.1. Let A be a finite, flat, local Zl-algebra with residue field k. 1 We say that A is Gorenstein if ExtA(k, A) ∼= k. Note that HomA(k, A) = 0 since A is torsion-free. Since A necessarily has Krull dimension 1, this implies that this definition is the same (at least for finite, flat, local Zl-algebras) as that of [Mat86, Section 18]. It is a general fact [Mat86, Theorem 21.3] that local complete intersection rings (and therefore regular local rings) are Gorenstein. In particular, any ring of the form Zl[x]/f(x) for a monic polynomial f(x) is Gorenstein. Gorenstein rings are also necessarily Cohen-Macaulay; see [Mat86, Theorem 18.1]. The following characterization of finite, flat, local, Gorenstein Zl-algebras is much more concrete and will be especially useful for us. For a finite, flat, local Zl-algebra A, we make HomZ (A, Zl) an A-module via af(x) = f(ax) for a A l ∈ and f HomZl (A, Zl). Note that HomZl (A, Zl) is isomorphic to A as a Zl-module, but they∈ need not be isomorphic as A-modules.

Lemma 1.2. Let A be a finite, flat, local Zl-algebra with maximal ideal m and residue field k. Then the following conditions are equivalent. (1) A is Gorenstein; (2) dimk(A/lA)[m] = 1;

(3) HomZl (A, Zl) is free of rank 1 as an A-module. Proof. To prove the equivalence of the ﬁrst two statements, consider the exact sequence (1.1) 0 A l A A/lA 0. → → → → Since HomA(k, A) = 0, we obtain from (1.1) an exact sequence

1 l 1 (1.2) 0 HomA(k, A/lA) Ext (k, A) Ext (k, A). → → A → A 1 But l kills k and thus kills ExtA(k, A); therefore (1.2) yields an isomorphism 1 HomA(k, A/lA) ∼= ExtA(k, A). HomA(k, A/lA) naturally identiﬁes with (A/lA)[m], from which the equivalence of (1) and (2) is clear.

141 142 B. GORENSTEIN LINEAR ALGEBRA

To prove the equivalence of the second two statements, we will ﬁrst show that dimk(A/lA)[m] = 1 is equivalent to HomFl (A/lA, Fl) being free of rank 1 over

A/lA. First assume that HomFl (A/lA, Fl) is free of rank 1 over A/lA. We have (1.3) (A/lA)[m] = Hom (A/lA, F )[m] = Hom (A/m, F ). ∼ Fl l ∼ Fl l The last term in (1.3) is easily seen to be a k-vector space of dimension 1, so that (A/lA)[m] is as well, as claimed. Next assume that dimk(A/lA)[m] = 1. We have an isomorphism

(1.4) HomF (A/lA, Fl) A/m = HomF (A/lA)[m], Fl . l ⊗A/lA ∼ l

A/lA[m] is a one-dimensional k-vector space, so HomFl (A/lA, Fl) A/lA k is as well. Nakayama’s lemma shows that any lift of a generator of this⊗ module to

HomFl (A/lA, Fl) will generate it as well; that is, HomFl (A/lA, Fl) is a cyclic A/lA- module. An easy Fl-dimension count together with (1.4) now shows that it must be free of rank 1 over A/lA. This establishes the asserted equivalence. To prove the lemma it now suﬃces to show that Hom (A/lA, F ) = A/lA is Fl l ∼ equivalent to Hom (A, Z ) = A. For this, note that Zl l ∼ HomZ (A, Zl) A A/lA = HomF (A/lA, Fl). l ⊗ ∼ l From here a Nakayama’s lemma argument and a Zl-rank count ﬁnish the proof.

Note that Lemma 1.2 says that Gorenstein Zl-algebras are in some sense those which are self-dual. To the best of my knowledge, this sort of result ﬁrst appeared in [Maz77, Chapter 2, Section 15].

2. Gorenstein traces and congruence elements

For the next two sections we fix a finite, flat, local, Gorenstein Zl-algebra A.

We will call any A-generator of HomZl (A, Zl) a Gorenstein trace for A. Note that the choice of a Gorenstein trace is determined up to multiplication by an element of A×. Let tr : A Zl be a choice of Gorenstein trace for A. Consider the ring → A Zl A, which we will always regard as an A-algebra via multiplication on the right⊗ factor of A. We have an isomorphism

HomZ (A, Zl) Z A = HomA(A Z A, A), l ⊗ l ∼ ⊗ l so that this module is free of rank 1 over A Z A. (In the A Z A-action on ⊗ l ⊗ l HomA(A Zl A, A), the left factor of A must act on the domain, but the right factor can⊗ act on either the domain or the range by A-linearity.) tr induces an A Z A-generator Tr = tr 1 : A Z A A of HomA(A Z A, A): ⊗ l ⊗ ⊗ l → ⊗ l Tr(a a0) = tr(a)a0. ⊗ Let ∆ : A Z A A be the diagonal map. Since Tr is a generator of ⊗ l → HomA(A Z A, A), we can write ∆ = ι Tr for a unique ι A Z A. We de- ⊗ l ∈ ⊗ l ﬁne the congruence element ηtr associated to tr to be ∆(ι). That is, ηtr is the image of 1 A under the maps ∈ ∆ α Tr α ∆ (2.1) A = HomA(A, A) ◦ HomA(A Z A, A) 7→ A Z A A. ∼ −→ ⊗ l −→ ⊗ l −→ We deﬁne the congruence ideal of A to be the A-ideal ηtrA; the next lemma shows that this ideal is independent of the choice of Gorenstein trace tr. 3. GORENSTEIN DUALITY 143

Lemma 2.1. For any u A×, ∈ 1 ηu tr = u− ηtr.

Proof. u tr is also a generator of HomZl (A, Zl), given by u tr(a) = tr(ua). The associated generator of HomA(A Zl A, A) is therefore Tr0 = (u 1) Tr. Writing 1 ⊗ 1 ⊗ 1 ∆ = ι Tr, we have ∆ = ι(u− 1) Tr0. Thus ηu tr = ∆(ι(u− 1)) = u− ηtr as ⊗ ⊗ claimed. For a definition of the congruence ideal for more general rings in a relative setting, see [Len97]. For us it will be critical to pin down the congruence element associated to a given Gorenstein trace, which more general formulations can not do. The next result is useful for giving conditions under which there is an exact sequence 0 A A A/ηA 0. → → → → Lemma 2.2. Let A be a finite, flat, local, Gorenstein Zl-algebra and let η be any congruence element for A. Then A is reduced if and only if η is a non-zero-divisor.

3. Gorenstein duality A key property of modules over Gorenstein rings is that it is possible to go between A-linear maps to A and Zl-linear maps to Zl, in the sense of the following lemmas. Lemma 3.1. Let M be a finitely generated free A-module. Fix a Gorenstein trace tr of A. Then the map f tr f is an isomorphism 7→ ◦ Hom (M,A) = Hom (M, Z ). A ∼ Zl l Proof. One immediately reduces to the case where M is free of rank 1, in which case this is just the definition of Gorenstein trace. Lemma 3.2. Let M be a finitely generated A-module. Fix a Gorenstein trace tr of A. Then the map f tr f is an isomorphism 7→ ◦ HomA(M,A Z Ql/Zl) = HomZ (M, Ql/Zl). ⊗ l ∼ l Proof. This follows from Lemma 3.1 on taking a resolution of M by free A-modules. Let M be a finitely generated free A-module. By Lemma 3.1 we can associate to a Gorenstein trace tr : A Zl an isomorphism → (3.1) Hom (M,A) = Hom (M, Z ). A ∼ Zl l We can use (3.1) to define a map

htr : EndZ M EndA M l → as the composition

EndZ M = HomZ (M, Zl) Z M = HomA(M,A) Z M l ∼ l ⊗ l ∼ ⊗ l HomA(M,A) A M = EndA M. ⊗ ∼ htr is a fairly remarkable map, as it associates to a Zl-linear endomorphism of M an A-linear endomorphism of M. The description of htr on A-linear endomorphisms of A will be of fundamental importance to us. 144 B. GORENSTEIN LINEAR ALGEBRA

Lemma 3.3. Let M be a ﬁnitely generated free A-module. Let tr be a Gorenstein trace for A. Then the endomorphism

htr EndA M EndZ M EndA M → l −→ of EndA M is multiplication by ηtr. Proof. As usual, we reduce immediately to the case M = A. Tracing through the maps above, we see that we must prove that the identity map in EndZl A maps to ι under the isomorphisms

(3.2) EndZ A = HomZ (A, Zl) Z A = A Z A, l ∼ l ⊗ l ∼ ⊗ l where ι A Zl A is such that ∆ = ι Tr. But this is clear after making the identiﬁcation∈ ⊗

(3.3) HomZ (A, Zl) Z A = HomA(A Z A, A) l ⊗ l ∼ ⊗ l and observing that the image of the identity map in (3.3) under (3.2) is precisely ∆. 4. Gorenstein pairings The next result again goes back to [Maz77, Chapter 2, Section 15]; it is yet another relationship between Gorenstein rings and duality. Let A be a finite, flat, local Zl-algebra and let M and N be finitely generated A-modules. We say that a pairing ψ : M Z N Zl ⊗ l → is A-hermitian if ψ(am, n) = ψ(m, an) for all a A, m M, n N. Equivalently, this means that the natural maps ∈ ∈ ∈

M HomZ (N, Zl) → l N HomZ (M, Zl) → l are maps of A-modules.

Lemma 4.1. Let A be a ﬁnite, ﬂat, local Zl-algebra and let H be a free A-module of rank 2. Then A is Gorenstein if and only if there exists an A-hermitian perfect pairing ψ : H Z H Zl. ⊗ l → Proof. First assume that such a ψ exists. Since ψ is perfect, there is an induced isomorphism H = Hom (H, Z ) ∼ Zl l of A-modules. Let A† denote the A-module Hom(A, Zl); we need to show that A† is a free A-module of rank 1. But this follows from the isomorphism

A† A† = HomZ (H, Zl) = H = A A ⊕ ∼ l ∼ ∼ ⊕ and the fact that A is ﬂat over Zl. For the other direction, let tr : A Zl be a Gorenstein trace and let x, y be a basis for H. One checks easily that the→ pairing ψ given by ψ(ax + by, cx + dy) = tr(ad bc) is perfect, A-hermitian and even skew-symmetric. This completes the − proof. We will call a pairing as in Lemma 4.1 a Gorenstein pairing. The next result give a method of factoring Gorenstein pairings. 5. SKEW-SYMMETRIC GORENSTEIN PAIRINGS 145

Lemma 4.2. Let A be a ﬁnite, ﬂat, local, Gorenstein Zl-algebra and let

ψ : H Z H Zl ⊗ l → be a Gorenstein pairing. Let tr : A Zl be any Gorenstein trace for A. Then the pairing ψ factors as →

ψ0 tr H Z H H A H A Zl ⊗ l ⊗ −→ −→ where ψ0 is an A-linear perfect pairing.

Proof. Since ψ is A-hermitian, it factors through some pairing H A H Zl. ⊗ → Lemma 3.1 shows that this pairing factors through an A-linear pairing ψ0 : H A ⊗ H A, and one checks immediately that ψ0 is perfect. → 5. Skew-symmetric Gorenstein pairings As the proof of Lemma 4.1 indicates, the most natural Gorenstein pairings are those that are skew-symmetric. In this section we investigate the linear algebra associated to such a pairing. Let A be a ﬁnite, ﬂat, local Zl-algebra and let H be a free A-module of rank 2. Let ψ : H Z H Zl ⊗ l → be a skew-symmetric Gorenstein pairing. ψ induces a map 2 (5.1) H Zl, ∧A → 2 and it is easily checked that any choice of isomorphism of A with AH turns (5.1) into a Gorenstein trace. Equivalently, we have the following lemma.∧

Lemma 5.1. Let A be a ﬁnite, ﬂat, local, Gorenstein Zl-algebra and let

ψ : H Z H Zl ⊗ l → be a skew-symmetric Gorenstein pairing. Let x, y be a ﬁxed A-basis of H such that ψ(x, y) = 1. Then the map tr : A Zl given by tr(a) = ψ(ax, y) is a Gorenstein trace. Furthermore, ψ factors as →

ψ0 tr H Z H H A H A Zl ⊗ l ⊗ −→ −→ where ψ0 : H A H A is the pairing ψ0(ax + by, cx + dy) = ad bc. ⊗ → − Proof. That the map tr is a Gorenstein trace is just the preceding argument made explicit. The factorization then follows by an easy computation. Recall that there are canonical decompositions 0 (5.2) EndA H = A End H ∼ ⊕ A 2 2 (5.3) H A H = Sym H. ⊗ ∼ ∧A ⊕ A Lemma 5.2. The map

(5.4) EndA H HomZ (H A H, Zl) → l ⊗ sending f : H H to the map gf (h h0) = ψ(h f(h0)) is an isomorphism of A-modules. Restricting→ to the direct summands⊗ of (5.2)⊗ and (5.3) induces isomorphisms 2 (5.5) A = HomZ ( H, Zl) ∼ l ∧A 146 B. GORENSTEIN LINEAR ALGEBRA

(5.6) End0 H = Hom (Sym2 H, Z ). A ∼ Zl A l Proof. Fix a basis x, y of H such that ψ(x, y) = 1. Let tr be the associated Gorenstein trace as in Lemma 4.2, so that ψ(ax+by, cx+dy) = tr(ad bc). Now let a b − f be an endomorphism of H with matrix c d with respect to x, y. The associated homomorphism

gf : H Z H A ⊗ l → is given by

(5.7) gf (αx + βy, γx + δy) = tr(αγc + αδd βγa βδb). − − That (5.4) is an isomorphism follows from (5.7) by an easy calculation using a basis for EndA H. The isomorphisms (5.5) and (5.6) also follow immediately on considering the cases a = d and a = d, b = c = 0. − Corollary 5.3. The isomorphism of Lemma 5.2 induces an isomorphism

EndA H Z Ql/Zl = HomZ (H A H, Ql/Zl). ⊗ l ∼ l ⊗ 6. Bilateral derivations In this section we give the basic theory of bilateral derivations as developed in [Maz]. We make no eﬀort to work in any level of generality which we will not need for our applications. Let be a commutative Zl-algebra and let M be an A Z -module. A bilateral derivation from to M is a Zl-linear map A ⊗ l A A : M D A → such that (βα) = (α 1) (β) + (1 β) (α) D ⊗ D ⊗ D for all α, β . Note that if the action on M factors through the diagonal map ∈ A ∆ : Zl , then a bilateral derivation is nothing more than a derivation in theA usual ⊗ sense.A → A

The fundamental example of a bilateral derivation is the map δ : Zl given by δ(α) = α 1 1 α. Note that the image of δ lies in the kernelA → AI ⊗of ∆;A one can show that ⊗δ : − ⊗I is the universal bilateral derivation. A → If M is an Z -module, deﬁne Mδ by A ⊗ l A Mδ = m M δ(α)m = 0 for all α . ∈ | ∈ A Mδ is canonically an -module via ∆. A Lemma 6.1. Let : M be a bilateral derivation and let a be an ideal of such that 1 a andDa A1 →annihilate M. Then the restriction of to a yields an A-module homomorphism⊗ ⊗ D A 2 ˜ : a/a Mδ. D → Proof. If β a, then the deﬁnition of a bilateral derivation shows that ∈ (6.1) (α 1) (β) = (βα) = (αβ) = (1 α) (β) ⊗ D D D ⊗ D for all α . Thus (a) Mδ. If also α a, then (6.1) shows that (αβ) = 0; thus (a2∈) A= 0. ThisD proves⊆ the lemma. ∈ D D 6. BILATERAL DERIVATIONS 147

Now let A be a finite, flat, local, reduced, Gorenstein Zl-algebra. Fix a Goren- stein trace tr for A and let η be the associated congruence element; η is a non-zero divisor since A is reduced. Let M and N be free A-modules of finite rank; M Z N ⊗ l is an A Z A-module in the obvious way. ⊗ l Lemma 6.2. There exists a unique A-module isomorphism

ν :(M Z N)δ ' M A N ⊗ l −→ ⊗ fitting into a commutative diagram (M Z N)δ / M Z N ⊗ l ⊗ l ν η M A N / M A N ⊗ ⊗ Proof. The uniqueness of such a ν is clear. To define ν it suffices to consider the case M = N = A. Consider the sequence

∆ (6.2) (A Z A)δ , A Z A A. ⊗ l → ⊗ l −→ Applying HomZ ( , Zl) to (6.2) yields a sequence l · ∆ f (6.3) HomZ (A, Zl) ◦ HomZ (A Z A, Zl) HomZ (A Z A)δ, Zl . l −→ l ⊗ l −→ l ⊗ l

The trace tr and Tr identify the first two terms with A and A Zl A, respectively. We claim that to define ν it suffices to prove that there is a commutative⊗ diagram

f (6.4) HomZ (A Z A, Zl) / HomZ (A Z A)δ, Zl l ⊗O l l ⊗O l Tr · ' ' ∆ A Z A / A ⊗ l Indeed, given (6.4) it follows immediately from (2.1) that (6.3) ﬁts into a commutative diagram

HomZ (A, Zl) ' / HomZ (A Z A)δ, Zl l O l ⊗O l tr · ' ' η A / A

' By duality we conclude that there is an isomorphism ν :(A Zl A)δ A such that ⊗ −→ ν ∆ A (A Z A)δ , A Z A A ←− ⊗ l → ⊗ l −→ is multiplication by η, as claimed. It thus suﬃces to construct (6.4). Note ﬁrst that for any α A Z A such ∈ ⊗ l that lα (A Z A)δ we must have α (A Z A)δ. It follows that (A Z A)δ is a ∈ ⊗ l ∈ ⊗ l ⊗ l Zl-module direct summand of A Zl A and thus that (6.5) ⊗ HomZ (A Z A)δ, Zl = HomZ (A Z A, Zl)/ HomZ A Z A/(A Z A)δ, Zl . l ⊗ l ∼ l ⊗ l l ⊗ l ⊗ l 148 B. GORENSTEIN LINEAR ALGEBRA

There is a commutative diagram (6.6) HomZ A Z A/(A Z A)δ, Zl / HomZ (A Z A, Zl) l ⊗ l O ⊗ l l ⊗O l Tr Tr · ' ' · δ(A) / A Z A ⊗ l by the deﬁnition of δ. The desired diagram (6.4) now arises as the cokernel of (6.6) via (6.5). Lemma 6.3. Let H be a free A-module of ﬁnite rank with an A-linear action of some group G. Assume also that every Jordan-Holder constituent of H Zl H has trivial G-invariants. Then there is a canonical isomorphism ⊗ 1 1 H (G, H Z H)δ = H G, (H Z H)δ . ⊗ l ∼ ⊗ l Proof. This is an easy argument on the level of cocycles; we omit the details.

Note that Lemma 6.3 applies in particular when H A k is absolutely irreducible of rank at least 2. ⊗

7. Torsion modules

In this section we collect some results on modules over finite, flat, local Zl- algebras. Let A be a such a Zl-algebra, with maximal ideal m and residue field k. If T is an A-module and a is an ideal of A, then we write T [a] for the a-torsion in T : T [a] = t T αt = 0 for all α a . ∈ | ∈ We will need the following simple facts about A-modules. Lemma 7.1. Let α A be a non-zero divisor. Then α divides some power of l in A and αA has finite∈ index in A.

Proof. Since A is finite over Zl and α is a non-zero divisor, α necessarily satisfies some monic linear equation of the form n n 1 α + an 1α − + + a1α + a0 = 0 − ··· with ai Zl and a0 = 0. Thus α divides a0, which proves the first statement. Since ∈ 6 a0A αA and a0A visibly has finite index in A (as A is free of finite rank over Zl), ⊆ the second statement is now clear as well. Lemma 7.2. Let T be an A-module and let t be a non-zero element of T annihilated by some power of l. Then there exists α A such that αt = 0 and αt T [m]. ∈ 6 ∈ Proof. Since t is l-power torsion, there is a largest power ln of l such that n t0 = l t = 0; thus t0 T [l]. Note that T [l] is a module over the artinian ring A/lA. 6 ∈ Let α1, . . . , αm be generators of m in A; they also generate the maximal ideal of A/lA and are therefore nilpotent in this ring. It follows that some power of α1 n1 1 annihilates t0; let n1 be the smallest such power, and set t1 = α1 − t0. Continuing n n1 1 nm 1 in this way, we obtain a non-zero element tm = αt where α = l α − α − . 1 ··· m This tm is killed by every generator of m, so it is clearly in T [m].

Lemma 7.3. Let T be a Zl-torsion A-module. If T [m] = 0, then T = 0. 7. TORSION MODULES 149

Proof. Suppose that T = 0 and let t be a non-zero element of T . By Lemma 7.2 there is some α 6 A such that αt is non-zero and annihilated by m. Thus T [m] = 0, which yields the∈ desired contradiction. 6

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