Arithmetic Duality Theorems Second Edition J.S. Milne Copyright c 2004, 2006 J.S. Milne. The electronic version of this work is licensed under a Creative Commons Li- cense: http://creativecommons.org/licenses/by-nc-nd/2.5/ Briefly, you are free to copy the electronic version of the work for noncommercial purposes under certain conditions (see the link for a precise statement). Single paper copies for noncommercial personal use may be made without ex- plicit permission from the copyright holder. All other rights reserved. First edition published by Academic Press 1986. A paperback version of this work is available from booksellers worldwide and from the publisher: BookSurge, LLC, www.booksurge.com, 1-866-308-6235, [email protected] BibTeX information @book{milne2006, author={J.S. Milne}, title={Arithmetic Duality Theorems}, year={2006}, publisher={BookSurge, LLC}, edition={Second}, pages={viii+339}, isbn={1-4196-4274-X} } QA247 .M554 Contents Contents iii I Galois Cohomology 1 0 Preliminaries............................ 2 1 Duality relative to a class formation . ............. 17 2 Localfields............................. 26 3 Abelianvarietiesoverlocalfields.................. 40 4 Globalfields............................. 48 5 Global Euler-Poincar´echaracteristics................ 66 6 Abelianvarietiesoverglobalfields................. 72 7 An application to the conjecture of Birch and Swinnerton-Dyer . 93 8 Abelianclassfieldtheory......................101 9 Otherapplications..........................116 AppendixA:Classfieldtheoryforfunctionfields............126 II Etale Cohomology 139 0 Preliminaries............................139 1 Localresults.............................148 2 Globalresults:preliminarycalculations..............163 3 Globalresults:themaintheorem..................176 4 Globalresults:complements....................188 5 Globalresults:abelianschemes...................197 6 Global results: singular schemes ..................205 7 Globalresults:higherdimensions.................208 III Flat Cohomology 217 0 Preliminaries............................218 1 Local results: mixed characteristic, finite group schemes . 232 2 Localresults:mixedcharacteristic,abelianvarieties........245 3 Global results: number field case ..................252 iii iv 4 Localresults:mixedcharacteristic,perfectresiduefield......257 5 Twoexactsequences........................266 6 Local fields of characteristic p ...................272 7 Localresults:equicharacteristic,finiteresiduefield........280 8 Globalresults:curvesoverfinitefields,finitesheaves.......289 9 Global results: curves over finite fields, N´eronmodels.......294 10 Localresults:equicharacteristic,perfectresiduefield.......300 11 Globalresults:curvesoverperfectfields..............304 AppendixA:Embeddingfinitegroupschemes..............307 AppendixB:Extendingfinitegroupschemes..............312 Appendix C: Biextensions and N´eronmodels..............316 Bibliography 328 Index 337 v Preface to the first edition. In the late fifties and early sixties, Tate (and Poitou) found some important du- ality theorems concerning the Galois cohomology of finite modules and abelian varieties over local and global fields. About 1964, Artin and Verdier extended some of the results toetale ´ cohomol- ogy groups over rings of integers in local and global fields. Since then many people (Artin, Bester, B´egueri, Mazur, McCallum, the au- thor, Roberts, Shatz, Vvedens’kii) have generalized these results to flat cohomol- ogy groups. Much of the best of this work has not been fully published. My initial purpose in preparing these notes was simply to write down a complete set of proofs before they were forgotten, but I have also tried to give an organized account of the whole subject. Only a few of the theorems in these notes are new, but many results have been sharpened, and a significant proportion of the proofs have not been published before. The first chapter proves the theorems on Galois cohomology announced by Tate in his talk at the International Congress at Stockholm in 1962, and describes later work in the same area. The second chapter proves the theorem of Artin and Verdier onetale ´ cohomology and also various generalizations of it. In the final chapter improvements using flat cohomology are described. As far as possible, theorems are proved in the context in which they are stated: thus theorems on Galois cohomology are proved using only Galois cohomology, and theorems onetale ´ cohomology are proved using onlyetale ´ cohomology. Each chapter begins with a summary of its contents; each section ends with a list of its sources. It is a pleasure to thank all those with whom I have discussed these questions over the years, but especially M. Artin, P. Berthelot, L. Breen, S. Bloch, K. Kato, S. Lichtenbaum, W. McCallum, B. Mazur, W. Messing, L. Roberts, and J. Tate. Parts of the author’s research contained in this volume have been supported by the National Science Foundation. Finally, I mention that, thanks to the computer, it has been possible to produce this volume without recourse to typist, copy editor1, or type-setter. 1Inevitably, the sentence preceding this in the original contained a solecism vi Preface to the second edition. A perfect new edition would fix all the errors, improve the exposition, update the text, and, of course, being perfect, it would also exist. Unfortunately, these conditions are contradictory. For this version, I have translated the original word- processor file into TEX, fixed all the errors that I am aware of, made a few minor improvements to the exposition, and added a few footnotes. Significant changes to the text have been noted in the footnotes. The number- ing is unchanged from the original (except for II 3.18). All footnotes have been added for this edition except for those on p 26 and p 284. There are a few minor changes in notation: canonical isomorphisms are often denoted ' rather than , and, lacking a Cyrillic font, I use III as a substitute for the Russian letter shah. I thank the following for providing corrections and comments on earlier ver- sions: Ching-Li Chai, Matthias F¨ohl, Cristian Gonzalez-Aviles, David Harari, Eugene Kushnirsky, Bill McCallum, Bjorn Poonen, Jo¨el Riou, and others. Since most of the translation was done by computer, I hope that not many new misprints have been introduced. Please send further corrections to me at [email protected]. 20.02.2004. First version on web. 07.08.2004. Proofread against original again; fixed many misprints and minor errors; improved index; improved TEX, including replaced III with the correct Cyrillic X. 01.07.2006. Minor corrections; reformatted for reprinting. vii Notations and Conventions We list our usual notations and conventions. When they are not used in a particu- lar section, this is noted at the start of the section. A global field is a finite extension of Q or is finitely generated and of finite transcendence degree one over a finite field. A local field is R, C, or a field that is locally compact relative to a discrete valuation. Thus it is a finite extension of Qp, Fp..T //,orR.Ifv is a prime of a global field, then jjv denotes the valuation at v normalized in the usual way so that the product formula holds, and Ov Dfa 2 K jjajv Ä 1g. The completions of K and Ov relative to jjv are b denoted by Kv and Ov: For a field K, Ka and Ks denote the algebraic and separable algebraic clo- sures of K,andKab denotes the maximal abelian extension of K. For a local un field K, K is the maximal unramified extension of K. We sometimes write GK s for the absolute Galois group Gal.K =K/ of K and GF=K for Gal.F=K/.By char.K/ we mean the characteristic exponent of K, that is, char.K/ is p if K has characteristic p ¤ 0 and is 1 otherwise. For a Hausdorff topological group G, Gab is the quotient of G by the closure of its commutator subgroup. Thus, Gab is ab ab the maximal abelian Hausdorff quotient group of G,andGK D Gal.K =K/. If M is an abelian group (or, more generally, an object in an abelian category) .m/ and m is an integer, then Mm and M are the kernel and cokernelS of multi- n plication by m on M . Moreover, M.m/ isT the m-primary component m Mm n and Mmdiv is the m-divisibleT subgroup n Im.m W M ! M/.Thedivisible c subgroup2 M of M is M . We write T M for lim M n and M for the div m m-div m m completion of M with respect to the topology defined by the subgroups of finite index (sometimes the subgroups are restricted to those of finite index a power of a fixed integer m, and sometimes to those that are open with respect to some topology on M ). When M is finite, ŒM denotes its order. A group M is of cofinite-type if it is torsion and Mm is finite for all integers m. As befits a work with the title of this one, we shall need to consider a great many different types of duals. In general, M will denote Homcts.M; Q=Z/,the group of continuous characters of finite order of M . Thus, if M is discrete torsion abelian group, then M is its compact Pontryagin dual, and if M is a profinite abelian group, then M is its discrete torsion Pontryagin dual. If M is a module D s over GK for some field K,thenM denotes the dual Hom.M; K /;whenM D is a finite group scheme, M is the Cartier dual Hom.M;Gm/. The dual (Picard 2This should be called the subgroup of divisible elements — it contains the largest divisible subgroup of M but it need not be divisible itself. A similar remark applies to the m-divisible subgroup. viii variety) of an abelian variety is denoted by At . For a vector space M , M _ denotes the linear dual of M . All algebraic groups and group schemes will be commutative (unless stated otherwise). If T is a torus over a field k,thenX .T / is the group Homks .Gm;Tks / of cocharacters (also called the multiplicative one-parameter subgroups). There seems to be no general agreement on what signs should be used in homological algebra. Fortunately, the signs of the maps in these notes will not be important, but the reader should be aware that when a diagram is said to commute, it may only commute up to sign.