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Dualities in Étale Cohomology Università degli studi di Milano Université de Bordeaux I Dualities in étale cohomology Alberto Merici supervised by Baptiste Morin Academic year 2016/2017 Introduction Amor ch’al cor gentil ratto s’apprende, prese costui de la bella persona che mi fu tolta; e ’l modo ancor m’offende Amor ch’a nullo amato amar perdona, mi prese del costui piacer sì forte, che come vedi ancor non m’abbandona. Dante Alighieri, Inferno, Canto V, vv. 100-105 Since its definition by Grothendieck and Deligne, étale cohomology has been seenasa "bridge" between arithmetics and geometry, i.e. something that generalizes purely geomet- ric aspects, as sheaf cohomology, and purely arithmetic ones, as Galois cohomology. One of the most important tools in order to calculate these invariants are duality theorems: • roughly speaking, if Λ is a ring, for a cohomology theory H with values in Λ-mod and a • corresponding compact supported cohomology theory Hc , a duality is a collection of perfect pairings r n−r n H × Hc Ï Hc n where Hc is canonically isomorphic to a "nice" Λ-mod A. The first example of duality theorem one meets throughout the study of algebraic topology is probably Poincaré duality for De Rham cohomology (see [AT11]): Theorem. If X is an oriented differential manifold of dimension n, then the wedge prod- uct induces a perfect pairing of R-vector spaces r n−r n ∼ ∫ H (X, dR) × Hc (X, dR) Hc (X, dR) = R (η, ψ) X η ∧ ψ On the other hand, in Galois cohomology one has the whole machinery of Tate dualities for finite, local and global fields: these are very important tools and I will recall themin the first chapter, mostly following [Mil06]. The aim of this mémoire is to generalize these theorems in the context of étale coho- mology: the second chapter is dedicated to the proof of the Proper Base Change theorem, which I will prove following [Del], and from that proof I will obtain a nice definition of a cohomology with compact support. Then I will deduce Poincaré duality on smooth curves i ii over algebraically closed fields, as it is done in[Del] and [Fu11]. This will follow from an appropriate definition of the cup product pairing coming from the machinery of derived categories. This approach leads almost immediately to a generalization of Poincaré duality on smooth curves over a finite field k, as it is done in [Mil16]. This is the link to arithmetics: in fact this theorem generalizes in some way to Artin-Verdier duality for global fields. The aim of the second part is then to prove Artin-Verdier duality for global fields as it is done in [Mil06], although here, in the case of a number field with at least one real em- bedding, we need to refine the definition of cohomology with compact support in ordetto include the real primes too. This can be done in different ways, and I will briefly explain the approach given by [Mil16]. In the appendix, I will recall some results that are needed. I will recall results on the cohomology of the Idèle group, mostly following [CF67] and [Neu13], then results on the cohomology of topoi and the definition of étale cohomology with some important theorems involved. Most of the theorems are proved in [Tam12] or [Sta]. Then I will recall the definition of derived categories and some results needed in the mémoire, following [Har]. Finally, I will give one powerful application of Poincaré duality for algebraically closed fields: Grothendieck-Verdier-Lefschetz Trace formula and the consequent rationality of L- functions on curves over finite fields. I will give an idea of what it is donein[Del]. iii Acknowledgements First of all, I would like to thank my family for always being there for me and for all the support they have shown throughout all my life: I would not be here without you. I thank mum and dad, for being the best parents I could ever imagine, my brother, for being the exact opposite of me, my grandmas and all my close and far relatives. A special thanks to my little cousin Sara, for being the strongest person I have ever known and I will probably ever know. Secondly, I would like to thank my advisor Baptiste Morin for the incredible amount of time and effort he spent for my career and for listening to all my exposés, introducing me to this crazy and beautiful part of mathematics. He will always be of great inspiration to me. I will also thank the Algant consortium for this great opportunity and all the professors I had the pleasure to meet, both in Milan and Bordeaux, and especially prof. Fabrizio An- dreatta, I owe to him my passion for arithmetic. Mathematics is for weirdos, so I thank all the weirdos I had the pleasure to meet in these years: From Milan I thank Davide, Ale, Mosca, Edo, Fabio and Chiara for all the briscoloni, the Algant team "Senza nome": Isabella, Riccardo, Luca and Massimo for being the best travel companion in this adventure, for "Cuori Rubati", for staying late at university drawing com- mutative diagrams and for all the Skype sessions, Fra, for reminding me that sometimes analysis is nice, and Ema, for reminding that most of the times it’s not, and last but not least, Lori, for being the beautiful person he always is. From Bordeaux, I thank Danilo, for always being there when I needed a friend, Margarita, Corentin, Félicien, Alexandre, Nero and Thibault, for this incredible year together in the IMB. A whole year alone in Bordeaux doing math can drive you insane, but luckily I met the right people to stay real: I thank every beautiful person I met this year, it would be impossible to mention everyone but I am sincerely happy that I have met all of you. I have to give a special thanks to MJ, for the beautiful trips we took together. I would like to thank my longtime friends, who are still trying to figure out what is my position in the world: I thank the "Omega": Fede and his motto "There’s more than one way to be an educated man", Ricca for being the craziest and most generous friend I will always have, and Giacomo for being the lovely jerk he will always be. I also thank Corrado and Stefano, for being the high school friendship that is here for staying, even if I am far away. Finally, a big thank goes to my roomies from Milan, for being the family every student far from home would dream of. And last but not least, I would like to thank my best friend Edoardo for the late night phone calls, all the stupid messages and the support he has given me throughout all my life. Contents 1 Preliminaries: Tate duality1 1.1 Local Tate duality......................................1 1.1.1 Tate cohomology groups.............................1 1.1.2 Duality relative to class formation.......................3 1.1.3 Dualities in Galois cohomology.........................8 1.2 Global Tate duality..................................... 15 1.2.1 P-class formation.................................. 16 1.3 Tate-Poitou.......................................... 20 1.3.1 Proof of the main theorem........................... 22 2 Proper Base Change 28 2.1 Step 1............................................. 29 2.2 Reduction to simpler cases................................ 32 2.2.1 Constructible Sheaves............................... 32 2.2.2 DÃľvissages and reduction to the case of curves.............. 35 2.3 End of the proof....................................... 36 2.3.1 Proof for ℓ = chark ................................ 36 2.3.2 Proof for ℓ ≠ char(k) ............................... 37 2.4 Proper support....................................... 38 2.4.1 Extension by zero................................. 38 2.4.2 Cohomology with proper support....................... 40 2.4.3 Applications..................................... 41 3 Geometry: PoincarÃľ duality 44 3.1 Trace maps.......................................... 44 3.2 PoincarÃľ duality of curves................................ 48 3.2.1 Algebraically closed fields............................ 48 3.2.2 Finite fields..................................... 53 4 Arithmetics: Artin-Verdier duality 55 4.1 Local Artin-Verdier duality................................ 55 4.2 Global Artin-Verdier duality: preliminaries...................... 60 4.2.1 Cohomology of Gm ................................ 61 4.2.2 Compact supported................................ 62 4.2.3 Locally constant sheaves............................. 65 iv CONTENTS v 4.3 Global Artin-Verdier duality: the theorem....................... 67 4.3.1 An application.................................... 76 5 Higher dimensions 77 5.1 Statement of the duality theorem............................ 77 5.2 Motivic cohomology.................................... 78 5.2.1 Locally logarithmic differentials......................... 78 5.2.2 Motivic cohomology................................ 80 Appendix A Global class field theory 81 A.1 AdÃĺle and IdÃĺle...................................... 81 A.2 The IdÃĺle class group................................... 84 A.3 Extensions of the base field................................ 87 A.4 Cohomology of the idÃĺle class group......................... 89 Appendix B Étale cohomology 91 B.1 Sheafification......................................... 91 B.1.1 Yoneda........................................ 92 B.2 Direct images........................................ 93 B.3 Cohomology on a site................................... 96 B.3.1 Čech cohomology................................. 96 B.3.2 Cohomology of Abelian sheaves........................ 98 B.3.3 Flasque Sheaves.................................. 99 B.3.4 Higher Direct Image............................... 101 B.4 Étale cohomology.....................................
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