Etale cohomology course notes Last update: April 22, 2014 at 10:27pm 2 Contents 1 Introduction5 1.1 Prerequisites.............................5 1.2 References...............................5 1.3 History of ´etalecohomology.....................6 2 Category theory7 1 Categories and functors.......................7 1.1 Categories...........................7 1.2 Functors............................8 2 Natural transformations and equivalences of categories......9 2.1 Natural transformations...................9 2.2 Equivalence of categories.................. 10 3 Representable functors, Yoneda's Lemma, and universal properties 10 3.1 Yoneda's Lemma....................... 10 3.2 Representable functors.................... 11 4 Limits and colimits.......................... 12 5 Special limits and colimits...................... 12 A Correspondences and adjunctions.................. 14 B The adjoint functor theorem..................... 15 3 Sheaves and the fundamental group 19 6 The category of sheaves on a topological space.......... 20 6.1 Examples........................... 20 6.2 The espace ´etal´e....................... 20 7 Etale spaces and sheaves....................... 20 7.1 The equivalence of categories................ 20 7.2 Sheafification......................... 23 8 Operations on sheaves........................ 24 8.1 Pushforward and pullback.................. 24 10 Further operations on sheaves.................... 25 10.1 Limits and colimits...................... 25 11 Even further operations on sheaves................. 26 11.1 Restriction.......................... 26 11.2 Stalks............................. 26 3 4 CONTENTS 11.3 Extension by the void.................... 26 11.4 Set theory........................... 26 12 Locally constant sheaves and path lifting.............. 27 12.1 Constant sheaves and locally constant sheaves....... 27 12.2 The homotopy lifting property............... 28 14 Uniform spaces............................ 30 14.1 Filters and uniformities................... 30 14.2 The topology of a uniform space.............. 31 14.3 Complete uniform spaces.................. 31 14.4 Completion.......................... 32 15 Categorical Galois theory...................... 33 15.1 Uniform groups........................ 33 15.2 Infinite Galois theory.................... 34 16 Pseudo-locally constant sheaves................... 35 18 Finite Galois theory......................... 37 19 The fundamental group....................... 39 19.1 Functoriality of the fundamental group........... 39 19.2 The universal cover...................... 39 19.3 The Hawai'ian earring.................... 40 A Sheaves of groups and torsors.................... 42 B Classification of torsors under locally constant groups...... 44 19.1 Crossed homomorphisms and semidirect products..... 44 19.2 Group objects and group actions.............. 44 19.3 Classification of torsors under pseudo-locally constant groups 46 C Fiber functors............................. 47 D The espace ´etal´evia the adjoint functor theorem......... 47 4 Commutative algebra 49 21 Affine schemes............................ 50 21.1 Limits of schemes....................... 50 21.2 Some important schemes................... 50 21.3 Topological rings....................... 51 21.4 The Zariski topology..................... 51 21.5 Why ´etalemorphisms?.................... 52 22 Smooth and ´etalemorphisms.................... 53 22.1 The functorial perspective.................. 53 22.2 The differential perspective................. 54 23 Homology of commutative rings................... 58 24 Homology of modules........................ 61 25 Flatness................................ 64 26 The equational criterion for flatness................ 65 30 Local criteria for flatness....................... 68 31 Flatness of ´etalemaps........................ 70 A Extending ´etalemaps........................ 73 B Completions of rings......................... 77 C Zariski's \Main Theorem"...................... 79 CONTENTS 5 D More perspectives on ´etalemaps.................. 81 31.1 The analytic perspective................... 81 31.2 The algebraic perspective.................. 81 31.3 Equivalence of the definitions................ 82 E Homology of modules........................ 83 31.1 Exact sequences....................... 85 F Cohomology of modules....................... 86 5 The ´etaletopology 89 32 Grothendieck topologies....................... 89 32.1 Examples........................... 90 32.2 Sheaves on Grothendieck topologies............ 91 32.3 More examples of Grothendieck topologies......... 92 33 Sheafification............................. 92 33.1 Topological generators.................... 92 33.2 Descent data......................... 93 34 Fpqc descent............................. 94 35 A rapid review of scheme theory.................. 96 35.1 A heuristic introduction................... 96 35.2 Schemes as functors..................... 98 36 The ´etaletopology.......................... 101 37 Henselization............................. 101 38 The ´etalefundamental group.................... 101 38.1 Covering spaces........................ 101 38.2 Locally constant sheaves................... 102 38.3 The topology on the ´etalefundamental group....... 102 6 Abelian categories and derived functors 103 39 Abelian categories.......................... 103 40 Resolution and derived functors................... 104 40.1 Injective and projective objects............... 104 40.2 Complexes.......................... 104 41 Spectral sequences.......................... 104 7 Cohomology of sheaves 105 42 Acyclic resolutions.......................... 105 42.1 Injective resolution...................... 105 42.2 Flaccid (flasque) resolution................. 105 42.3 Soft resolution........................ 105 42.4 Partitions of unity and de Rham cohomology....... 106 43 Cechˇ cohomology........................... 106 44 Compactly supported cohomology................. 107 44.1 The compactly supported cohomology of the real line.. 107 45 Proper base change.......................... 107 46 Leray spectral sequence....................... 108 46.1 K¨unnethformula....................... 108 6 CONTENTS 46.2 Homotopy invariance of cohomology............ 108 46.3 The cohomology of spheres................. 109 46.4 The cohomology of complex projective space....... 109 47 Poincar´eduality........................... 110 48 Lefschetz fixed point theorem.................... 110 8 Grothendieck topologies 111 49 Sieves, functors, and sheaves..................... 111 50 Morphisms of sites.......................... 112 51 Group cohomology.......................... 113 52 (*) Cohomology in other algebraic categories........... 113 52.1 Groups............................ 113 52.2 Rings............................. 113 52.3 Commutative rings...................... 113 52.4 Hyper-Cechˇ cohomology................... 113 53 Fibered categories and stacks.................... 114 9 Schemes 115 54 Solution sets as functors....................... 115 55 Solution sets as spaces........................ 115 56 Quasi-coherent modules....................... 115 10 Properties of schemes 117 57 Flatness................................ 117 58 Smooth, unramified, and ´etalemorphisms............. 117 59 (*) Weakly ´etalemorphisms..................... 117 11 Curves 119 60 Riemann surfaces........................... 119 61 Riemann{Roch............................ 119 62 Serre duality............................. 119 63 The Jacobian............................. 119 12 Abelian varieties 121 64 Lattices in complex vector spaces.................. 121 65 The dual abelian variety....................... 121 66 Geometric class field theory..................... 121 13 Topologies on schemes 123 67 Faithfully flat descent........................ 123 68 The ´etaletopology.......................... 123 69 Other topologies........................... 123 69.1 The Zariski topology..................... 123 69.2 (*) The pro-´etaletopology.................. 123 69.3 (*) The infinitesimal site................... 123 CONTENTS 7 14 Etale cohomology 125 70 Constructible sheaves and `-adic cohomology........... 125 71 Etale cohomology in low degrees.................. 125 72 Etale cohomology and colimits................... 125 73 Cup product............................. 126 15 Etale cohomology of points 127 74 Group cohomology.......................... 127 75 Hilbert's theorem 90......................... 127 76 The Brauer group.......................... 127 77 Tsen's theorem............................ 127 16 Etale cohomology of curves 131 78 Calculation.............................. 131 79 Poincar´eduality for curves...................... 131 17 Base change theorems 133 80 Smooth base change......................... 133 80.1 Auslander{Buchsbaum formula............... 133 80.2 Purity of the branch locus.................. 134 8 CONTENTS Chapter 1 Introduction 1.1 Prerequisites The essential prerequisites for this course are comfort with point set topology and commutative algebra. Here is a partial list of commutative algebra concepts we will use without review: 1. commutative rings, 2. modules under commutative rings, 3. localization, 4. polynomial rings, 5. tensor product, 6. kernel, cokernel, and image. We will use a lot of homological algebra, but we will review what we use; study- ing homological algebra concurrently might work well. Comfort with the theory of schemes will be assumed as little as possible. I'll do my best to review what we use, but there are bound to be some