Lectures on Etale Cohomology J.S. Milne Version 2.10 May 20, 2008 These are the notes for a course taught at the University of Michigan in 1989 and 1998. In comparison with my book, the emphasis is on heuristic arguments rather than formal proofs and on varieties rather than schemes. The notes also discuss the proof of the Weil conjectures (Grothendieck and Deligne). BibTeX information @misc{milneLEC, author={Milne, James S.}, title={Lectures on Etale Cohomology (v2.10)}, year={2008}, note={Available at www.jmilne.org/math/}, pages={196} } v2.01 (August 9, 1998). First version on the web; 197 pages. v2.10 (May 20, 2008). Fixed many minor errors; changed TeX style; 196 pages Please send comments and corrections to me at math0 at jmilne.org. The photograph is of the Dasler Pinnacles, New Zealand. Copyright c 1998, 2008, J.S. Milne. Single paper copies for noncommercial personal use may be made without explicit permis- sion from the copyright holder. Contents Contents 3 I Basic Theory 7 1 Introduction . 7 2 Etale Morphisms . 16 3 The Etale Fundamental Group . 25 4 The Local Ring for the Etale Topology . 31 5 Sites . 38 6 Sheaves for the Etale Topology . 41 7 The Category of Sheaves on Xet........................ 49 8 Direct and Inverse Images of Sheaves. 57 9 Cohomology: Definition and the Basic Properties. 63 10 Cechˇ Cohomology. 68 11 Principal Homogeneous Spaces and H 1.................... 74 12 Higher Direct Images; the Leray Spectral Sequence. 79 13 The Weil-Divisor Exact Sequence and the Cohomology of Gm. 82 14 The Cohomology of Curves. 87 15 Cohomological Dimension. 102 16 Purity; the Gysin Sequence. 105 17 The Proper Base Change Theorem. 111 18 Cohomology Groups with Compact Support. 115 19 Finiteness Theorems; Sheaves of Z`-modules. 119 20 The Smooth Base Change Theorem. 122 21 The Comparison Theorem. 125 22 The Kunneth¨ Formula. 130 23 The Cycle Map; Chern Classes . 133 24 Poincare´ Duality . 139 25 Lefschetz Fixed-Point Formula. 142 II Proof of the Weil Conjectures. 145 26 The Weil Conjectures . 145 27 Proof of the Weil Conjectures, except for the Riemann Hypothesis . 148 28 Preliminary Reductions . 155 29 The Lefschetz Fixed Point Formula for Nonconstant Sheaves. 157 30 The MAIN Lemma . 170 31 The Geometry of Lefschetz Pencils . 178 32 The Cohomology of Lefschetz Pencils . 180 33 Completion of the Proof of the Weil Conjectures. 187 34 The Geometry of Estimates . 191 Index 195 3 Notations and conventions The conventions concerning varieties are the same as those in my notes on Algebraic Geom- etry. For example, an affine algebra over a field k is a finitely generated k-algebra A such that A kal is has no nonzero nilpotents for one (hence every) algebraic closure kal of ˝k k — this implies that A itself has no nilpotents. With such a k-algebra, we associate a ringed space Specm.A/ (topological space endowed with a sheaf of k-algebras), and an affine variety over k is a ringed space isomorphic to one of this form. A variety over k is a ringed space .X; OX / admitting a finite open covering X Ui such that .Ui ; OX Ui / D[ j is an affine variety for each i and which satisfies the separation axiom. We often use X to denote .X; OX / as well as the underlying topological space. A regular map of varieties will sometimes be called a morphism of varieties. For those who prefer schemes, a variety is a separated geometrically reduced scheme X of finite type over a field k with the nonclosed points omitted. If X is a variety over k and K k, then X.K/ is the set of points of X with coordinates 1 in K and XK or X is the variety over K obtained from X. For example, if X =K D Specm.A/, then X.K/ Hom .A; K/ and XK Specm.A K/. In general, X.K/ D k-alg D ˝k is just a set, but I usually endow X.C/ with its natural complex topology. A separable closure ksep of a field k is a field algebraic over k such that every separable polynomial with coefficients in k has a root in ksep. Our terminology concerning schemes is standard, except that I shall always assume that our rings are Noetherian and that our schemes are locally Noetherian. Our terminology concerning rings is standard. In particular, a homomorphism A B ! of rings maps 1 to 1. A homomorphism A B of rings is finite, and B is a finite A- ! algebra, if B is finitely generated as an A-module. When A is a local ring, I often denote its (unique) maximal ideal by mA. A local homomorphism of local rings is a homomorphism 1 f A B such that f .mB / mA (equivalently, f .mA/ mB ). W ! D Generally, when I am drawing motivation from the theory of sheaves on a topological space, I assume that the spaces are Hausdorff, i.e., not some weird spaces with points whose closure is the whole space. X YX is a subset of Y (not necessarily proper); X def YX is defined to be Y , or equals Y by definition; D X YX is isomorphic to Y ; X YX and Y are canonically isomorphic (or there is a given or unique isomorphism) ' Prerequisites and References Homological algebra I shall assume some familiarity with the language of abelian cate- gories and derived functors. There is a summary of these topics in my Class Field Theory notes pp 69–76, and complete presentations in several books, for example, in Weibel, C.A., An Introduction to Homological Algebra2, Cambridge U.P., 1994. We shall not be able to avoid using spectral sequences — see pp 307–309 of my book on Etale Cohomology for a brief summary of spectral sequences and Chapter 5 of Weibel’s book for a complete treatment. 1Our terminology follows that of Grothendieck. There is a conflicting terminology, based on Weil’s Foun- dations and frequently used by workers in the theory of algebraic groups, that writes these the other way round. 2 n n p On p 9, Weibel defines C Œp C . The correct original definition, universally used by algebraic Dn n p and arithmetic geometers, is that C Œp C C (see Hartshorne, R., Residues and Duality, 1966, p26). Also, in Weibel, a “functor category” need notD be a category. 4 Sheaf theory Etale cohomology is modelled on the cohomology theory of sheaves in the usual topological sense. Much of the material in these notes parallels that in, for example, Iversen, B., Cohomology of Sheaves, Springer, 1986. Algebraic geometry I shall assume familiarity with the theory of algebraic varieties, for example, as in my notes on Algebraic Geometry (Math. 631). Also, sometimes I will men- tion schemes, and so the reader should be familiar with the basic language of schemes as, for example, the first 3 sections of Chapter II of Hartshorne, Algebraic Geometry, Springer 1977, the first chapter of Eisenbud and Harris, Schemes, Wadsworth, 1992, or Chapter V of Shafarevich, Basic Algebraic Geometry, 2nd Edition, Springer, 1994. For commutative algebra, I usually refer to Atiyah, M., and MacDonald, I., Introduction to Commutative Algebra, Addison-Wesley, 1969. Etale cohomology There are the following books: Freitag, E., and Kiehl, R., Etale Cohomology and the Weil Conjecture, Springer, 1988. Milne, J., Etale Cohomology, Princeton U.P. 1980 (cited as EC). Tamme, Introduction to Etale Cohomology, Springer. The original sources are: Artin, M., Grothendieck Topologies, Lecture Notes, Harvard University Math. Dept. 1962. Artin, M., Theor´ emes` de Representabilit´ e´ pour les Espace Algebriques´ , Presses de l’Universite´ de Montreal,´ Montreal,´ 1973 Grothendieck, A., et al., Seminaire´ de Geom´ etrie´ Algebrique.´ SGA 4 (with Artin, M., and Verdier, J.-L.). Theorie´ des topos et cohomologie etale´ des schemas´ (1963–64). Springer Lecture Notes 1972–73. SGA 4 1/2 (by Deligne, P., with Boutot, J.-F., Illusie, L., and Verdier, J.-L.) Cohomologie etale´ . Springer Lecture Notes 1977. SGA 5 Cohomologie l-adique et fonctions L (1965–66). Springer Lecture Notes 1977. SGA 7 (with Deligne, P., and Katz, N.) Groupes de monodromie en geom´ etrie´ algebriques´ (1967–68). Springer Lecture Notes 1972–73. 1 Except for SGA 4 2 , these are the famous seminars led by Grothendieck at I.H.E.S.. I refer to the following of my notes. FT Field and Galois Theory (v4.20 2008). AG Algebraic Geometry (v5.10 2008). AV Abelian Varieties (v2.00 2008). CFT Class Field Theory (v4.00 2008). Acknowledgements I thank the following for providing corrections and comments for earlier versions of these notes: Martin Bright; Sungmun Cho; Carl Mautner and others at UT Austin; Behrooz Mirzaii; Sean Rostami; Steven Spallone; Sun Shenghao; Zach Teitler; Ravi Vakil. Comment. The major theorems in etale´ cohomology are proved in SGA 4 and SGA 5, but often under unnecessarily restrictive hypotheses. Some of these hypotheses were removed later, but by 5 proofs that used much of what is in those seminars. Thus the structure of the subject needs to be re-thought. Also, algebraic spaces should be more fully incorporated into the subject (see Artin 1973). It is likely that de Jong’s resolution theorem (Smoothness, semi-stability and alterations. Inst. Hautes Etudes´ Sci. Publ. Math. No. 83 (1996), 51–93) will allow many improvements. Finally, such topics as intersection cohomology and Borel-Moore homology need to be added to the exposition. None of this will be attempted in these notes. 6 Chapter I Basic Theory 1 INTRODUCTION For a variety X over the complex numbers, X.C/ acquires a topology from that on C, and so one can apply the machinery of algebraic topology to its study.