These course notes from spring 2010 are extremely rough, so caveat lector. 1 An Introduction to Etale Cohomology Donu Arapura November 25, 2012 Contents I Introduction3 0 Review of schemes4 0.1 Coordinate rings...........................4 0.2 Spectrum...............................5 0.3 Affine schemes............................6 0.4 Schemes................................7 0.5 Localization..............................8 0.6 Morphisms..............................9 0.7 Schemes as functors......................... 11 1 Differential Calculus of Schemes 13 1.1 Etale´ maps.............................. 13 1.2 K¨ahlerdifferentials.......................... 15 1.3 Flat maps............................... 17 1.4 Extension to schemes......................... 19 2 Etale´ fundamental group 20 2.1 Profinite groups and Galois theory................. 20 2.2 Etale´ fundamental group....................... 21 2.3 First homology............................ 22 3 Etale´ topology 24 3.1 Grothendieck topologies....................... 24 3.2 Sheaves................................ 25 3.3 Faithfully flat descent........................ 26 3.4 Schemes modulo equivalence relations............... 28 4 Cohomology 29 4.1 Exact sequences............................ 29 4.2 Stalks in the ´etaletopology..................... 30 4.3 Cohomology.............................. 31 4.4 Torsors................................ 33 4.5 Galois cohomology.......................... 35 1 5 Cohomology of Curves 38 5.1 Picard group............................. 38 5.2 Etale cohomology of Gm ....................... 40 5.3 Constructible Sheaves........................ 42 6 Cohomology of surfaces 44 6.1 Finiteness of Cohomology...................... 44 6.2 Divisors on a surface......................... 45 6.3 The Lefschetz trace formula..................... 47 6.4 Riemann hypothesis for curves................... 49 7 Comparison with classical cohomology 51 7.1 The comparison theorem for curves................. 51 7.2 The comparison theorem for surfaces................ 53 7.3 Base change.............................. 54 8 Main Theorems 55 2 Part I Introduction 3 Chapter 0 Review of schemes In the classical approach to algebraic geometry, one considers the basic objects, called varieties, as sets of solutions to a polynomial equations in affine or pro- jective space. There are however some limitations: • One would like to be able to construct spaces by gluing simpler pieces together, as one does in other parts of geometry and topology. • When working over nonalgebraically closed fields { which is important for applications to number theory and also within algebraic geometry { there may not be any points at all. • Even in classical arguments, one sometimes has to keep track of multi- plicities. For example, one might view the double line y2 = 0 as the different from the x-axis. So one would like a framework where this kind of distinction makes sense. Grothendieck's theory schemes gave a solution to all of these problems. Of course there is a price to be paid in the extra abstraction. Here is a quick overview. See Hartshorne's text [H] for more details. The ultimate source is [EGA]. In a nutshell, a scheme is built by gluing together simpler pieces called affine schemes. Affine schemes are determined by the rings of functions on them. 0.1 Coordinate rings In classical algebraic geometry, the basic objects can be thought of sets of points. Over an algebraically closed field this is reasonable, but consider the equations X : x2 + y2 = −1 Y : x2 + y2 = 3 These have no solutions in 2 and 2 respectively, and therefore no solutions AR AZ=4Z over . Since they both define the empty set in 2 , should they be regarded as Z AZ 4 the same? The answer is no. An isomorphism should be given by an invertible transformation x 7! f(x; y); y 7! g(x; y), given by a pair of integer polynomials, which takes the first equation to the second. In other words, what we really want is an isomorphism of rings 2 2 2 2 O(X) = Z[x; y]=(x + y + 1) =∼ Z[x; y]=(x + y − 3) = O(Y ) Such an isomorphism would guarantee a bijection between the sets X(R) =∼ Y (R) of solutions to x2 + y2 + 1 = 0 and x2 + y2 + 3 = 0 respectively, for any ring R. Taking R = R shows that this impossible. n In general, the affine \variety" X ⊆ AR, over some ring R, defined by polynomials fi = 0 is completely determined by the coordinate ring S = O(X) = R[x1; : : : xn]=(f1; : : : fN ) The set of solutions X(R0) in any R-algebra R0 can be identified with 0 HomR-alg(S; R ) because any homomorphism h : S ! R0 is uniquely determined by the point 0 0 (h(x1); : : : h(xn)) 2 X(R ). When R = R is an algebraically closed field, we can identify this with the set of maximal ideals in S by Hilbert's nullstellensatz. All the other familiar constructions: dimension, tangent spaces, ... can be read off from S as well. 0.2 Spectrum Before going further, it may be useful to have a basic model from analy- sis/topology. Given a compact Hausdorff space X, let C(X) denote the set of continuous complex valued functions. This is a commutative ring (always with 1 for us). When endowed with the sup norm, it becomes a so called unital C∗-algebra. (Since this is just for motivation, we won't bother writing down the axioms.) A special case of Gelfand duality says that we can reconstruct X from C(X). This is what we want to understand. Given a commutative ring R, let Max R denote the set of maximal ideals of R. This is the maximal ideal spectrum. We make this into a topological space by equipping it with the Zariski topology: the closed sets are the sets fm j I ⊆ mg for ideals I ⊂ R. To every x 2 X, define mx = ff 2 C(X) j f(x) = 0g ∼ This is a maximal ideal such that C(X)=mx = C. The reconstruction result now follows from the next theorem. Theorem 0.2.1. The map X ! Max C(X) is a homeomorphism. 5 Given a continuous map of spaces f : X ! Y , we get a homomorphism C(Y ) ! C(X) given by g 7! g ◦f. Thus C(X) can be regarded as contravariant functor, or equivalently a covariant functor of the category of C∗-algebras with arrows reversed. This is called the opposite category. The full form of Gelfand duality says Theorem 0.2.2 (Gelfand). The functor X 7! C(X) induces an equivalence between the category of compact Hausdorff spaces and the opposite category of commutative unital C∗-algebras. Now let us return to algebraic geometry. Following Grothendieck, we can make the bold leap Affine Algebraic Geometry = (Commutative Algebra)opposite We need to understand how to get a geometric object out of a general com- mutative ring. Given a such ring R, we have already defined Max R. The problem is that given a homomorphism of rings h : R ! S, there is in general no natural way to get a map Max S ! Max R. If however, we relax the con- ditions from maximal ideals to prime ideals, then there is a way. Let Spec R denote the set of prime ideals of R. It is called the spectrum of R. We do get a map Spec S ! Spec R given by p 7! h−1p. So now we have a set Spec R. We need to give it more structure. Proposition 0.2.3. There is a topology, called the Zariski topology, on Spec R whose basic open sets are D(f) = fp 2 Spec R j f2 = pg with f 2 R. The closed sets are V (I) = fp 2 Spec R j I ⊂ P g for ideals I ⊂ R. The map Spec S ! Spec R defined above is easily seen to be continuous. We make an observation that will be useful below. If R[1=f] = R[x]=(xf − 1) is the localization of R at f. Then Lemma 0.2.4. The maps Spec R[1=f] ! Spec R and Spec R=I ! Spec R are injective and the images are D(f) and V (I) respectively. We now have a contravariant functor from the category of commutative rings to the category of topological spaces. However, there is still not enough structure to set up the equivalence given above. The missing ingredient is the structure sheaf. 0.3 Affine schemes The intuition coming form Gelfand duality is bit too coarse for what happens next. A better model for us comes from manifolds. The basic example is X = Rn. Consider the collection C1 real valued functions on X and its open subsets. The key thing to observe is that the C1 condition is local. This means that a function is C1 if and only if its restriction to the neighbourhood of every point is C1. Or equivalently this means that f 2 C1(X) if and only for any 6 1 open cover fUig, fjUi 2 C (Ui). There are plenty of other classes of functions with this property. Abstracting this leads to the notion of a sheaf. Given a topological space X, a presheaf of sets (groups, rings...), F, is a collection of sets (groups, rings...) F(U) for each open set U ⊆ X, together with maps (homomorphisms...) ρUV : F(U) !F(V ) for each pair U ⊆ V such that 1. ρUU = id 2. ρWV ◦ ρVU = ρWU whenever U ⊆ V ⊆ W . We usually denote ρUV (f) = fjV . A presheaf is called a sheaf if for any open cover fUig of an open U ⊆ X and sections fi 2 F(Ui) such that fijUi\Uj = fjjUi\Uj , there exists a unique f 2 F(U) satisfying fi = fjUi . It will help to think of a scheme, when we get to it, as something like a C1 manifold. In additional to a topological space, we have a sheaf of commutative rings. The building blocks are given as follows: Theorem 0.3.1.