Étale homotopy theory

Transcription

1 Étale homotopy theory Tomer Schlank Contents 1 Introduction and overview ( ) Hour Hour Introduction to pro-categories and hypercovers ( ) Hour Hour More on hypercovers, and the original definition of the étale homotopy type ( ) Hour Hour [LECTURE TITLE] ( ) Hour Hour [LECTURE TITLE] ( ) Hour Hour [LECTURE TITLE] ( ) Hour Hour [LECTURE TITLE] ( ) Hour Hour [LECTURE TITLE] ( ) Hour Hour [LECTURE TITLE] ( ) Hour

2 9.2 Hour Étale homotopy theory: Applications ( ) Hour Hour Étale homotopy theory and knot theory ( ) Hour Hour More étale knot theory ( ) Hour Hour Introduction and overview ( ) 1.1 Hour 1 We begin with three motivating questions. 1. Let R be a ring, and let X be a scheme over R. Does X admit any R-points? (The first special case is R = Q.) 2. Let F be a number field, and write Γ un F = Gal(F un /F ) be the Galois group of the maximal unramified extension of F. For a given finite group G, does there exist a surjective homomorphism Γ un F G? 3. Given an algebraic group G and a scheme X, can we find nice invariants to classify G-torsors over X? (The first special case is G = GL n and X a C-scheme, i.e. a complex variety, so that we re asking about algebraic vector bundles of dimension n.) In this course, our answers will generally have a certain flavor: no. That is, we will construct obstructions to the existence of positive answers to the first two questions. Of course, we will be using the techniques of étale homotopy theory. In a sense, as with everything else this goes back to Grothedieck: whereas we usually take a ring and take its associated scheme, he observed that we can take the étale site (or topos) of this scheme, and extract notions like fundamental group and cohomology. However, it was Artin and Mazur who realized that we could actually associate a space to our ring (actually, a pro-space), from which we can extract invariants like higher homotopy groups. This was the birth of étale homotopy theory. We take a moment to explain why this might be useful for understanding the above motivating questions. We ll stick with the first one, which will be our main example throughout this course. So, suppose we have X Spec R. We would like to show that there do not exist sections of this structure map. We would like to think of both of these things as spaces of some sort. As topologists, we look to the analogy with a map of spaces X S with homotopy fiber F (which we ll assume are all even CWcomplexes). Of course, there s a classical obstruction theory to the existence of a section S X, which probably goes back to Eilenberg. Let s assume for simplicity that π 0 (F ) = and that π 1 (F ) is abelian. Then there s a natural obstruction class o 1 H 2 (S, π 1 (F )), and if this is nonzero then there cannot exist a section. Roughly, we should consider this as coming from the CW-complex structure of S, as we attempt to construct our section by working inductively up the n-skeleta of S for varying n. By our assumptions, we can certainly (even essentially uniquely) lift the 0-skeleton of S, and moreover we can lift its edges since F 2

3 is connected. So, the first obstruction is at trying to extend over the 2-cells of S. Moreover, if these vanish, then there s a higher obstruction class o 2 H 3 (S, π 2 (F )), and so on. If all of these obstructions vanish and we re in a reasonably nice situation, then we do indeed obtain a section to the map X S. We point out that actually these groups π n (F ) form local systems on S. Of course, if S is simply connected then this must be trivial, but to be more truthful we should perhaps replace these coefficient groups with the associated local coefficient systems, which we ll denote π n (X/S). Of course, from here it turns out that just as always in topology, where we have an obstruction theory we also have a classification theory given by changing dimensions just a bit. Namely, let Sec(X/S) be the space of sections. Then there is a spectral sequence H s (S, π t (X/S)) π t s (Sec(X/S)). However, note that this abutment is not simply H t s (Sec(X/S)). (We re lying here a bit; of course we need to be talking about basepoints everywhere.) As we all know, homotopy groups are very difficult to compute, so we can perhaps ask instead for the homology groups, which should (as always) be thought of as an abelian version. Namely, there s an abelianization Sec(X/S) Sec Z (X/S), and an associated spectral sequence H s (S, H t (X/S)) π t s (Sec Z (X/S)). This is actually quite useful, since we re generally in the game of showing that Sec(X/S) is empty, so it suffices to show that Sec Z (X/S) is empty. Now, everything we just said for spaces we want to do for schemes. In fact, we ll do this for any map of sites (including the étale sites of schemes). In this context we want to have a notion of H s (S, π n+1 (X/S)). Now, luckily a site is precisely the most general possibly gizmo for which we can take cohomology. But what about the coefficients? In fact, we ll have π n+1 (X/S) Sh(S), the category of sheaves on S. Of course, it s bad form to have homotopy groups lying around without them being associated to a homotopy type. That is, we d like these sheaves π n+1 (X/S) to be the homotopy groups of some object. This should be a sheaf of spaces on S. Roughly speaking, we will encode this as op Sh(S), a simplicial sheaf on S. Thus, our mission should we choose to accept it is to construct this sheaf of spaces over S. Let s write Sp(S) = (Sh(S)) op, the category of spaces over S. Now, let s start in the easiest possible case. Suppose we have X Spec C. Since C is algebraically closed, then the étale site is given by (Spec C)ét =, a single point. Thus, Sp((Spec C)ét ) = Sp(Set) = Set op = sset. Thus, we should have X/ Spec C sset....except remember that we were supposed to have profinite homotopy groups. If this thing is just a space, then its homotopy groups will not come with a topology. Thus, we actually will somehow want X/ Spec C Pro(sSet). In fact, we will be able to realize this in the easiest possible way: in general, we will want X/S Pro(Sh(S)) op. (Then, our cohomology will be continuous cohomology.) The pro- story will mostly just be a technical device; for the most part, we ll be able to pretend that our pro-spaces are just spaces. 1.2 Hour 2 So, given a site S we will define a functor Sites/S Pro(Sp(S)), the realization X/S of X S. Of course, there is the final object S S, and this will have S/S S, the terminal sheaf. Then, suppose we have a section of X S, i.e. a commutative diagram S X ============== Then we have a map S X/S in Pro(Sp(S)). We can consider this as a point in the derived mapping space dmap( S, X/S ). We call this the space of homotopical sections, and denote it Sec h (X/S); this should be thought of as some avatar of the set of sections. Then of course, by what we ve said above, we have S. 3

4 a well-defined map X(S) π 0 (Sec h (X/S)); this should be thought of as an approximation (and so for instance if the target is empty, then so is the source). We will have a spectral sequence H s (S, π t ( X/S )) π s t (Sec h (X/S)). Since we mentioned it before, we mention again that we ll also have Sec h (X/S) Sec Zh (X/S), and a spectral sequence H s (S, H t ( X/S )) π t s (Sec Zh (X/S)). Note that the relative information is not necessarily contained in the absolute information. Here s an easy example. Take A 1 C A1 C by the squaring map. As is quite believable, A1 C is contractible. So the map of absolute spaces is a map of contractible spaces and hence must be trivial, but the relative map will have fiber which is one point at the origin and two points everywhere else. (This information will be contained in the stalks.) Everything here has been pretty general, so we want to give some intuition for everything that s going on. What follows will be a very theoretical sketch of the sorts of methods we ll use. We consider the story as really beginning with the following theorem. Let X be a paracompact space, and suppose that U = {U i } i A is a good cover of X. Then there is a weak equivalence N(U) X. Recall that the cover U is called good if for every finite subset J A, the intersection U J = j J U j is a (possibly empty) disjoint union of contractible subsets. Let us also recall the nerve of a cover. There are two definitions: a naive one, and a less naive one. We begin with the naive one. For each open set we have a vertex; for each intersection of two open sets we add an edge; for each three-fold intersection we add a triangle; etc. To make this more formal, note that U = i A U i admits a natural map U X, and then we have the Cech construction π 0 (U X U X U) π 0 (U X U) π 0 (U). We denote this by π 0 (Ǔ) (and the same thing without the π 0 s by Ǔ ). Then, note that we have the zig-zag const(x) Ǔ π 0 (Ǔ). We claim that both maps are weak equivalences. The first one is already a levelwise weak equivalence; for the second, we can see that the levelwise homotopy fibers are just products of the original homotopy fiber. So, we do indeed have X π 0 (Ǔ). To generalize this we need the following notions: open subsets, intersections, open covers, and good open covers. Of course, the first three are precisely encoded by the notion of a (Verdier) site: this is a category with fiber products and coproducts and a notion of covering maps. For instance, for a scheme X the étale site is the category of étale maps to X, i.e. maps of schemes f : Y X such that f is flat, quasifinite (i.e. has all fibers are finite), and has nondegenerate differential. The morphisms in this category are the étale maps between the sources of such maps. (This is the algebraic analog of a map which is locally a finite cover. For instance, away from characteristic 2, the squaring map G m G m is étale.) Given an object U X, we can then construct its associate Cech simplicial object U X ( +1). In general, an object X of a site is called connected if it is not a coproduct of two nonempty objects. A site S is called locally connected if every object is a coproduct of connected objects. In this case, we have a connected components functor π 0 : S Sets. (This isn t totally obvious; one needs some sort of unique factorization properties for functoriality. Note that in any site, coproducts are disjoint.) Theorem 1 (SGA 4). If X is a smooth scheme, then Xét is locally connected. Now, to deal with the notion of good covers, the key observation is that we don t actually want to throw away nontrivial covers, we just want to think of all covers as approximations, and finer covers are better approximations. Of course, we don t actually want to take a limit; we want to think of this inverse system itself as the limiting object. This is not unlike the way in which one completes a metric space: the Cauchy sequences themselves are their own limits. But there s a problem: the covers don t quite form an inverse system. (The problem is that étale covers have automorphisms.) We ll deal with this next time. 4

5 2 Introduction to pro-categories and hypercovers ( ) 2.1 Hour 1 We now dig more deeply into the notion of pro-category. We begin with the following definition. Definition 1. A cofiltered (or directed) category is a small category I such that: 1. I ; 2. for all a, b I, there is some c I with hom(c, a) and hom(c, b) ; 3. for any pair of parallel arrows a b, there is some arrow c a which equalizes them. Example 1. Take the objects of I to be indexed by Z, with a unique map from m to n whenever m n (and no maps otherwise). Example 2. Take the objects of I to be N, the positive integers, with hom(a, b) = whenever b a and hom(a, b) = otherwise. Then, the satisfaction of condition 2 is just the fact that any two positive integers have a common multiple. Example 3. Any small category with an initial object is cofiltered. We will need one other definition. Definition 2. A cofinite directed poset is a nonempty poset A such that 1. for all a, b A, there is some c a, b; 2. for all a A, the set (which we call the reysha) is finite. R a = {b A : b a} Now, given a category C we make the following definition. Definition 3. The category Pro(C) of pro-objects in C has its objects the functors F : I C for some directed category I. Given X : I C and Y : J C, we define hom Pro(C) (X, Y ) = lim j J colim i I hom C (X i, Y j ). We emphasize that the indexing category of an object should not be considered as an essential part of the data. For instance, if we throw away the front part of a directed diagram, we get the same object. Heuristically speaking, we might say that we only care about what happens in our objects in the limit (in in the limit/colimit sense). Let us now make this precise. Definition 4. A functor F : I J is called cofinal if any of the following equivalent conditions hold: 1. For every functor G : J Sets, the map is an isomorphism. lim J G lim I G F 2. For every functor G : J E (for any category E), the map is an isomorphism (whenever the limits exist). lim J G lim I G F 5

6 3. For every j 0 J, the comma category F /j0 is nonempty and connected. (Recall that the objects of F /j0 are pairs (i I, F i α α α j 0 ), and the morphisms from (i 1 I, F i 1 j0 ) to (i 2 I, F i 2 j0 ) are β those maps i 1 i2 such that the diagram F i 1 F β F i 2 commutes.) α 1 j 0 α2 Example 4. If F : I picks out an initial object, then F is cofinal. Then, we have the following result. Lemma 1. If F : J I is a cofinal functor of directed categories and X : I C is any object of Pro(C), then there is a natural isomorphism F X = X in Pro(C). Loosely speaking, reindexing by a cofinal directed subdiagram doesn t change the pro-object. We now have the following simplifying fact. Proposition 1. If I is directed then there is some cofinite directed poset A and a cofinal functor F : A I. Proof sketch. We take A to be the set of all finite subcategories of I which have a unique initial object, ordered by inclusion. The functor F takes each such subcategory to its initial object. This is cofinal by the directedness of I. This suggests (a proof of) the following fact. Lemma 2. A category I is direct iff every functor F : D I from a finitely generated category D can be extended to a functor F : D I. (Here, D denotes the category obtained by adjoining an initial object to D.) All of this means that we can reindex any object of Pro(C) by a cofinite directed poset. We will use this frequently. We now talk a bit more about morphisms in Pro(C). One natural way to contruct a map in Pro(C) is the following. If X, Y Pro(C) happen to have the same indexing category I and we have a natural transformation f : X Y (i.e. a morphism in Fun(I, C)), we obtain a morphism in Pro(C) which is actually defined at every level i I. This is often called a strict map. Of course, this is not how all morphisms arise; most pairs objects of Pro(C) don t have the same indexing set. However, we have the following fact, which implies that every morphism in Pro(C) is isomorphic to a strict one. Proposition 2. The natural functor is an equivalence of categories. Pro(Arrow(C)) Arrow(Pro(C)) 6

7 Proof sketch. Suppose we have X : I C and Y : J C, and suppose we are given f lim j J colim i I hom C (X i, Y j ). Given any j 0 J and i 0 I, this hom-set fits into a diagram lim j J colim i I hom C (X i, Y j ) pj 0 colimi I hom C (X i, Y j0 ) q i0 Then, consider the set hom C (X i0, Y j0 ). {(i I, j J, f i,j ) : f i,j hom C (X i, Y j ), q i (f i,j ) = p j (f)}. In fact this can be considered as a category A f, where the morphisms are the evident commutative diagrams. Then, the heart of the proof lies in the fact that A f is directed and has cofinal projection maps A f I and A f J. Thus, we can reindex X and Y over A f. Of course, this is sufficiently natural that this gives a natural transformation; it remains to check that this gives an isomorphic map to the one we started with. 2.2 Hour 2 We reiterate that this is functorial, which would not be true if we simply took the obvious choice I J of a category which is cofinal over both I and J. In general, we wouldn t know which map to choose from any given X i to any given Y j. The point is that the construction of A f is precisely the choice of a map. In fact, more generally when A is finite and loopless (i.e. no objects have nontrivial endomorphisms) then there is an equivalence L A : Pro(C A ) (Pro(C)) A. The point is that there are no relations in A, so we can lift freely. A lift along this functor is called a level representation. Of course, note that a functor I C A is equivalent to a functor A I C. This suggests the following result. Theorem 2 (Isaksen). If A is a cofinite directed poset, then L A is essentially surjective. If additionally C is small, then this is an equivalence. Something about the proof. The nice thing about cofinite directed posets is that we can work inductively. Given an object, we show that if we ve already constructed a level representation on its reysha, then we can extend the level representation to that object too. Corollary 1. Let A be a directed category, and suppose F : A Pro(C) is any diagram. Then F has a limit in Pro(C). That is, regardless of the sorts of limits we may have had or not had in C, the category Pro(C) has all directed limits, without actually doing anything in the category C itself. This is proved first by assuming that A is a cofinite directed poset, and then lifting to a level representation (i.e. an object of Pro(C A )). (This is more or less a diagonal argument : in a tower of towers, the diagonal sub-tower is a natural choice of cofinal subdiagram.) This has the following general implications. Corollary 2. If F : C D is any functor, then the natural prolongation functor Pro(F ) : Pro(C) Pro(D) preserves all directed limits. 7

8 Corollary 3. If C has finite limits, then Pro(C) has all limits. Proof sketch. We first show that we can compute finite limits in Pro(C). It suffices to show that we can compute pullbacks. Let P B be the walking pullback diagram. Then we have functors (Pro(C)) P B Pro(C P B ) lim Pro(C). So Pro(C) has finite limits. But then it has both finite limits and directed limits, hence it has all limits. (This is the same statement as that if we have arbitrary products and finite limits then we have all limits.) Corollary 4. If F : C mcd is a functor of categories with finite limits that preserves them, then Pro(F ) : Pro(C) Pro(D) preserves all limits. Note that (perhaps sidestepping set-theoretic issues), this implies that in this case Pro(F ) admits a left adjoint. This is a nice version of the adjoint functor theorem. A special case is when F : C Set is left-exact (i.e. respects finite limits); then Pro(F ) : Pro(C) Pro(Set) has a left adjoint L F. Moreover, L F ( ) Pro(C) has that for any c 0 C, hom Pro(C) (L F ( ), c 0 ) = hom Pro(Set) (, F (c 0 )) = F (c 0 )). Thus, a functor which respects limits is pro-representable. On the other hand, the Yoneda lemma implies that we have an embedding Pro(C) Set Cop, and this lands on left-exact presheaves. In fact, Pro(C) is precisely the category of left-exact presheaves on C. In our remaining time, we return to homotopy theory. Recall that if U X is an étale cover, then we obtain π 0 (Ǔ) sset. This gives rise to a functor π 0 ( ˇ ) : Covét (X) sset, which is a decent approximation to the étale homotopy type functor...but there are a few problems. First of all, we would hope that the cohomology of this space is the étale cohomology of X. As cohomology is a contravariant functor, the natural thing is to consider colim U Covét (X) H (π 0 (Ǔ), A) (for some A Ab). Now, recall that π 0 (Ǔ) comes from the Cech construction Thus, we have π 0 ((Ǔ) X ( +1) ). H (π 0 (Ǔ), A) = Ȟ (X, A). This does indeed give the correct answer (i.e. étale cohomology) when X is quasiprojective over a noetherian ring, but in general this is not what we want. To solve this, we must pass to hypercovers. Recall that the goodness of a cover would say that passing to π 0 on the Cech construction is an equivalence. But moreover, the fibers of a good cover (i.e. of the map down to the original object) are contractible Kan complexes. Thus, the idea of hypercovers is to replace the Cech construction with something (a simplicial scheme, it will turn out) in which this is true. Definition 5. An étale hypercover is a simplicial object U in the étale site Ét(X) of X, such that: 1. U 0 X is a cover; 2. U n (U ) n is a cover for all n. 8

9 Before giving a few examples, we explain what we mean in the second condition. Let s begin with n = 1. Then we have U 1 (U ) S0. What we mean by this is two vertices of U, i.e. U 0 U 0. At n = 2, we have that (U ) 2 is given by three edges whose vertices agree in the appropriate places, i.e. U 1 U0 U 1 U0 U 1 U0, where the final arrow is meant to indicate that the last fiber product is taken with the first copy of U 1. Another way of describing this is the following. Note that we have the embedding n of objects [m] for m n. This induces a truncation functor tr n : C op Then, the n th coskeleton is the right Kan extension C op n. cosk n : C op n C op, and then we can define (U ) n = (cosk n 1 (U )) n. We will return to this next time, but we first go back to the main point. The first correction is that we will replace Covét (X) with the appropriate category of hypercovers. However, this replacement will not be cofiltered. There are at least four ways of fixing this; we will go over these in turn over the coming lectures. 3 More on hypercovers, and the original definition of the étale homotopy type ( ) 3.1 Hour 1 Recall that last time we were talking about constructing hypercovers; this is a generalization of the Cech construction for a map U X; namely, it is a simplicial object in the étale site. Recall that the key property of the Cech construction is that the fiber of U X ( +1) X is F ( +1), where F = fib(u X), and this is a contractible Kan complex. So, recall that we have the full subcategory k of objects [m] for m k. This gives us the k th truncation functor tr k : C op C op k. If C has limits and colimits then so do these two functor categories, since they re computed objectwise. This implies that the functor tr k has both a left and a right adjoint. These are denoted sk k tr k cosk k (where recall that the horizontal line points to the right adjoint). Let s consider the case C = Set. We first point out that there s an abuse of notation (which we find slightly annoying but ultimately worthwhile): we have an adjunction sk k tr k cosk k tr k of endofunctors on sset, but we denote this again by sk k cosk k. Now, the k th skeleton construction is well-known, and essentially consists in throwing away all nondegenerate simplices of dimension greater than k. On the other hand, by the adjunction, (cosk k (X)) n = hom( n, cosk k (X)) = hom(sk k ( n ), X). 9

10 So if n k, then this doesn t do anything: we again recover X n. Then, at n = k +1 we obtain hom( n, X). To understand the case n = k + 2 (and hence all cases n > k), observe that hom( n+1, tr n (cosk k (X))) = (cosk k (X)) n+1 so the way we construct the coskeleton inductively at level k + 1 and above is by filling in the appropriate dimensioned simplices in the unique possible way. All in all, this is really just killing homotopy groups, and we have the following result. Proposition 3. If we X is a Kan complex, then X cosk k (X) induces isomorphisms on π l for l < k, and π l (cosk k (X)) = 0 for l k. (That is, this is just a functorial Postnikov construction.) Now, the nice thing about Kan complexes is that we can compute their homotopy groups simply by mapping off of n+1 ; we don t need to consider all possible triangulations of the sphere. (This is just a co/fibrancy argument.) Since sk k n = sk k n if n > k, we get that the restriction map is an isomorphism. hom( n, cosk k (X)) hom( n, cosk k (X)) Example 5. Let us begin with k = 1, which has 1 =. Hence, cosk 1 (X) = pt. Next, we have (cosk 0 (X)) n = hom(sk 0 ( n ), X) = hom( n+1 pt, X) = (X 0) (n+1). Taking n to be variable, we see that this is always contractible unless X was empty to begin with; that is, cosk 0 kills all homotopy groups besides the ( 1) st (which is pt if X is nonempty and is if X is empty). Exercise 1. Show that if X = cosk 2 (X), then X is a Kan complex iff it s the nerve of a groupoid. We now recall the definition of a hypercover. Definition 6. A hypercover of X is an object U (Ét(X)) op such that: 1. U 0 X is a cover, and 2. the map U n+1 (cosk n (U )) n+1 is a cover. (In fact, we can take the first condition to be the special case of the second condition where we take n = 1.) Example 6. Suppose k is an algebraically closed field, and let X = Spec k. Then, Ét(X) FinSets. Hence, a hypercover of X is a special sort of object U (FinSets) op. By the adjunction, we can rewrite the second condition as referring to the map hom( n+1, U ) hom( n=1, U ), which we are now requiring to be surjective (since the covers in FinSets are the surjections). This is asking for all lifts n U n. This is exactly the condition that U is a contractible Kan complex. 10

11 Example 7. Let k be a field and let k sep denote its separable closure. As always, let Γ = Γ k = Gal(k sep /k). Then, Ét(Spec k) is the category of finite sets with a continuous action of Γ (that is, the stabilizer of every point should be open). (Explicitly, given an étale algebra A over k, the associated set is the set of ring homomorphisms A k sep.) A map X Y in this category is a cover if it is surjective (as a map of sets). We hence obtain the following useful fact: a hypercover of Spec k is an object in (Finite Continuous Γ-Sets) op whose underlying object in (FinSets) op sset is a contractible Kan complex. So, what can we do with hypercovers? First of all, notice that if f : X Y is a map of schemes and U Y is a hypercover, then we obtain a hypercover f U X (since the property of being étaleis preserved under base change). In particular, given any geometric point f : Spec k X, a hypercover U X yields a contractible Kan complex f U (FinSet) op. Thus, a hypercover should be thought of as an algebraically parametrized family of contractible Kan complexes. As we will see, in fact we can think of a hypercover as some sort of acyclic Kan fibration of our scheme; we can also think of it as a resolution. Now, what does it mean for F to be an étale sheaf (of abelian groups, say) over X? Recall that given a cover U X we can apply the Cech construction followed by F to get a cosimplicial abelian group F (U) F (U X U). Turning this into a complex and taking cohomology, this is (by definition) Cech cohomology. On the other hand, we can replace Cech(U) with any hypercover and everything still goes through, so we can in fact define the Cech cohomology of any hypercover. Theorem 3 (Verdier). There is a canonical isomorphism colim (U X) HypCov(X) Ȟ U (X, F ) = H ét(x, F ) from the colimit over all hypercovers of X of the associated Cech cohomology to the derived functor cohomology of X. The original proof of this fact uses spectral sequences, but we will recover it from general facts about model categories (namely, by considering hypercovers as acyclic Kan fibrations). We have one last fact about hypercovers that we d like to Suppose X is a (paracompact) space, and U X is a hypercover in the category Open(X). We can still give a definition of a good hypercover: we demand that in every level, we have a disjoint union of contractible spaces. Now, we always have X U, and the goodness implies that we also have U π 0 (U ). Example 8. Let s consider S 2, covered by two hemispheres. Their intersection isn t contractible, but it s coverable by two contractible sets. This is a more economical way of getting a good (hyper)cover of S 2 than using four different open sets. 3.2 Hour 2 Recall that we had a functor Cov(X) sset given by (U X) π 0 (Ǔ ). We can of course enhance this to HypCov(X) sset by U π 0 (U ). Now, Verdier s theorem above essentially tells us that if we try to compute the cohomology of this system of spaces, for a constant sheaf we re actually going to get the correct answer. However, we would like this to be true for more general sheaves, but the problem is that HypCov(X) isn t directed, so we don t get a pro-object in sset. This will suggest our final modification of the category. But let us first explain why HypCov(X) isn t directed. First of all, we have the identity cover, so this is nonempty. Then, given H, H HypCov(X), it s not hard to check that H X H is again a hypercover; this lives over both H and H, so we always have refinements. But the problem is with equalizers: given any two parallel maps, we re supposed to have a map to the source which equalizes them. 11

12 Example 9. Suppose X = Spec k. Say A and B are Kan-contractible simplicial sets. Then there can be two maps A B with equalizer A (e.g. the maps EZ/2 EZ/2 given by the identity and the flip map on the free Z/2-set (Z/2)/e). So of course these maps can have no equalizer. The way to fix this is to observe that our category is actually enriched in FinsSet, since Ét(X) has finite coproducts. Now, when we re taking cohomology, we only care about the inverse system of the hypercovers. So if we ever have a refinement of a hypercover, we can throw away the original one. In particular, if X is a quasicompact scheme, we can always throw away all but finitely many of the open sets in each level. This implies that we can give the following definition. Definition 7. Two maps f, g : H H in HypCov(X) are strictly homotopic if there is an appropriate map F : H 1 H. We say that f and g are homotopic if they are equivalent under the closure of this relation to an equivalence relation. (This is essentially a fibrancy problem in H ; there s no way to replace the horn 2 1 = Λ by a line 1.) Theorem 4 (Artin Mazur). The category ho(hypcov(x)) is directed. That is, any pair of parallel arrows H H have a map H H such that the two compositions are homotopic (though not necessarily equal). Of course, this comes with a price: we only have a functor ho(hypcov(x)) ho(sset) (since π 0 isn t well-defined among weak equivalence classes of hypercovers). We now finally come to Artin & Mazur s original definition: the étale homotopy type of X, denoted Ét(X), is the object ho(hypcov(x)) ho(sset) in Pro(ho(sSet)). Of course, we now know that this isn t such a good idea: we really wouldn t like to end up with a diagram in the homotopy category of spaces. Eventually we ll lift it to something better, but it turns out that we can do quite a bit already just with this. Proposition 4. This construction enjoys the following properties. 1. If A is a (locallly) constant sheaf of abelian groups on X, then there is a natural isomorphism H sing(ét(x), A) = H ét(x, A). 2. If x X is a geometric point, then there is a natural isomorphism π 1 (Ét(X), Ét(x)) = πét 1 (X, x). The first fact is basically Verdier s theorem. The cohomology of a pro-space is by definition just the pro-system of cohomologies. However, if the sheaf isn t locally constant, then the left side doesn t actually mean anything; really we need to use the second fact to define a local coefficient system. Example 10. Let s compute Ét(Spec k). Of course, HypCov(X) is now a category of contractible Kan complexes, and applying π 0 does nothing, so Ét(Spec k) is indeed contractible. This is good. 12

13 Example 11. Here s another property we d like to explore. Let X/C be noetherian. Then there is a natural map X(C) Ét(X) (where the source has the analytic topology), and this induces an isomorphism on profinite completions. (There is a notion of the profinite completion of a space.) This implies that the profinite fundamental groups will agree, and that we ll get an isomorphism in cohomology with finite coefficients. The proof of this runs by construction a site which refines both X an and Xét, which we ll denote X an,ét ; the maps here are just local homeomorphisms, i.e. maps which are étale in the analytic sense. Then, the map X an,ét X an induces an equivalence on homotopy types, and the map X an,ét Xét induces an equivalence after profinite completion; this basically comes from the classical SGA result H (X(C), A) = Hét (X, A). However, it turns out that the étale homotopy type is usually already profinitely complete; at least, this is true in the most interesting cases. Definition 8. A scheme X is called geometrically unibranched if for every point x X, the integral closure of the local ring O X,x is still local. (In particular, if this isn t an integral domain, then X won t be geometrically unibranched anyways.) Example 12. The intersection of the x- and y-axes isn t geometrically unibranched. Proposition 5. X is geometrically unibranched iff for every étale cover Y X, Y is connected iff Y is irreducible. Here is a theorem that we will prove next time. Theorem 5. If X is connected, geometrically unibranched, and noetherian, then Ét(X) is pro-finite. (That is, if U X is a hypercover, then π 0 (U ) has only finite homotopy groups.) In particular, if X is smooth (or at least geometrically unibranched), then we can compute Ét(X) as the profinite completion of X(C). So an interesting question is: Is this just because we didn t make the right definition? Is it possible to recover the complex-analytic topology itself? It turns out that the answer is no. In 64, Serre constructed a smooth projective variety X/k (for k a number field) with two embeddings σ 1, σ 2 : k C such that X σ1 (C) and X σ2 (C) are inequivalent. In particular, he showed that they have non-isomorphic fundamental groups. Of course, their profinite completions are the same, but this shows that it s impossible to obtain π 1 of the complex points from any algebraic construction. In 74, Abelson gave an example where X σ1 (C) and X σ2 (C) have isomorphic (finite!) fundamental groups, but are nevertheless not homotopy equivalent. Even more recently, Charls gave an example where the two rings H (X σi (C), R) (with the cup product) are non-isomorphic. This is quite surprising, because if we replace R with C l then we do get isomorphisms. 4 [LECTURE TITLE] ( ) 4.1 Hour 1 Let us recall the construction from last time. Suppose X is a scheme. then we have Ét(X) : HypCov(X)/ Ho(sSet); this defines an object Ét(X) Pro(Ho(sSet)). (Recall that on hypercovers, we took the homotopy relation by modding out by crossing with the 1-simplex 1.) The reason we passed to homotopy categories is that maps of hypercovers don t always have equalizers. 13

14 This is of course not enough to make us homotopy theorists happy, but it suffices for a great many purposes. For instance, suppose we have some Y = {Y i } i I Pro(Ho(sSet)), and take any A Ab. Then we can define H n (Y ; A) := colim I H n (Y i ; A). Similarly, we can obtain {H n (Y i ; A)} i I Pro(Ab). One might naturally wonder: Why with cohomology can we simply take a colimit, whereas with homology we need to retain the pro-structure? There will actually be a number of reasons for this, but the first one we can state already: taking a colimit commutes with short exact sequence, but taking a limit doesn t. What about homotopy groups? Well, suppose we re given some basepoint y 0 Y for our object Y Pro(ho(sSet)). (Of course, a choice of basepoint is just a map from the terminal object; this is equivalent to choosing compatible basepoints for all spaces in the system. Actually, we should be careful here; we might want to take basepoints before passing to the homotopy category.) Then we obtain π n (Y, y 0 ) {π n (Y, y 0 )} Pro(Grp). Of course, we can obtain a pointed object Ét(X, x 0)Pro(sSet)) from a geometric point x 0 of our chosen scheme X. Now, the upshot of what we ve gotten so far is that there is a natural isomorphism H n (Ét(X); A) = H ń et(x; A). Similarly, if we have a geometric point x 0 X, we have a natural isomorphism π 1 (Ét(X, x 0)) = π et 1 (X, x 0 ). (The latter was originally defined as follows. Let Rev(X) denote the category of finite (not just quasifinite) étale connected covering maps Y X. Take a geometric point x 0 X, and consider the fiber functor F x0 : Rev(X) Set, which sends f : Y X to f 1 (x 0 ). Then, we ask for automorphisms of the functor F x0. This can be considered as a pro-group (by looking at the resulting action on each F x0 (Y ), and considering this as a pro-system), and indeed each of these is finite by assumption. So in the end we obtain a pro-group. Note that this really does need us to choose a basepoint, in order to identify our fibers. (Otherwise, the entire point of covering spaces is that you can t say which fiber is which!) This issue is related to the same construction in topology. One can consider the fundamental group to be the automorphism group of a universal cover. But to choose a universal cover, you need to choose a basepoint.) From here, we can also simply define This is exciting! πét n (X, x 0 ) = π n (Ét(X, x 0)). However, we can see that this does indeed depend on basepoints, and this ends up being something of a problem. For instance, this manifests itself if one is dealing with a pointless pro-space. For instance, take the tower N N where each map is given by n n + 1. This has no maps in from the terminal pro-space, but nor is it isomorphic to the empty pro-space (since the latter admits no maps at all!). Now, let us work in Ho(sSet conn. ) for simplicity. Here, we have a Whitehead theorem: a map is an equivalence iff it induces isomorphisms on all homotopy groups. This is no longer true in Pro(Ho(sSet conn ))! Example 13. Let X be your a based connected homotopy type with infinitely many nontrivial homotopy groups. Then let s take the coskeletal cofiltration (a/k/a the Postnikov tower) X P 2 X P 1 X. 14

15 This is evidently an isomorphism on homotopy groups. On the other hand, this cannot be an isomorphism in Pro(Ho(sSet conn )): if so, there would exist a map P n X X for some sufficiently large n such that the composition X P n X X is an equivalence. But this is impossible by our assumption. (In fact, a space is pro-equivalent to its Postnikov tower iff it s truncated.) So, what we re going to do is force Whitehead s theorem to be true. Suppose that X = {X i } i I Pro(Ho(sSet)), i.e. X : I Ho(sSet). Then we define X : N I Ho(sSet) by (n, i) cosk n (X i ). Now, note that there s a natural map X X, and this is a π -isomorphism. following. In fact, we have the Proposition 6. Given f : X Y in Pro(Ho(sSet conn )), the following are equivalent: 1. f : X Y is an isomorphism. 2. f : π (X) π (Y ) is an isomorphism. 3. cosk n (X) cosk n (Y ) is an isomorphism for all n. Of course, this looks a whole lot like a localization. And that s what it is! We will work almost exclusively with Ét (X). anyways.) (It ends up not making a huge difference in practice, Now, suppose that k is a field and let X = Spec k. Recall that Ét(X) = FinSetΓ, for Γ = Γ k = Gal(k sep /k). Hence, HypCov(X) consists of those contractible Kan complexes which are levelwise finite and come with a continuous action of Γ. (This is given by U π 0 (U ). But what does this really mean? If A/k is an étale algebra, then we can write this as i L/k, and then we see that the set {A ksep has Γ-action, and the quotient by this action exactly picks out the L i. Hence, π 0 (U ) = U /Γ.) 4.2 Hour 2 We need the following result. Lemma 3. Let U be a levelwise-finite Γ-sset (i.e. a hypercover of X = Spec k), such that U = cosk n (U ) for some n. Then there is some normal open subgroup H Γ and a Γ-equivariant map cosk 0 (Γ/H) U. Thus, in the case of a field, we can refine every hypercover by a Cech cover. (We only need the map, not that it s itself a cover.) Proof of lemma. Notice that by the adjunction, a map cosk 0 (Γ/H) U = cosk n (U ) is equivalent to a map tr n (cosk 0 (Γ/H)) tr n (U ). Let us set H = stab Γ (u); u U i for 0 i n note that the set of u over which we re indexing is finite; moreover this are stable under conjugation (since g(stab Γ (u))g 1 = stab Γ (g u)), i.e. H is normal. Moreover, let us write Y = Γ/H. 15

16 So, choose some y 0 Y. Then we can send this to any point x 0 U 0, since stab Γ (y 0 ) stab Γ (x 0 ). Now, how must we map Y Y U 1? Each orbit should be thought of as a line segment, i.e. it has d 0 and d 1 in Y ; since U is a contractible Kan complex, we can choose (essentially uniquely!) an edge in U 1 with boundary the corresponding points of U 0. We can continue to lift in this way. Example 14. All of this indicates that to compute the étale homotopy type of a field, it suffices to restrict to these Cech hypercovers. If we write G = Γ/H, then the associated Cech construction is G ( +1) = EG; we want to quotient by Γ, but this action factors through that of G, and so we just get G\EG = BG. Thus, finally Ét(Spec k) = {B(Γ/H)} H Γ, and this implies that π n (Ét(Spec k)) = { Γk, n = 1 0, n 1. That is, étalehomotopy-theoretically, Spec k is a classifying space for Gal(k sep /k). This shouldn t be surprising: the étale cohomology of a field is well-known to be isomorphic to the continuous cohomology of its Galois group. Example 15. Let us consider G m over a field k = k of characteristic 0. Recall that we have the covers µ n : G m G m ; these have π 0 (Cech(µ n )) = Bµ n. Hence, Ét(G m) = {Bµ n } n N (with indexing diagram the divisibility poset). Example 16. Let k = k be a field of characteristic 0. To study P 1, we take the following hypercover. Take U 0 = A 1 A 1, as usual. This has self-intersection U 0 P 1 U 0 = A 1 A 1 G m G m. But we refine this to have degree-n maps on both copies of G m. This proceeds upwards analogously, taking 1-coskeleta everywhere; call this U n. Then we obtain π i (π 0 (Cech(U n ))) 0, i = 0 = 0, i = 1 Z/n, i = 2. (In fact, we can also see that π 3 = Ẑ; this is generated by an étale version of the Hopf fibration.) Before leaving the world of Artin Mazur for greener pastures, we prove one final nontrivial result, the pro-finiteness theorem. Proposition 7. Let X be a connected scheme which is geometrically unibranched (recall that this means that an étale cover is irreducible iff it s connected). Then for every hypercover U X, π n (π 0 (U )) is finite. Proof. Note that X id X is étale, so X is itself irreducible; thus it has a generic point f : η X. Given U, let us consider the hypercover f U η. We claim that π 0 (f U ) π 0 (U ). This comes directly from the geometrically unibranched condition: irreducibility can be checked over the generic point. But then, η = Spec(K(X)), for K(X) the function field of X. Thus, we can consider f U Γ K(X) -FinsSet. However, we won t get all possible hypercovers in this way. (For instance, the generic point of the affine line is Spec(k(t)) A 1 k, and the source of this has the étale cover Spec(k(t)[ t]), but this is ramified at 0 and hence isn t pulled back from an étale cover of A 1 k.) Now, we claim we can assume that X = Spec k for some field k. Then we can assume further that U = cosk n+1 (U 0 ). This implies, by finiteness, that the action factors through a quotient Γ Γ/H to a 16

17 finite group. Hence, it suffices to prove that if U is a levelwise-finite contractible Kan complex with an action of a finite group G, then π n (U /G) is finite. If this were simply connected, we could just prove that the homology groups are finite. So let s prove this first, and then prove the general case next time. So, we compute H ( ; Z) by taking the homotopy groups of the sset {ZU n /G} n (by Dold Kan). We have a diagram ZU n /G ZU n ZU n /G as follows. The first map is given by summing over orbits; the size of an orbit is G / H (for H the stabilizer), so we re summing over H points in the fiber. (This is the transfer map.) The second is the quotient. Hence, the composition is the multiplication-by-n map. But this factors through a contractible sset, so these homology groups must all have finite exponent. [...] 5 [LECTURE TITLE] ( ) 5.1 Hour 1 We recall that we were in the middle of proving the profiniteness theorem: X is an irreducible geometrically unibranched scheme and U X is a hypercover, and we re trying to show that π n (π 0 (U )) are all finite. We take a minute to review the Kan Dold correspondence. If A is an abelian category, we have an equivalence of categories N : sa C 0 (A) : K. This is defined by and then the differential is simply given by N(A) k = k ker(d i ) A k, i=1 d N(A) 0 k N(A)k 1. This might seem odd when we already have the unnormalized chain complex, which is just C(A) k = A k with d = ( 1) i d i. But in fact the normalized chains have better properties, and it turns out that the normalized chains are actually a deformation retract of the unnormalized ones. Let s look at the specific case A = Ab, which is particularly instructive. Then this composes with the further (levelwise) adjunction Z{ } : sset sab : U, so we can talk about the set of n-simplices: (K(C )) n = (U(K(C ))) n = hom sset ( n, U(K(C ))) = hom sab (Z{ n }, C ). Of course this just picks out the elements of C n. It s not hard to see that this composite U K actually lands in Kan complexes. More generally, given X sset we have the unit X U(ZX ), and it turns out that the homotopy groups of the target are precisely the homology groups of the source. Moreover, π n (U (ZX )) = H n (N(ZX )) H n ( X ), and so applying homotopy to this map precisely gives the Hurewicz map π n (X) H n (X). 17

18 End of proof. Now, let s return to the profiniteness theorem. We assume that our hypercover U X is bounded (i.e. U = cosk n (U )), which is legal since we re only working one homotopy group at a time. Now, we take the pullback along the generic point to get the pullback hypercover f (U ) U and this induces η f X, π 0 (f U ) π 0 (U ). Now, a hypercover of η = Spec(K(X)) is just a levelwise-finite Γ-simplicial set (for Γ = Γ K(X) = Gal(K(X)/K(X)) whose underlying simplicial set is a contractible Kan complex; moreover, we have that π 0 (U ) = U /Γ. Since we assume U = cosk n (U ), we may assume that its action of Γ factors through some finite quotient Γ G. So, it suffices to prove the statement when G is finite and X s(g-set) is a levelwise-finite contractible Kan complex, namely that π k (X /G) is finite for all k 0. Now, by Serre s C-theory, in the simply-connected case it suffices to show that the Z-homology groups are finite. Now, we have the projection ZX Z(X /G). But note that we also have a map in the other direction: define Z(X /G) ZX by taking the equivalence class [σ] of a simplex σ to g G gσ. (This is called the transfer map.) Of course, the composition Z(X /G) ZX Z(X /G) is given by σ G σ. So, the induced composition in homology needs to be zero since it factors through something with no homology (above dimension 0). So G needs to be the zero map on all H k (Z(X /G)) for k > 0; since it s finitely generated as well, it must in fact be finite. (Of course, X /G is connected since X was.) So that settles the simply-connected case. To deal with the non-simply-connected case, we first prove the following result: the map X /G pt has the right lifting property for 2-horns Λ 2 i 2 (i.e., it s Kan in level 1 ). Now, we know by assumption that X is Kan. So if we can first lift the horn Λ 2 i X /G to X, then we get the lift of the 2-simplex to X, and then we re done. So, let e 1 and e 1 be the two edges of Λ 2 i in X /G, meeting at vertex v. Then we can take arbitrary lifts ẽ 1 and ẽ 2 ; these may not meet at their preimages of v, but by assumption we can act on ẽ 2 by G so that they actually do. And that s it. (This trick won t work in higher dimensions, because there s too much compatibility that we require.) where So, now let a (X /G) 0 be a vertex. We claim that there is a natural sexseq 1 K G φ π 1 (X /G, a) 1, K = b X 0 stab G (b). (This is normal since this generating set is already stable under conjugation: g(stab G (b))g 1 = stab G (g b).) This is given by the following: take each g G to an edge l g : a ga; this projects to a loop l g at a (X /G) 0. If we choose another path l g : a ga, then there s a homotopy between them since X is a contractible Kan complex. This story translates around different preimages of a, so this map is well-defined. 18

19 This is surjective since given a loop at a X /G we can lift it to a path beginning at a, and then its end must land at something in the same orbit as a and so we have an edge l g connecting them. To compute the kernel K, first observe that K stab G (a) since then we can take the degenerate path at a. Otherwise, let b X 0 and choose a path l ab : b a. This induces an isomorphism π 1 (X /G, a) = π 1 (X /G, b), which commutes with the respective projections from G via the various definitions of φ. So we see that all stabilizers must be contained in K. Let s call the subgroup of G generated by the stabilizers K 0 K for the moment; we aim to show that K 0 = K. Now, observe that we can define a map f : X 0 G/K 0 of G-sets (since for transitive G-sets there s a map iff the stabilizer of the source is a subconjugate of that of the target). This implies that there s a map X cosk 0 (G/K 0 ); recall that K 0 G is normal so in fact this quotient is a group let s call it H and the target is just EH = ( H H H) (the usual bar construction). So now we have a map X EH, and quotienting by G gives This induces a map X /G EH/G = EH/H = BH = B(G/K 0 ). G π 1 (X /G) G/K 0... and eventually Tomer claims that X /K has a free H-action, and the quotient X /G is compatible with the diagram X /K EH /G X /G /G BH, and so it follows that X /K is simply-connected in fact, this left arrow is the universal cover, and moreover it follows that π 1 is finite. But now since the homotopy groups of the universal cover agree above dimension 1, we re done. 5.2 Hour 2 So let s get back to the bigger picture. We have that for a connected, geometrically unibranched variety X/C, the map X(C) Ét(X) is just the profinite completion. (This is proved inductively by Postnikov towers, proving it first for Eilenberg MacLanes.) In particular, π1 et (X) = π 1 (Ét(X)) = π 1(X(C)). Now, if π 1 (X(C)) = 0 and moreover all π n (X(C)) are finitely generated, we get that π n (Ét(X)) = π n(x(c)). (We don t get lim 1 terms, by the profiniteness theorem.) 19