Main references, mil80 milne, etale cohomology bs14 bhatt, scholze, the proetale topology for schemes pdf. The subscript l is in honor of steve lichtenbaum, who. A related problem is to understand the connectivity. One of the technical reasons that the category works better is that schemes morally become \locally contractible. Gfset fset the forgetful functor, then g is profinite if and only if the. To accomplish these tasks, we define and study the proetale topology, which is a. Number theory learning seminar 20162017 the seminar will meet wednesdays 1. Etale cohomology, site, proetale, constructible sheaf, fundamental group. Changyeon chough ibs center for geometry and physics. Monodromy of heckeinvariant subvarieties 293 the image of n. In the 1940s, weil observed that some of his results on the numbers of points on certain. To accomplish these tasks, we define and study the proetale topology, which is a grothendieck topology on schemes that is closely related to the etale topology, and yet better suited for infinite constructions typically encountered in. The proetale fundamental group universiteit leiden. Pro etale sheaves march 20, 2018 throughout this lecture, we let c denote an essentially small coherent category with disjoint coproducts for example, a small pretopos.
A secondary goal is to show how using the proetale topology simplifies the introduction of l adic. Commelin may 6, 2015 1 introduction smooth morphisms are the analogue in algebraic geometry of what submersions are in diffential geometry. The proetale topology for schemes by bhargav bhatt. Indeed, unlike in classical topologies such as the zariski or etale topology, sheaves on the pro etale site of a point are not merely sets. Citeseerx document details isaac councill, lee giles, pradeep teregowda. Evan jenkinss notes of a seminar on etale cohomology click on the pdf icons. For example, if xis irreducible, then the groups hr.
An essential foundational result is that this site is locally contractible in a wellde. The proetale topology for algebraic stacks 3 the collection is a fpqc covering of y. Assume x 0 is closed in x and every point specializes to a unique point of x 0. In mathematics, especially in algebraic geometry, the etale homotopy type is an analogue of the homotopy type of topological spaces for algebraic varieties roughly speaking, for a variety or scheme x, the idea is to consider etale coverings and to replace each connected component of u and the higher intersections, i. To accomplish these tasks, we define and study the pro \ etale topology, which is a grothendieck topology on schemes that is closely related to the \ etale topology, and yet better suited for. The weiletale topology for number rings article pdf available in annals of mathematics 1702 april 2005 with 30 reads how we measure reads.
The pro etale site of spec k speck identifies with the category of profinite continuous g gsets. Now, it turns out that this is actually already a sheaf. Pyknoticity versus cohesiveness the ncategory cafe. Here is a summary of delignes second paper on the weil conjectures weil2. Etale homotopy theory after artinmazur, friedlander et al. Bhatt, scholze choose to work with weakly etale maps instead of pro etale. In the previous lecture, we proved that pro c is also a coherent category with disjioont coproducts. Z, computed using the zariski topology, are zero for all r0. We give a new definition of the derived category of constructible q. Friedlanders etale ktheory duggerisakens sums of squares formulas rational points and homotopy. So long as the map f f from the pro etale site of x x to the pro etale site of the point point has a fully faithful inverse image f f\ast, i believe that the. Comparisons etale realizations of motivic spaces algebraic cycles and etale cobordism. It generalizes the fundamental groups of schemes introduced by grothendieck in sga1 and sga3. We consider the category gsets whose objects are sets endowed with a left action of g and whose morphisms are gequivariant maps.
Introduction the purpose of this paper is to serve as the. This definition works well for spaces such as real and complex manifolds, but gives undesirable results for an algebraic variety with the zariski topology. A variant, the proetale site locally contractible in some sense is discussed in. This topos, denoted pt, is called the punctual topos. The proetale topology was suggested by scholze and then fully developed in. The small zariski site of a scheme x is the category xzar which is the full subcategory of schx of objects u. In order to motivate the construction of this paper we explain in section8why classi. Proetale site ii compare the proetale topology with the classical etale topology bs sec. The pro etale and vtopologies notes by tony feng for a talk by eugen hellman april 5, 2016 1 the pro etale topology the pro etale topology is a topology on the category perf of perfectoid spaces.
The arxiv notes of a minicourse given by a fine expositor, antoine ducros, which also cover analytical aspects of etale cohomology used for berkovich spaces. In particular, for a large class of nice topological groups, continuous cohomology agrees with the proetale cohomology. Xof nite presentation is etale, if it is at and the diagonal y. In particular, the automorphism group of an object x of x is given by. There is a topology, called the zariski topology, on specr whose basic open sets are df fp2specrjf 2pgwith f2r. The aim of these notes is to explain this construction. The proetale and vtopologies notes by tony feng for a talk by eugen hellman april 5, 2016 1 the proetale topology the proetale topology is a topology on the category perf of perfectoid spaces. The homogeneous coordinate ring of a projective variety, 5. Changyeon chough ibs center for geometry and physics 77 cheongamro, namgu, pohang gyeongbuk, 37673, republic of korea. This is great if one is quite pro cient in algebra, but not so good. Moreover, we define a refined fundamental group of schemes, which is large enough to see all lisse q. H, the procategory of the homotopy category of cwcomplexes. What seemed like a bug of the pro etale site is actually a feature for us here, as clausen explained to the lecturer. Xgto be a jointly surjective set of open embeddings.
Then the site h ic is just pro c with the trans nite topology. This amounts to extracting lots of ppower roots of units in the tower v. He then gives a quick description of what bhatt and scholze are doing and why it is a good thing. Verdier in order to explain weils insight weil 1 that, for polynomial equations with integer coef. An important property of the category perf which makes this theory possible is that it has all inverse limits with a noid transition functions. A gentle introduction to homology, cohomology, and sheaf.
Bhargav bhatt, peter scholze, the proetale topology for schemes arxiv. We denote objects over f qwith subscript 0, suppression of. Homotopy exact sequence for the proetale fundamental group. Note that the small pro etale topology on x of 2, 4. In the introduction to the pro etale topology we argued.
We extend this result from schemes to algebraic stacks. This years seminar will focus on etale cohomology, the goal being to understand laumons proof of the main theorem of delignes weil ii paper that gave a powerful and vast generalization of the riemann hypothesis over finite fields. Pro etale cohomology 5 welistsomeelementaryproperties. We refer to it as the big pro etale topology on xand denote by schx. Proetale cohomology helene esnault, lars kindler, lei zhang march 17, 2014 introduction the etale cohomology theory which was initially suggested by grothendieck in 1960s plays a very important role in modern algebraic geometry. The corresponding category of covers of a scheme x consists of schemes y x that are etale and satisfy the valuative criterion of properness. These course notes from spring 2010 are extremely rough.
Exercise sheet department of mathematics winter semester 1819 prof. We will restrict our discussion mostly to the small pro. In the introduction to the pro etale topology we argued that. The pro etale topology was suggested by scholze and then fully developed in. Constructible sheaves i explain functoriality of the proetale site under open and closed immersions and. Ill complement the list of well known books on the subject by some freely available documents, which i find userfriendly. The etale topology is one of the tools which is used in algebraic geometry for purely geometrical as well as for arithmetic questions. So long as the map f f from the proetale site of x x to the proetale site of the point point has a fully faithful inverse image f f\ast, i believe that the. In the introduction to the pro etale topology we argued that adding ltered limits to the site made z l and q lsheaves work better.
To accomplish these tasks, we define and study the pro etale topology, which is a. 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. The proetale topology for schemes oberseminar of the ag schmidt. More concretely, every scheme x admits a pro etale covering fu i. A gentle introduction to homology, cohomology, and sheaf cohomology. We note that the idea to extract many ppower roots is common to all known proofs of comparison theorems in padic hodge theory. An important invariant of x, x is the topological fundamental group px, x. The proetale fundamental group of a scheme was introduced by bhatt and scholze. The pro etale site pro et of a point is the category of pro nite sets. The proetale fundamental group wouter zomervrucht, december 16, 2014 1. Pdf the weiletale topology on schemes over finite fields. The proetale site of spec k speck identifies with the category of profinite continuous g gsets.