Workshop on motives, galois representations and cohomology around the langlands program. The work is one possible formulation of quillens ideas about algebraic ktheory, and i am very. Using the etale topology to define the sheaves to be used in cohomology results in etale cohomology, the original driving force for the development of the concept of the grothendieck topos. Etale cohomology theoryrevised editio nankai tracts in mathematics book 14 kindle edition by lei fu. Its a quick introduction to the relation between galois theory, covering spaces, cohomology, and higher categories.
Abelian sheaves, mostly in the context of etale topology. The cobordism classes of manifolds form a ring that is usually the. Sheaf theory etale cohomology is modelled on the cohomology theory of sheaves in the usual topological sense. This definition of cohomological dimension is highly nonstandard and it is not obvious whether or not it agrees with the standard one from topos theory.
Download it once and read it on your kindle device, pc, phones or tablets. Algebraic geometry i shall assume familiarity with. The topological questions on compact lie groups, once they have been reduced to algebraic questions on lie algebras, suggest a certain number of purely algebraic objects, which may be constructed in relation with any lie. The functor of global sections is left exact and its derived functors where is a. Evan jenkinss notes of a seminar on etale cohomology click on the pdf. The collection of functors from topological spaces to abelian groups which assign cohomology groups of ordinary cohomology e. 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 mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups associated to a topological space, often defined from a cochain complex. In mathematics, the etale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by grothendieck in order to prove the weil conjectures. In 1904 schur studied a group isomorphic to h2g,z, and this group is known as the schur multiplier of g. Istrodlction in this paper i shall describe a method of associating a spectrum, and hence a cohomology theory, to a category with a compositionlaw of a suitable kind. Weil cohomology theories and the weil conjectures 3 proposition 1. Weil etale cohomology and special values of lfunctions. All schemes are taken to be noetherian and separated 1 introduction the development of etale cohomology was motivated by work on the weil conjectures, which state that localzeta functions x.
Jan 24, 2017 in cohomology in algebraic geometry we have introduced sheaf cohomology and cech cohomology as well as the concept of etale morphisms, and the grothendieck topology see more category theory. Generalized etale cohomology theories springerlink. Thus a covering for the etale topology is a surjective family of etale morphisms uix. In more detail, a generalized cohomology theory is a sequence of contravariant functors h i for integers i from the category of cwpairs to the category of abelian groups, together with a natural transformation d. Etale morphisms, etale fundamental group, the local ring for the etale topology, sheaves for the etale topology, direct and inverse images of sheaves, cohomology. The cohomology theory from the zariski topology is not suitable for several reasons. The relative picard group and suslins rigidity theorem 47 lecture 8. Y is a generically nite, surjective morphism of degree. Etale cohomology of fields and galois cohomology theories. Topics include nonabelian cohomology, postnikov towers, the theory of nstu, and ncategories for n 1 and 2. The essentials of etale cohomology theory springerlink. Some questions from the audience have been included.
The padic hodge theory of semistable galois representations. Ill complement the list of well known books on the subject by some freely available documents, which i find userfriendly. By matthew emerton and mark kisin let p be a prime number, fp the. This is a cohomology theory defined for spaces with involution, from which many of the other ktheories can be derived. It is defined in the standard manner by means of derived functors. Then the category of sheaves of abelian groups on is an abelian category with a sufficient collection of injective objects. Some versions of cohomology arise by dualizing the construction of homology. A generalized etale cohomology theory is a theory which is represented by a presheaf of spectra on an etale site for an algebraic variety, in analogy with the way an ordinary spectrum represents a cohomology theory for spaces. Group theory a concise introduction to the theory of groups, including the representation theory of finite.
Etale cohomology theory nankai tracts in mathematics. Etale cohomology pro and con todays lecture has a somewhat schizophrenic goal. Perfectoid spaces stand at the crossroads where topology, galois theory, and the study of equations by means of congruences meet. Hopefully we will be able to flesh out more of this interesting theory in future posts. A guide to etale motivic sheaves joseph ayoub abstract. It also ventures into deeper waters, such as the role of posets and brations. Sheaves and homotopy theory daniel dugger the purpose of this note is to describe the homotopytheoretic version of sheaf theory developed in the work of thomason 14 and jardine 7, 8, 9. Thus a covering for the etale topology is a surjective family of etale morphisms u ix.
Nevertheless, the cohomology of a space, which is obtained by dualizing its simplicail chain complex, carries important additional structure. The remaining talks, given in the category theory seminar at chicago, were more advanced. Etale cohomology and galois representations tom lovering june 22, 2012 abstract in this essay we brie y introduce the main ideas behind the theory of studying algebraic varieties over a number eld by constructing associated galois representations, and see how this can be understood naturally. This chapter summarizes the theory of the etale topology on schemes, culmi nating in the results on ladic cohomology that are needed in. This book gives new and complete proofs of both thomasons. A generalized etale cohomology theory is a theory which is represented by a. Most of the definitions and results from sheaf theory generalize quite. In the third part we define the motivic cohomology of a variety in this setting, a setting where it carries cohomology operations. Ct0408177 to define a cohomology theory that inherits the important properties of etale cohomology while allowing greater flexibility with the. One of the main reasons that it fails to give the results that we would want is that if xis a complex variety and fis a constant sheaf then hi zarx. Lectures on local cohomology university of illinois at. One must be over an algebraically closed eld here or happy with schemes to avoid confusion. Cobordism studies manifolds, where a manifold is regarded as trivial if it is the boundary of another compact manifold.
One of these conditions, the dimension axiom eilenbergsteenrod 52, i. An etale cover uis just a collection of etale open sets whose image is the whole of x. Also the cohomology groups hrxzar,f vanishes when fis a constant sheaf, or r dim x. Much of the material in these notes parallels that in, for example, iversen, b. Examples include etale cohomology and etale k theory. Etale cohomology tom sutherland october, 2008 note. The ereader files are formatted for viewing on ereaders they have double the number of pages. Bordism and cobordism theories edit cobordism studies manifolds, where a manifold is regarded as trivial if it is the boundary of another compact manifold. So once we have such a cover, we can x some abelian group of coe cients and compute the cech cohomology h u.
Use features like bookmarks, note taking and highlighting while reading etale cohomology theoryrevised editio nankai tracts in mathematics book 14. Notice that, while the terminology generalized cohomology is standard in algebraic topology with an eye towards stable homotopy theory, it is somewhat unfortunate in that there are various other and further generalizations of the axioms that all still deserve to be and are called cohomology. Unit lfunctions 1 unit lfunctions and a conjecture of katz. Generalized etale cohomology theories john jardine springer. For this reason, the existence of the \true a guide to motivic sheaves 3 abelian category of motives is still considered to be an. The cropped files have had their margins cropped may be better for viewing on gadgets. Mike shulmans extensive appendix x5 clari es many puzzles raised in the talks. We recall the construction, following the method of morel and voevodsky, of. Test case for an absent theory michael harris department of mathematics columbia university perfectoid prologue its not often that contemporary mathematics provides such a clearcut example of concept formation as the one i am about to present. Etale cohomology is an important branch in arithmetic geometry. Peter scholzes introduction of the new notion of perfectoid space.
Lectures on local cohomology craig huneke and appendix 1 by amelia taylor abstract. Etale cohomology theoryrevised editio nankai tracts in. Oct 11, 2016 etale cohomology pms33, volume 33 ebook written by james s. Download for offline reading, highlight, bookmark or take notes while you read etale cohomology pms33, volume 33. In this post, we give one important application of these ideas, related to the ideas discussed in galois groups. Cohomology can be viewed as a method of assigning richer algebraic invariants to a space than homology. This book covers the main materials in sga 1, sga 4, sga 4 12 and sga 5 on etale cohomology theory, which includes decent theory, etale fundamental groups, galois cohomology, etale cohomology, derived categories, base change theorems, duality, and ladic cohomology. The main application of this sheaf cohomology lies in. The first two parts develop a motivic homotopy theory, and are joint work with fabien morel. In order to get a modkuvalued re nement of kxhlg we observe that the ktheory functor for calgebras has a factorization. Etale cohomology fixes this problem by defining a cohomology from the covers of a space. Lei fu, etale cohomology theory is also nice and has not been. Workshop on motives, galois representations and cohomology. Pdf etale cohomology of schemes and analytic spaces.
For example, if x is a variety over c, then we have hrxet,l. Definition and the basic properties, cohomology of curves, cohomological dimension, purity. These underly a lot of deep relations between topics in topology and algebraic number theory, which in turn constitute an important part of. Let xbe a smooth, connected, ndimensional projective variety. Construction in this section, we go through the construction of the categories da ets. Etale cohomology of schemes and analytic spaces article pdf available in lecture notes in mathematics springerverlag 2119 january 2011 with 84 reads how we measure reads. One of the main reasons that it fails to give the results. Algebraic geometry i shall assume familiarity with the theory of algebraic varieties, for. Evan jenkinss notes of a seminar on etale cohomology click on the pdf icons.
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