The aim of this ergebnisbericht is to develop as selfcontained as possible and as short as possible grothendiecks 1adic cohomology theory including delignes monodromy theory and to present his original proof of the weil conjectures. 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. One of the main reasons that it fails to give the results. Galois cohomology is the derived functor see more on chain complexes and the hom and tensor functors of the fixed module functor. Equivariant group cohomology and brauer group cegarra, a. Its a quick introduction to the relation between galois theory, covering spaces, cohomology, and higher categories. The next two chapters concern the basic theory of etale sheaves and elementary etale cohomology, and are followed by an application of the cohomology to the study of the brauer group. The work is one possible formulation of quillens ideas about algebraic ktheory, and i am very. Oct 11, 2016 the next two chapters concern the basic theory of etale sheaves and elementary etale cohomology, and are followed by an application of the cohomology to the study of the brauer group. Brauer groups and etale cohomology in derived algebraic. 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.
For more information about the weil conjectures, see appendix c of hartshornes algebraic geometry 6. Deligne succeded in proving the weil conjectures on the basis of grothendiecks ideas. In this course we are going to follow closely sga1 and sga4 to develop an abstract framwork of fundamental groups and cohomology theory. On the one hand we construct an \etale topological. 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. Lectures on etale cohomology harvard mathematics department. Number theory learning seminar 20162017 the seminar will meet wednesdays 1. May 01, 2020 buy introduction to etale cohomology universitext on free shipping on qualified orders.
The first thing to remark is that the zariski topology is not entirely satisfactory. Then the category of sheaves of abelian groups on is an abelian category with a sufficient collection of injective objects. Etale cohomology is one of the most important methods in modern algebraic the book gives a short and easy introduction into the world of. We intended especially to provide a complete introduction to etale and ladic cohomology theory including the monodromy theory of lefschetz pencils. Download pdf topicsinergodictheorypms44 free online. In this post, we give one important application of these ideas, related to the ideas discussed in galois groups. Topics include nonabelian cohomology, postnikov towers, the theory of nstu, and ncategories for n 1 and 2. New edition available here etale cohomology is an important branch in arithmetic geometry. The cohomology theory from the zariski topology is not suitable for several reasons. Etale cohomology tom sutherland october, 2008 note. 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. Etale cohomology of fields and galois cohomology theories.
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. Weil cohomology theories and the weil conjectures 3 proposition 1. 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. We also show that lichtenbaum cohomology, in contrast to the usual motivic cohomology, compares well with integral cohomology theories. In 1904 schur studied a group isomorphic to h2g,z, and this group is known as the schur multiplier of g. We show that there is a stable homotopy theory of profinite spaces and use it for two main applications. Sheaf theory etale cohomology is modelled on the cohomology theory of sheaves in the usual topological sense. The ereader files are formatted for viewing on ereaders they have double the number of pages. Stephen lichtenbaum conjectured that there exists a new cohomology theory, called weil etale cohomology, for arbitrary arithmetic schemes x, such that its cohomologies are connected to the zeta function of x.
In comparison with my book, the emphasis is on heuristic arguments rather than formal proofs and on varieties rather than schemes. Also the cohomology groups hrxzar,f vanishes when fis a constant sheaf, or r dim x. The relative picard group and suslins rigidity theorem 47 lecture 8. After a detailed analysis of the cohomology of curves and surfaces, professor milne proves the fundamental theorems in etale cohomology those of base change. We establish various fundamental facts about brauer groups in this setting, and we provide a com. By this reason in this paper the bounded cohomology theory of groups is discussed in thecaseofgeneralcoef. Artin and verdier on their duality theorem for number fields. Etale cohomology fixes this problem by defining a cohomology from the covers of a space.
Evan jenkinss notes of a seminar on etale cohomology click on the pdf icons. The functor of global sections is left exact and its derived functors where is. More generally, there is a connection between the etale cohomology of a field and galois cohomology, an important part of algebraic number theory that we now define. Let xbe a smooth, connected, ndimensional projective variety. 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. 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. An etale open set of x is an etale map ux which is the algebraic version of a local diffeomorphism. The cropped files have had their margins cropped may be better for viewing on gadgets. It also ventures into deeper waters, such as the role of posets and brations. Examples include etale cohomology and etale k theory. Ill complement the list of well known books on the subject by some freely available documents, which i find userfriendly. This book gives new and complete proofs of both thomasons.
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 local. Wellpresented and chosen this will be a most welcome addition to the algebraic geometrists library. Cohomology can be viewed as a method of assigning richer algebraic invariants to a space than homology. Etale cohomology is one of the most important methods in modern algebraic geometry and number theory. Examples include etale cohomology and etale ktheory. Etale homotopy theory after artinmazur, friedlander et al. Of course, all the central ideas are due to the people who created the theory, especially grothendieck and deligne. It is defined in the standard manner by means of derived functors. The focus of the cohomology theory of groups is on the cohomology with nontrivial coef. It has, in the last decades, brought fundamental new insights in arithmetic and algebraic geometric problems with many applications and many important results. About 1958, grothendieck defined the etale topology of a scheme, and the theory of etale cohomology was worked out by him with the.
Cohomology in the finite topology and brauer groups. Intro to mixed hodge modules pdf file published in cycles, motives and shimura varieties, tifr 2010 cohomology support loci for local systems and higgs bundles talks at u. Etale cohomology stacks project columbia university. The pdf files are formatted for printing on a4letter paper. What is etxle best textbook or book for studying etale cohomology. Zeta functions and arithmetic of abelian varieties cassels, dwork, shimura, serre pdf 3.
Every algebraic cycle of codimension gives a cohomology class in dimension, which makes it possible to construct a theory of chern classes cf. Introduction to etale cohomology by gunter tamme, available at book depository with free delivery worldwide. Knapp, advanced algebra, digital second edition east setauket, ny. 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. Y is a generically nite, surjective morphism of degree. The remaining talks, given in the category theory seminar at chicago, were more advanced. Etale cohomology of constructible sheaves is used to set up adic cohomology and to prove the weil conjecture on the zetafunction. 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. Adams, stable homotopy and generalized homology, univ.
Lubin, serre, and tate on formal groups and the lifting of abelian varieties. Mike shulmans extensive appendix x5 clari es many puzzles raised in the talks. Generalized etale cohomology theories springerlink. Algebraic geometry i shall assume familiarity with the theory of algebraic varieties, for. Etale cohomology theoryrevised editio nankai tracts in. 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. This chapter summarizes the theory of the etale topology on schemes, culmi nating in the results on ladic cohomology that are needed in the construction of. We use a new approach based on the relationship between eigenspaces and etale cohomology groups over the cyclotomic. The systematic use of etale cohomology greatly simplifies the proof and allows to generalize the classical result about the minuseigenspace to all odd eigenspaces. Etale cohomology is an important branch in arithmetic geometry. Some versions of cohomology arise by dualizing the construction of. We finish by focusing on the version of poincare duality for the etale cohomology of curves, which is one of the good properties that the theory. Kang kahlerde rham cohomology and chern classes comm.
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