Ingeneral,xkisjustaset,butiusuallyendowxcwithitsnaturalcomplex topology. Etale cohomology theoryrevised editio nankai tracts in mathematics book 14 kindle edition by lei fu. More generally, there is etale generalized cohomology theory with coefficients in sheaves of spectra on the etale site jardine 97. Let xbe a paracompact space, and suppose that u fu ig i2a is a \good cover of x. 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. 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 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. Etale cohomology theory by lei fu, 9789814307727, available at book depository with free delivery worldwide. Much of the material in these notes parallels that in, for example, iversen, b. Mazurs paper, notes on etale cohomology of number fields. Etale cohomology is an important branch in arithmetic geometry. The first identification is very general it is true for any cohomology theory on a site.
An excursion into etale cohomology 3 proposition 6. If x admits a weak n eron model xover the valuation ring of k, then sx equals the adic euler characteristic of the special ber of x. Some versions of cohomology arise by dualizing the construction of homology. Fu, algebraic geometry, tsinghua university press and springer.
What looked paradoxical in this image was the following observation about tran. Newest etalecohomology questions mathematics stack. Mazurs original paper artins book on grothendieck topology original version or latexed one tammes book on etale cohomology this is a detailed exposition of artins book. The correspondence between higgs bundles and local systems can be viewed as a hodge theorem for nonabelian cohomology. Grothendieck was the rst to suggest etale cohomology 1960 as an attempt to solve the weil conjectures. These course notes from spring 2010 are extremely rough. Macdonald, introduction to commutative algebra, addisonwesley publishing company, 1969. Fu, etale cohomology theory, revised edition, world scienti. The theory of motives is envisioned to be a universal cohomology theory that will unify the various known cohomology theories e. Etale cohomology theory nankai tracts in mathematics. However, it was artin and mazur who realized that we could actually associate a space to our.
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. For example,ifx specma,thenxkhom kalga,kandx k specma. The idea, which goes back to grothendieck, is that the motive of an algebraic variety xshould be. Thus cx can be regarded as contravariant functor, or equivalently a covariant functor of the category of c. In this article we will find more number fields \k\ such that. The topological etale site x et of xis given by 1the category of etale xspaces, i. Sheaf theory etale cohomology is modelled on the cohomology theory of. Etale cohomology theory nankai tracts in mathematics nankai tracts in mathematics hardcover 9789814307727. Pdf brauer groups and \etale cohomology in derived. But about discrete valuation, i have found two different definitions. Cohomology can be viewed as a method of assigning richer algebraic invariants to a space than homology.
A read is counted each time someone views a publication summary such as the title, abstract, and list of authors, clicks on a figure, or views or downloads the fulltext. Criteria for cohomological descent 43 references 67 introduction in classical cech theory, we \compute or better. Fujiwara, who proved delignes conjecture using his theory. Galois cohomology of fontaine rings max planck society. New post fulltext search for articles, highlighting downloaded books, view pdf in a browser and download history correction in our blog. This book covers the main materials in sga 1, sga 4, sga 4 12 and sga 5 on etale c. This is a graduate level course in etale cohomology theory on 7.
Cohomology of topological groups with applications to the. Etale homotopy theory tomer schlank contents 1 introduction and overview 9420 2. One of the most important tools in order to calculate these invariants are duality theorems. Lei fu, etale cohomology theory is also nice and has not been. For the case of good reduction, one gets the following period isomorphisms involving the barsotttate. Sheaf theory etale cohomology is modelled on the cohomology theory of sheaves in the usual topological sense. The prerequisites for reading this book are basic algebraic.
Any open immersion is a local isomorphism, which proves 1. Fu, etale cohomology theory, revised edition, world scientific, 2015. In 1904 schur studied a group isomorphic to h2g,z, and this group is known as the schur multiplier of g. Lectures on etale cohomology v ifx isavarietyoverk andk. Evan jenkinss notes of a seminar on etale cohomology click on the pdf. 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. The gros topos of topological spaces we follow the foundational conventions of gav72 based on the axiom of the existence of universes in set theory. This term, we will talk about etale cohomology theory. A simpson correspondence for abelian varieties in positive. The theory is hence called the nonabelian hodge theory, and sometimes is called the simpson correspondence. System upgrade on tue, may 19th, 2020 at 2am et during this period, ecommerce and registration of new users may not be available for up to 12 hours. In page 3 of lei fu s etale cohomology theory, there is a notion called discrete valuation ring as follows. Etale cohomology theory nankai tracts in mathematics by lei fu.
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