Algebraic geometry i shall assume familiarity with. I will be following several sources, but most seriously will be lei fus etale cohomology, milnes etale cohomology, and sga 4. Generalized etale cohomology theories springerlink. Pdf etale cohomology of schemes and analytic spaces. 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. Like many places in math, there are many natural reasons why etale cohomology or other suitable replacements for the zariski topology is a desirable thing to study. Let m be a model a small elementary substructure of m. Due to my own lack of understanding of the materials, i have inevitably introduced. The essentials of etale cohomology theory springerlink. Jan 24, 2017 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. Galois cohomology is the derived functor see more on chain complexes and the hom and tensor functors of the fixed module functor. Jan 04, 2011 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.
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. Topological gauge theories and group cohomology robbert dijkgraaf institute for theoretical physics, university of utrecht, the netherlands edward witten school of natural sciences, institute for advanced study, olden lane, princeton, n. Some versions of cohomology arise by dualizing the construction of homology. Roughly speaking, this is k theory with the negative dimensional parts killed off. This note is based on the 3hour presentation given in the student seminar on winter 2014. 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. One of the eilenbergsteenrod axioms for a cohomology theory is the dimension axiom. Grothendieck was the rst to suggest etale cohomology 1960 as an attempt to solve the weil conjectures. For this, in homological language, we need the fact that cig. Cohomology can be viewed as a method of assigning richer algebraic invariants to a space than homology. This is a cohomology theory defined for spaces with involution, from which many of the other ktheories can be derived. Galois cohomology is the derived functor see more on chain complexes and the hom and tensor functors of. 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 gives new and complete proofs of both thomasons. Etale cohomology of almost strongly minimal groups will johnson april 7, 2014 1 the conjectural picture let tbe an almost strongly minimal theory. Weil cohomology theories and the weil conjectures weil realized that the rationality and the functional equation part of the weil conjectures would follow from the existence of a cohomology theory with suitable properties. 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. Adams, stable homotopy and generalized homology, univ.
Lecture notes on motivic cohomology carlo mazza vladimir voevodsky charles a. Generalized cohomology theories encyclopedia of mathematics. Much of the material in these notes parallels that in, for example, iversen, b. One of the authors frielander and dwyer, using the etale cohomology of grothendieck, gave a modn topological ktheory, called etale ktheory, which led to the work of suslin and voevodsky on the motivic homology of algebraic cycles, which is the main focus of this book. But weil had postulated such a theory to show the weil conjectures for varieties over nite elds by a xpoint formula, as it is known in topology. It gives a sheaf theory and cohomology theory with properties very close to those arising from the complex topology.
References for etale cohomology and related topics fall 2011. One of the most important uses of cohomology is that it converts short exact sequences of gmodulesto long exact sequences of abelian groups. Etale homotopy theory after artinmazur, friedlander et al. Its a quick introduction to the relation between galois theory, covering spaces, cohomology, and higher categories. The essentials of etale cohomology theory springerlink skip. Whitehead observed that it is fruitful to omit the dimension axiom completely. The most important examples of generalized cohomology theories are theory and the various cobordism theories. The analogous statement for classical chow groups is known to. 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. We establish various fundamental facts about brauer groups in this setting, and we provide a com. Weibel on his 65th birthday abstract we discuss the structure of integral.
The solution was found by grothendieck, together with m. The earliest results identifiable as galois cohomology had been known long before, in algebraic number theory and the arithmetic of elliptic curves. The generalized homology theories dual to cobordisms are the bordisms cf. The normal basis theorem implies that the first cohomology group of the additive group of l will vanish.
Every algebraic cycle of codimension gives a cohomology class in dimension, which makes it possible to construct a theory of chern classes cf. Etale cohomology theory is also nice and has not been mentioned yet. The collection of functors from topological spaces to abelian groups which assign cohomology groups of ordinary cohomology e. For any prime 6 chark this provides cohomology groups hix. We show that the beilinsonlichtenbaum conjectures for motivic cohomology implies the quillenlichtenbaum conjectures for algebraic k theory. Lecture notes on motivic cohomology carlo mazza vladimir voevodsky charles weibel american mathematical society clay mathematics institute clay mathematics monographs volume 2. We then get a nice cohomology theory by considering the rightderived functor of this global sections functor restricted to the subcategory of abelian schemes. Brauer groups and etale cohomology in derived algebraic geometry. The relative picard group and suslins rigidity theorem 47 lecture 8. 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. Examples include etale cohomology and etale ktheory. The remaining talks, given in the category theory seminar at chicago, were more advanced.
The essentials of etale cohomology theory springerlink skip to main content. We will basically follow milec, chapter i,ii,iii,v and millec, sections 114. We show that the beilinsonlichtenbaum conjectures for motivic cohomology implies the quillenlichtenbaum conjectures for algebraic ktheory. Serre, that illuminates some of the difficulties in constructing a weil cohomology. Etale cohomology theory nankai tracts in mathematics. Such a cohomology theory is nowadays called a weil cohomology theory. Given an arbitrary finitedimensional vector space f over r, we denote by cm, v the vector space of all glinear alternating functions defined on vim with values in v. The latter property makes it in general computable. Topics include nonabelian cohomology, postnikov towers, the theory of nstu, and ncategories for n 1 and 2. Etale cohomology of fields and galois cohomology theories. Brauer groups and etale cohomology in derived algebraic.
One of the authors frielander and dwyer, using the etale cohomology of grothendieck, gave a modn topological k theory, called etale k theory, which led to the work of suslin and voevodsky on the motivic homology of algebraic cycles, which is the main focus of this book. 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. Like many places in math, there are many natural reasons why etale cohomology or other suitable replacements for. Sheaf theory etale cohomology is modelled on the cohomology theory of sheaves in the usual topological sense. Etale cohomology is an important branch in arithmetic geometry. Mar 01, 2014 i will be following several sources, but most seriously will be lei fus etale cohomology, milnes etale cohomology, and sga 4. The notion of a sheaf over a topological space x generalize that of a bundle over x. This theory is a generalisation of cech theory to allow fairly general morphisms, and was used originally to compute the cohomology of an arbitrary complex variety in terms of smooth projective ones by deligne. Etale cohomology tom sutherland october, 2008 note. Algebraic geometry i shall assume familiarity with the theory of algebraic varieties, for. Etale cohomology of constructible sheaves is used to set up adic cohomology and to prove the weil conjecture on the zetafunction.
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