We show that the beilinsonlichtenbaum conjectures for motivic cohomology implies the quillenlichtenbaum conjectures for algebraic k theory. The remaining talks, given in the category theory seminar at chicago, were more advanced. Stacks project chapters on hypercoverings, cohomology of sheaves, and cohomology on sites. 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. Examples include etale cohomology and etale k theory. Generalized etale cohomology theories springerlink. B is agmodule homomorphism, then for each i 0, there is an induced. I will be following several sources, but most seriously will be lei fus etale cohomology, milnes etale cohomology, and sga 4.
Brauer groups and etale cohomology in derived algebraic geometry. 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. 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. Topics include nonabelian cohomology, postnikov towers, the theory of nstu, and ncategories for n 1 and 2. Due to my own lack of understanding of the materials, i have inevitably introduced. Some versions of cohomology arise by dualizing the construction of homology. Cohomology can be viewed as a method of assigning richer algebraic invariants to a space than homology. Etale cohomology of almost strongly minimal groups will johnson april 7, 2014 1 the conjectural picture let tbe an almost strongly minimal theory.
The analogous statement for classical chow groups is known to. The most important examples of generalized cohomology theories are theory and the various cobordism theories. We will basically follow milec, chapter i,ii,iii,v and millec, sections 114. 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. Etale cohomology is an important branch in arithmetic geometry. Much of the material in these notes parallels that in, for example, iversen, b. We then get a nice cohomology theory by considering the rightderived functor of this global sections functor restricted to the subcategory of abelian schemes. We show that the beilinsonlichtenbaum conjectures for motivic cohomology implies the quillenlichtenbaum conjectures for algebraic ktheory. These are the notes for a course taught at the university of michigan in w89 as math 732and in w98 as math 776. Weibel on his 65th birthday abstract we discuss the structure of integral. Roughly speaking, this is k theory with the negative dimensional parts killed off. 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.
The latter property makes it in general computable. 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. Roughly speaking, this is ktheory with the negative dimensional parts killed off. 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. 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.
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. We establish various fundamental facts about brauer groups in this setting, and we provide a com. Etale cohomology theory is also nice and has not been mentioned yet. Mar 01, 2014 i will be following several sources, but most seriously will be lei fus etale cohomology, milnes etale cohomology, and sga 4. Lecture notes on motivic cohomology carlo mazza vladimir voevodsky charles a. 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. Grothendieck was the rst to suggest etale cohomology 1960 as an attempt to solve the weil conjectures. Etale cohomology of fields and galois cohomology theories. Since the identity is an etale morphism, we can compute the global sections of an. Brauer groups and etale cohomology in derived algebraic. Etale cohomology tom sutherland october, 2008 note. Algebraic geometry i shall assume familiarity with.
The essentials of etale cohomology theory springerlink. Etale homotopy theory after artinmazur, friedlander et al. Galois cohomology is the derived functor see more on chain complexes and the hom and tensor functors of. Every algebraic cycle of codimension gives a cohomology class in dimension, which makes it possible to construct a theory of chern classes cf. This book gives new and complete proofs of both thomasons. For any prime 6 chark this provides cohomology groups hix. This is a cohomology theory defined for spaces with involution, from which many of the other ktheories can be derived. One of the eilenbergsteenrod axioms for a cohomology theory is the dimension axiom. 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. Its a quick introduction to the relation between galois theory, covering spaces, cohomology, and higher categories. 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.
The essentials of etale cohomology theory springerlink skip. Galois cohomology is the derived functor see more on chain complexes and the hom and tensor functors of the fixed module functor. One of the most important uses of cohomology is that it converts short exact sequences of gmodulesto long exact sequences of abelian groups. 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. 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. Examples include etale cohomology and etale ktheory. Adams, stable homotopy and generalized homology, univ. Like many places in math, there are many natural reasons why etale cohomology or other suitable replacements for. Such a cohomology theory is nowadays called a weil cohomology theory. The normal basis theorem implies that the first cohomology group of the additive group of l will vanish. For this, in homological language, we need the fact that cig. The generalized homology theories dual to cobordisms are the bordisms cf.
The relative picard group and suslins rigidity theorem 47 lecture 8. This note is based on the 3hour presentation given in the student seminar on winter 2014. The earliest results identifiable as galois cohomology had been known long before, in algebraic number theory and the arithmetic of elliptic curves. References for etale cohomology and related topics fall 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. Etale cohomology of constructible sheaves is used to set up adic cohomology and to prove the weil conjecture on the zetafunction. Etale cohomology theory nankai tracts in mathematics. Algebraic geometry i shall assume familiarity with the theory of algebraic varieties, for. The essentials of etale cohomology theory springerlink skip to main content.
Pdf etale cohomology of schemes and analytic spaces. The collection of functors from topological spaces to abelian groups which assign cohomology groups of ordinary cohomology e. 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. Whitehead observed that it is fruitful to omit the dimension axiom completely. The notion of a sheaf over a topological space x generalize that of a bundle over x. Serre, that illuminates some of the difficulties in constructing a weil cohomology. Generalized cohomology theories encyclopedia of mathematics. Let be an dimensional vector bundle over, orientable see orientation in a generalized cohomology theory, and let be its thom space. Lecture notes on motivic cohomology carlo mazza vladimir voevodsky charles weibel american mathematical society clay mathematics institute clay mathematics monographs volume 2. Sheaf theory etale cohomology is modelled on the cohomology theory of sheaves in the usual topological sense. It gives a sheaf theory and cohomology theory with properties very close to those arising from the complex topology. 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.
1274 1174 101 934 1527 1161 476 838 927 1025 1201 474 1036 64 801 469 1164 648 382 1208 154 658 705 385 1458 573 952 1153 708 1453 1397 1529 625 742 926 1386 1044 1567 194 127 77 915 1407 1397 555 158 935 1465 1132 254