Examples include etale cohomology and etale ktheory. This result reflects a geometric construction of complex k theory and suggests that a group theoretic description of a certain cohomology theory for bg might eventually lead to a geometric construction of the cohomology theory. The latter property makes it in general computable. In this paper, we study the indecomposability which is also a grouptheoretic property of profinite groups of various anabelian profinite groups. Etale cohomology is an important branch in arithmetic geometry.
Fu, etale cohomology theory, revised edition, world scienti. Given a presheaf of sets g, we define the free abelian presheaf on g, denoted zg. An oxmodule is called free if it is isomorphic to a direct sum of some. A homotopy lierinehart resolution and classical brst cohomology kjeseth, lars, homology, homotopy and applications, 2001. 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. More generally, there is etale generalized cohomology theory with coefficients in sheaves of spectra on the etale site jardine 97. Etale cohomology theory nankai tracts in mathematics by lei fu. The most important examples of generalized cohomology theories are theory and the various cobordism theories. C czzk is a free exterior algebra with klgenerators. Cohomology theories in algebraic geometry cohomology theories in motivic stable homotopy theory andreas holmstrom universitetet i oslo 12 sep 2012. Etale cohomology theory by lei fu overdrive rakuten.
These are the notes for a course taught at the university of michigan in w89 as math 732and in w98 as math 776. Etale cohomology lei zhang february 8, 2017 introduction in this course we are going to follow closely sga1 and sga4 to develop an abstract framwork of fundamental groups and cohomology theory. This book gives new and complete proofs of both thomasons. 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. For example, the first cohomology group of a complex elliptic curve is a free. Show that two compactly supported functions f and g di. 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 theory lei fu chern institute of mathematics, nankai university, china. 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. A very nice feature of fus work is the inclusion of some relevant. This sections states the classical formulation of the eilenbergsteenrod axioms due to eilenbergsteenrod 52, i. The investigation of the path cohomology from this framework, leads to a subtle version of poincare lemma for digraphs, which follows from the construction of the cw complex. 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.
From 2 on, the word linear will usually be dropped, since more general. Generalized etale cohomology theories john jardine springer. 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. On the other hand, we present an approach of using sheaf theory to reformulate digraph cohomologies. The generalized homology theories dual to cobordisms are the bordisms cf. Galois cohomology is the derived functor see more on chain complexes and the hom and tensor functors of the fixed module functor. By using the solution of the problem, one immediately sees that h3 r. Examples include etale cohomology and etale k theory. Classically, it is wellknown that various anabelian profinite groups, i. Cohomology theory article about cohomology theory by the.
Download it once and read it on your kindle device, pc, phones or tablets. By author lei fu chern institute of mathematics, nankai university, china. Cohomology theories in motivic stable homotopy theory andreas holmstrom universitetet i oslo. This book covers the main materials in sga 1, sga 4, sga 4 12 and sga 5 on etale c mathematics nonfiction. Such a cohomology theory is nowadays called a weil cohomology theory. For a locally constant sheaf on the complement of a divisor in x flat over s, deligne and laumon proved that the universal local acyclicity is equivalent to the local constancy of swan conductors. Etale cohomology theory nankai tracts in mathematics.
Serre, that illuminates some of the difficulties in constructing a weil cohomology. In the second chapter all necessary cohomology theory for lie algebras. A generalized etale cohomology theory is a theory which is represented by a. Every algebraic cycle of codimension gives a cohomology class in dimension, which makes it possible to construct a theory of chern classes cf.
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. In this article, assuming the universal local acyclicity, we show an analogous result of the continuity of. 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. Evan jenkinss notes of a seminar on etale cohomology click on the pdf icons.
Proetale cohomology helene esnault, lars kindler, lei zhang march 17, 2014 introduction the etale cohomology theory which was initially suggested by grothendieck in 1960s plays a very important role in modern algebraic geometry. Etale cohomology theoryrevised editio nankai tracts in mathematics book 14 kindle edition by lei fu. Use features like bookmarks, note taking and highlighting while reading etale cohomology theoryrevised editio nankai tracts in mathematics book 14. The essentials of etale cohomology theory springerlink skip to main content. The essentials of etale cohomology theory springerlink skip to. Note that, however, the category of chow motives does not give a weil cohomology theory since it is not abelian. Abelianizations of derivation lie algebras of the free associative algebra and the free lie algebra morita, shigeyuki, sakasai, takuya, and suzuki, masaaki, duke mathematical journal, 20. Etale cohomology tom sutherland october, 2008 note. New edition available hereetale cohomology is an important branch in arithmetic geometry. Etale cohomology of constructible sheaves is used to set up adic cohomology and to prove the weil conjecture on the zetafunction. Lei fu etale cohomology is an important branch in arithmetic geometry.
Cohomology theories in motivic stable homotopy theory. The collection of functors from topological spaces to abelian groups which assign cohomology groups of ordinary cohomology e. C2,c1,c0 are the free abelian groups generated by the set of faces. For instance, we prove that the etale fundamental group of the configuration space of a. Sep 22, 2014 on the other hand, we present an approach of using sheaf theory to reformulate digraph cohomologies. This result reflects a geometric construction of complex ktheory and suggests that a group theoretic description of a certain cohomology theory for bg might eventually lead to a geometric construction of the cohomology theory. The brown representability theorem says that for any reduced cohomology theory e. Continuous etale cohomology continuous hypercohomology. Generalized etale cohomology theories springerlink. Etale cohomology theoryrevised editio nankai tracts in.
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. For each group gand representation mof gthere are abelian groups hng,m and hng,m where n 0,1,2,3. I of x such that fui is a free oui module, if and only if f is flat, i. 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. 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. Generalized cohomology theories encyclopedia of mathematics. Etale cohomology theoryrevised editio and millions of other books are available for amazon kindle. Fu, etale cohomology theory, revised edition, world scientific, 2015. New edition available here etale cohomology is an important branch in arithmetic geometry.
More abstractly, via the classical model structure on topological spaces, these structures are seen to serve as presentations for certain homotopy pushouts. This book covers the main materials in sga 1, sga 4, sga 4 12 and sga 5 on etale c. Lectures on local cohomology craig huneke and appendix 1 by amelia taylor abstract. Contemporary trends in algebraic geometry and algebraic topology. 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. Grothendieck was the rst to suggest etale cohomology 1960 as an attempt to solve the weil conjectures. Specifically, we determine the proportion of twists. Macdonald, introduction to commutative algebra, addisonwesley publishing company, 1969. Any cohomology theory which is a1invariant and satis. We study the parity of 2selmer ranks in the family of quadratic twists of a fixed principally polarised abelian variety over a number field. The investigation of the path cohomology from this framework, leads to a subtle version of poincare lemma for digraphs, which follows from the. The first identification is very general it is true for any cohomology theory on a site and has nothing to. The essentials of etale cohomology theory springerlink.