This is to some extent a continuation of an earlier question of mine. The combination of theory and examples, together with the. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields. We extend the framework of entire cyclic cohomology to the equivariant context.
Automorphisms in positive characteristic 20 references 23 0. Because a cyclic group is abelian, each of its conjugacy classes consists of a single element. Journal of algebra 11, 598612 1969 on the cohomology of finite semigroups william r. On the cohomology of polycyclicbyfinite groups martin lorenz maxplancklnstitut ffir mathematik, gottfriedclarenstr. This section is a collection of generally known facts about graded and filtered rings and modules. This is the first book to deal comprehensively with the cohomology of finite groups. For example, g a nilpotent group or g a frobenius group where p. The most general results involve the definitions of the cyclic homology of cyclic chain complexes and the notions of cyclic and cocyclic spaces so precise statements will be postponed until w 3. Lectures on the cohomology of finite groups alejandro adem.
Also, many of the hyperbolic groups exhibit exotic properties, like kazhdans property t 18, 5. Commutative algebra is used extensively in the cohomology of groups. We give a simple algebraic recipe for calculating the components of the bv operator. We recall that an elementary abelian pgroup is a group isomorphic to zpz. Recent developments in the cohomology of finite groups.
Cohomology ring of a finite group 225 summation with respect to that index. Aug 01, 2010 im pretty sure it follows from hodge theory and the finitedimensionality of the harmonic forms. This book is purely algebraic and concentrates on cyclic homology rather than on cohomology. The presentations we will give for the examples in chapter6are simpler than what is in the literature to date. Galois cohomology math 2510 2 the case of finite cyclic groups before we move on to the case of galois cohomology, let us specialize to the situation of the cohomology of cyclic groups.
In this series of lectures, i concentrate on nite groups, but i also discuss the cohomology of nite group schemes, compact lie groups, pcompact groups, in nite discrete groups and pro nite groups. 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 spaces of cohomology type 0, 0, we show that every psubgroup of g is either cyclic or a generalized quaternion group. These are notes based on lectures given at the summer school interactions between homotopy theory and algebra, which was held at the university of chicago in the summer of 2004. Historically, the subject has been one of significant interaction between algebra and topology and has directly led to the creation of such important areas of mathematics as homo logical algebra and algebraic ktheory. In this paper, we show that if a finite group g acts freely on a space of type 0, 1, where n is odd, then g cannot contain. This section describes magma functions for computing the first cohomologygroup of a finite group with coefficient in a finite not necessarily abelian group. The cohomology of the finite general linear group 101 and observe that if t is cyclic, xi is a generator for h2t,zp as i, and nf then both are isomorphisms. For idimensional characters there is a wellknown isomorphism. A finite group gacts freely on a finite complex xhomotopy equivalent to a sphere if and only if every abelian subgroup in gis cyclic. It was in 1945 that eilenberg and maclane introduced an algebraic approach which included these groups as special cases. Hodge decomposition for finite cyclic coverings 12 5. Connes as the cohomological structure involved in the computation of indices of elliptic operators cf.
We compute the cohomology of fms with arbitrary coe. If g is finite, however, then homology and cohomology seem to have similar properties rather than dual ones. The largest finite group that is also a sporadic simple group, i. In 1904 schur studied a group isomorphic to h2g,z, and this group is known as the schur multiplier of g. The audience consisted of graduate students and postdocs and my task was to introduce them to the subject. The cohomology of groups has, since its beginnings in the 1920s and 1930s, been the stage for significant interaction between algebra and topology and has led to the creation of important new fields in mathematics, like homological algebra and algebraic ktheory. Finite groups slawomir klimek, wltold kondracki, and andrzej lesniewski department of mathematics, harvard university, cambridge, ma 028, u. The first chapter deals with the intimate relation of cyclic theory to ordinary hochschild theory. Given a projective resolution and a set of r cocycles that represents a homogeneous system of parameters of the integral cohomology ring, the. Nico department of mathematics, tulane university, new orleans, louisiana 70118 communicated by sounders maclane received november 8, 1967 given a semigroup s and an ismodule d, we define cohomology by hs, d ext5z, d as is usual for monoids.
The cohomology groups with coefficients in an abelian group which we may treat as a module over a unital ring, which could be or something else are given by. Cohomology of finite pgroups of fixed nilpotency class. A cyclic group of order n therefore has n conjugacy classes. Im pretty sure it follows from hodge theory and the finitedimensionality of the harmonic forms.
Quillen 20, and if equivariant homology and cohomology theories. Introduction finite groups can be studied as groups of symmetries in di. Group homology and cohomology can be treated uniformly for some groups, especially finite groups, in terms of complete resolutions and the tate cohomology groups. The university of georgia algebra research group cohomology of finite groups. In this section, we shall consider a central extension c g q of finite pgroups with c a cyclic pgroup and show that under mild assumptions, the cohomology algebra h. Now that im all cleared up on what it means for a finite group to have periodic cohomology, i have another question.
Moreover, if hai n, then the order of any subgroup of hai is a divisor of n. Homology and cohomology are usually thought of as dual to one another. In case a is commutative and is nite, our results are complete and given in terms of the singular cohomology. The cohomology of semigroups exhibits some distinctively different characteristics from that of groups. In 1932 baer studied h2g,a as a group of equivalence classes of extensions. Pdf on the bv structure of the hochschild cohomology of. You can interpret cyclic cohomology as ext groups using representations of the cyclic category. To send this article to your dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. The fundamental theorem of abelian groups states that every finitely generated abelian group is a finite direct product of primary cyclic and infinite cyclic groups. Bernhard keller, on the cyclic homology of exact categories, journal of pure and applied algebra 6 1999. Introduction for any finite group g one can consider the integral cohomology ring hg, z and the ring rg of unitary characters cf.
If this is the first time you use this feature, you will be asked to authorise cambridge core to connect with your account. Finally hg, a will refer to the socalled unmodified cohomology groups. Discussion of group characters and group cohomology of finite groups includes. Cohomology of finite monogenic selfdistributive structures.
We study the periodic cyclic homology groups of the crossproduct of a nite type algebra a by a discrete group. Some versions of cohomology arise by dualizing the construction of. On the cohomology of finite semigroups sciencedirect. Group cohomology of finite cyclic groups groupprops. It attempts to single out the basic algebraic facts and techniques of the theory. On the way we develop general tools for studying the cohomology of shelves. However, i learned a neat elementary proof that id like to discuss. Thirdly, the cohomology of any finite polyhedron can be embedded into the cohomology of a hyperbolic group 18, 0. Cohomology can be viewed as a method of assigning richer algebraic invariants to a space than homology. F p is determined up to a finite number of possibilities by the cohomology algebra h. The connection with posets, mobius inversion convolution algebras and fundamental theorem of calculus is well known 4 and the elementary tip of the iceberg. I was wondering if it was possible to distill this 2periodic resolution somehow out of the standard barresolution above in some natural way. Venkov received 10 junuary 1972 let g be a finite group and let hg hg, zpz be its mod p cohomology ring. Finite monogenic shelves fms, of which laver tables and cyclic racks are the most famous examples, form a remarkably rich family of structures and play an important role in set theory.
Bernhard keller, on the cyclic homology of exact categories, journal of pure and applied algebra 6 1999, 156, pdf. The following text is an expanded version of my lectures. Finite group cohomology louisiana state university. Products of finite type algebras jacek brodzki, shantanu dave, and victor nistor abstract. Masoud khalkhali, a short survey of cyclic cohomology, arxiv1008. The cyclic cohomology groups of a a are the cohomology groups of the dual cochain complex. Hochschild and linckelmann cohomology 5 denotes a cyclic group of order 2. Let g be a group with normal sylow psubgroup p such that for any k. Bernhard keller, on the cyclic homology of ringed spaces and schemes, doc. 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. In contrast to the simple result that for a finite group g every element of hg, d for n 0 has order dividing the order of g, we find in corollary 3.
The cohomology groups with coefficients in the ring of integers are given as follows. The point is that a finite group gsatisfies the group. We study the periodic cyclic homology groups of the crossproduct of a. Essential cohomology of finite groups alejandro adem anddikran karagueuzian mathematics department university of wisconsin madison, wi 53706 usa x0 introduction one of the main di culties in calculating or understanding the mod pcohomology of a nite group gis the fact that in many instances nontrivial cohomology classes. In september 2016 i gave 5 introductory lectures on cyclic cohomology and some of its applications in impan warsaw, during the simons semester in noncommutative geometry. In this example the topological notion is connected to the grouptheoretic hypothesis via the concept of groups with periodic cohomology. Also, many of the hyperbolic groups exhibit exotic properties, like kazhdans property. Some historical background this book deals with the cohomology of groups, particularly finite ones. Commutative algebra in the cohomology of groups dave benson abstract.
We will then demonstrate that our results verify and complete results. Gruenberg received i1 may 1984 revised 19 november 1984 let kg be the group algebra of a polycyclicbyfinite group g over a field k. Let w be the weyl group of gl,q which we will take to be the group of linear transformations permuting 33. Big picture view from 10,000 ft representations and cohomology the ongoing project of the algebra vrg the enormous theorem.
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