Hochschild-serre spectral sequence pdf

A lyndon hochschild serre spectral sequence 3 arising from functoriality are maps of commutative s0algebras. We consider the lyndon hochschild serre spectral sequence with if, coefficients for a central extension with kernel cyclic of order a power of p and arbitrary discrete quotient group. This book is entirely based on the lecture courses delivered within the noncommutative geometry and quantum groups project sponsored by the. In this section the ground ring is an arbitrary field k of characteristic zero. So, suppose that we have two abelian categories aand b. The first degree of 4 is just the degree of hochschild serre filtration of the cohomology class of hg representing 9. Spectral sequences via examples the graduate journal of. Let me try and push infres as far as it can reasonably go. In this note we give two analogous spectral sequences in the cohomology theory. Read hochschild serre spectral sequences of split extensions and cohomology of some extraspecial p groups, journal of pure and applied algebra on deepdyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. G k\subset g a normal subgroup and a a a left g gmodule. Serre, who needed a spectral sequence in homology that applied to path space fibrations, whose total spaces are almost never locally compact, thus was unable to use the original leray spectral sequence and so derived a related spectral sequence whose cohomological variant agrees, for a compact fiber bundle on a wellbehaved space with the. The hochschild serre spectral sequence it is well past time for an example. The algebra is kept fairly simple here, and most of the discussion is about computing the homology of a double complex, and constructing the lyndon hochschild serre spectral sequence.

For this spectral sequence d 2 and d 3 are known, and we give a description for d 4. We have drawn largely on a paper of hochschild and mostow 4 which treats the case of gmodules which are real vector spaces, but our emphasis is quite different. The hochschildserre spectral sequence springerlink. Hochschild serre spectral sequence kevin buzzard june 11, 20 last modi ed 110620. These computations are used to determine a value of n su. Pdf the relative hochschildserre spectral sequence and. The hochschildserre spectral sequence is a spectral sequence that expresses group cohomology by a special case of the grothendieck spectral sequence. Introduction algebraic topology focuses on the properties of spaces and maps between. This can be done in many ways, but can be done \canonically if ahas enough injectives, and we get a les. Lyndonhochschildserre spectral sequence application. For details on the plain serre spectral sequence see at atiyahhirzebruch spectral sequence and take e h r e h r to be ordinary cohomology. Group cohomology and the hochschild serre spectral sequence. We consider the lyndon hochschild serre spectral sequence with modp coefficients for a central extension with kernel cyclic of order a power of p and arbitrary discrete quotient group. Then we want to know how how to extend this to a les.

Hochschildserre spectral sequence of algebras converging as an algebra 10. We can then form a knlocal ehh nadams resolution of ehk n in the stable category of ehk nmodules, again in the sense of 8. Ken browns book, cohomology of groups also gives a fairly readable introduction to spectral sequences. In this section we fix a graded lie algebra, l l i i, with universal enveloping algebra ul. Transfer map and hochschildserre spectral sequences. For these spectral sequences the second and third differentials are known. A homologically graded spectral sequence e er consists of a. This spectral sequence has the desired abutment and the e 2term involves the total right derived functor of h. Contents 1 introduction 1 2 the group cohomology and the group extensions 3 3 the calabi invariants and the euler class 5 4 the connection cochains and the extension classes 7 5 the lyndon hochschild serre spectral sequence and the veterm exact sequence 9 1 introduction. In mathematics, especially in the fields of group cohomology, homological algebra and number theory, the lyndon spectral sequence or hochschildserre spectral sequence is a spectral sequence relating the group cohomology of a normal subgroup n and the quotient group gn to the cohomology of the total group g. If h is a normal subgroup of a group g, the lyndon hochschild serre spectral sequence relates the cohomology of h and of gh to that of g.

Let g be a pgroup with frattini subgroup z of order p, and let erg be the hochschildserre spectral sequence corresponding to the central extension. Lectures on the cohomology of finite groups 3 2 using joins, we may construct a model for egwhich is functorial in g, namely eg colim i g. A generalization of the lyndonhochschildserre spectral sequence for polynomial cohomology a dissertation submitted to the faculty of purdue university by bobby william ramsey, jr. As a particular example, we have the hochschildserre spectral sequence for the cohomology of groups. Transfer map and hochschildserre spectral sequences, journal. This will be useful in showing that a spectral sequence degenerates at e2, just by showing that the displaced spectral sequence e satis. A users guide to spectral sequences, second edition mathematics. Gysin sequence and hochschildserre spectral sequence. We consider the lyndonhochschildserre spectral sequence with if, coefficients for a central extension with kernel cyclic of order a power of p and arbitrary. Serre spectral sequences and applications emory computer.

Applications of group cohomology university of texas at austin. Associated to a fibration there is the serre spectral sequence 10. Thanks to laurent berger and david harari for pointing out independent blunders in earlier versions. Two spectral sequences associated with a gsheaf and a normal subgroup of g are given together with an application. A differential in the lyndonhochschildserre spectral sequence. In many cases our cohomology agrees with that constructed by. Transfer map and hochschild serre spectral sequences transfer map and hochschild serre spectral sequences minh, pham anh 199510 00. Lyndonhochschildserre spectral sequence for not normal. Categories of module spectra we use the framework of 8 in this paper. We prove the above results by using basic facts about the hochschildserre spectral sequence for relative lie algebra cohomology, and most of the paper consists of. A differential in the lyndonhochschildserre spectral. Pdf the relative hochschildserre spectral sequence and the. I have a hopefully simple question about the hochschild serre spectral sequence which may just be a simple question about general spectral sequences.

We will give the most important general family of examples. The calabi invariant and the lyndonhochschildserre spectral. We construct a new localglobal spectral sequence for thomasons nonconnective ktheory, generalizing the quillen spectral sequence to possibly nonregular schemes. Convergence proofs 12 references 15 this chapter contains those results about spectral sequences that we used earlier. Group cohomology and the hochschild serre spectral sequence 12 10. We consider the lyndon hochschild serre spectral sequence with f n coefficients for a central extension with kernel cyclic of order a power of p and arbitrary discrete quotient group. We consider the lyndon hochschild serre spectral sequence with coefficients in the field of p elements for central extensions in which the kernel is cyclic of order a power of p. View enhanced pdf access article on wiley online library html view. Niveau spectral sequences on singular schemes and failure of generalized gersten conjecture paul balmer abstract. Cartan leray and lyndonhochschildserre spectral sequences, are. We consider the lyndonhochschildserre spectral sequence with modp coefficients for a central extension with kernel cyclic of order a power of p and arbitrary discrete quotient group.

For this spectral sequence the second and third differentials are known, and we give a description for the fourth differential. If m is a left right ulmodule then restriction of the action of ul to l defines a left right representation of l in m, and this correspondence is a bijection 21 a. If n is a normal subgroup of a group g, and m is an abelian group with an. A generalization of the lyndonhochschildserre spectral sequence.