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. 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. For this spectral sequence d 2 and d 3 are known, and we give a description for d 4. Pdf the relative hochschildserre spectral sequence and. For this spectral sequence the second and third differentials are known, and we give a description for the fourth differential.
The additive structure of the spectral sequence and the structure constant algebra h. Jan 01, 2012 we prove the above results by using basic facts about the hochschild serre spectral sequence for relative lie algebra cohomology, and most of the paper consists of proving these facts using the. In this note we give two analogous spectral sequences in the cohomology theory. 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 these spectral sequences the second and third differentials are known. I have a hopefully simple question about the hochschild serre spectral sequence which may just be a simple question about general spectral sequences. A primer on spectral sequences university of chicago. 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. We will give the most important general family of examples. 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. Hochschild serre spectral sequence kevin buzzard june 11, 20 last modi ed 110620.
A generalization of the lyndonhochschildserre spectral sequence for polynomial cohomology a dissertation submitted to the faculty of purdue university by bobby william ramsey, jr. We construct a new localglobal spectral sequence for thomasons nonconnective ktheory, generalizing the quillen spectral sequence to possibly nonregular schemes. We consider the lyndonhochschildserre spectral sequence with if, coefficients for a central extension with kernel cyclic of order a power of p and arbitrary. The hochschildserre spectral sequence is a spectral sequence that expresses group cohomology by a special case of the grothendieck spectral sequence. In many cases our cohomology agrees with that constructed by. Two spectral sequences associated with a gsheaf and a normal subgroup of g are given together with an application. 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. These computations are used to determine a value of n su. Group cohomology and the hochschild serre spectral sequence 12 10. In this section we fix a graded lie algebra, l l i i, with universal enveloping algebra ul. Transfer map and hochschild serre spectral sequences transfer map and hochschild serre spectral sequences minh, pham anh 199510 00. This spectral sequence has the desired abutment and the e 2term involves the total right derived functor of h. 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.
A differential in the lyndonhochschildserre spectral. The first degree of 4 is just the degree of hochschild serre filtration of the cohomology class of hg representing 9. Introduction algebraic topology focuses on the properties of spaces and maps between. Associated to a fibration there is the serre spectral sequence 10. This will be useful in showing that a spectral sequence degenerates at e2, just by showing that the displaced spectral sequence e satis. Computing group cohomology rings from the lyndonhochschild.
Convergence proofs 12 references 15 this chapter contains those results about spectral sequences that we used earlier. A generalization of the lyndonhochschildserre spectral sequence. Transfer map and hochschildserre spectral sequences. Transfer map and hochschildserre spectral sequences, journal. Group cohomology and the hochschild serre spectral sequence. As a particular example, we have the hochschildserre spectral sequence for the cohomology of groups. Serre spectral sequences and applications emory computer. This book is entirely based on the lecture courses delivered within the noncommutative geometry and quantum groups project sponsored by the. 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. G k\subset g a normal subgroup and a a a left g gmodule. Applications of group cohomology university of texas at austin. Gysin sequence and hochschildserre spectral sequence. Good description of fis then an effective tool to determine the images of the transfer map, especially to the solving process of hochschild serre spectral sequence. Ken browns book, cohomology of groups also gives a fairly readable introduction to spectral sequences.
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. Thanks to laurent berger and david harari for pointing out independent blunders in earlier versions. The hochschild serre spectral sequence it is well past time for an example. 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. 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.
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. The hochschildserre spectral sequence springerlink. In this section the ground ring is an arbitrary field k of characteristic zero. A homologically graded spectral sequence e er consists of a. This can be done in many ways, but can be done \canonically if ahas enough injectives, and we get a les. Let me try and push infres as far as it can reasonably go. 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.
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 n is a normal subgroup of a group g, and m is an abelian group with an. Hochschildserre spectral sequence of algebras converging as an algebra 10. Categories of module spectra we use the framework of 8 in this paper. Then we want to know how how to extend this to a les.
View enhanced pdf access article on wiley online library html view. 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. Pdf the relative hochschildserre spectral sequence and the. So, suppose that we have two abelian categories aand b. 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. Niveau spectral sequences on singular schemes and failure of generalized gersten conjecture paul balmer abstract. Spectral sequences via examples the graduate journal of. A lyndon hochschild serre spectral sequence 3 arising from functoriality are maps of commutative s0algebras. Lyndonhochschildserre spectral sequence for not normal. The calabi invariant and the lyndonhochschildserre spectral. A differential in the lyndonhochschildserre spectral sequence. 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. Lyndonhochschildserre spectral sequence application. A users guide to spectral sequences, second edition mathematics.