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. Whenever i see the sequence written down, it has always been in the following form. The hochschild serre spectral sequence is a spectral sequence that expresses group cohomology by a special case of the grothendieck spectral sequence. Stefan and guichardet have provided lyndonhochschildserre type spectral sequences which converge to the hochschild cohomology and ext groups of a smash product. The purpose of this paper is to study the map fin some. A lyndon hochschildserre spectral sequence 3 arising from functoriality are maps of commutative s0algebras.
Cartan leray and lyndonhochschildserre spectral sequences, are. Spectral sequences for the cohomology rings of a smash product cris negron abstract. It follows from 1 and 2 that the entry of every subsequent page of the spectral sequence is also zero. Hochschildserre spectral sequence kevin buzzard june 11, 20 last modi ed 110620. Most spectral sequences we will encounter will be biregular. The hochschildserre spectral sequence is of general utility, and one of the goals of this paper is to provide a careful treatment of these facts for the literature. A generalization of the lyndonhochschildserre spectral sequence. A generalization of the lyndonhochschildserre spectral. As a particular example, we have the hochschildserre spectral sequence for the cohomology of groups. Gysin sequence and hochschildserre spectral sequence. Serre spectral sequences are a powerful theoretical and computational tool with numerous applications to algebraic topology. The hochschildserre property for some p adic analytic. Here, precise statements for the following spectral sequences are given. Jan 01, 2012 the hochschild serre spectral sequence is of general utility, and one of the goals of this paper is to provide a careful treatment of these facts for the literature.
For every maximal subgroup k of g, we study the map t e. Let me try and push infres as far as it can reasonably go. We use this to identify lmodules as ulmodules and conversely. Serre spectral sequence of a bration, lyndonhochschildserre spectral sequence of a short exact sequence of groups, atiyahhirzebruch spectral sequence of a bration. Miller spectral sequence converging to the mod p stable homology of a space. The functor homogeneous cochain complex is a resolving functor in the sense of gr, x2. This spectral sequence has the desired abutment and the e 2term involves the total right derived functor of h. A users guide to spectral sequences, second edition mathematics. Thanks to laurent berger and david harari for pointing out. The hochschildserre spectral sequence springerlink.
Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields. Let hand kbe closed subgroups of the extended morava stabilizer group gnand suppose that his normal in k. We construct a strongly convergent spectral sequence h ckh. Pdf the relative hochschildserre spectral sequence and. Indeed, a main application is the computation of various homotopy groups of spheres, and we will demonstrate this by the calculation of.
Alternatively, we can make the assumption that irreducible galois representa tions are attached to homology eigenvectors only in degrees in the \cuspidal range. In this section we fix a graded lie algebra, l l i i, with universal enveloping algebra ul. The first degree of is just the degree of hochschildserre filtration of the cohomology class of hg representing tc. We consider the lyndonhochschildserre 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. Two spectral sequences associated with a gsheaf and a normal subgroup of g are given together with an application. 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. Note that our spectral sequence 1 is potentially interesting even for local schemes, where the. All examples in these notes are based on these spectral sequences. Let \h \subseteq g\ be an inclusion of padic lie groups. We consider the lyndon hochschildserre spectral sequence with if, coefficients for a central extension with kernel cyclic of order a power of p and arbitrary discrete quotient group. Derived functors and sheaf cohomology contemporary.
A differential in the lyndonhochschildserre spectral sequence. I have a hopefully simple question about the hochschild serre spectral sequence which may just be a simple question about general spectral sequences. In appendix a, we will present the singular homology with integer. Thanks for contributing an answer to mathematics stack exchange. When h is normal or even subnormal in g, the hochschildserre spectral sequence implies that any topological gmodule whose hcohomology vanishes in all degrees also has vanishing gcohomology. Moving this page from lyndon hochschild serre spectral sequence to lyndon hochschild serre s. Our results are applied to get some mod p cohomology classes of extra. Using this result we deduce a similar formula for the serre spectral. Also, the construction of the relative hochschildserre spectral sequence in bw uses the identi. To synthesize, let us consider a general dimension function 3 dim.
We will give the most important general family of examples. Serre spectral sequences and applications emory computer. Spectral sequences via examples the graduate journal of. Group cohomology and the hochschild serre spectral sequence. Ken browns book, cohomology of groups also gives a fairly readable introduction to spectral sequences. It turns out that 5 is compatible with the hochschildserre filtrations on ck and. Hochschild serre spectral sequence kevin buzzard june 11, 20 last modi ed 110620.
Categories of module spectra we use the framework of 8 in this paper. Stefan and guichardet have provided lyndon hochschild serre type spectral sequences which converge to the hochschild cohomology and ext groups of a smash product. We consider the lyndonhochschildserre spectral sequence with if, coefficients for a central extension with kernel cyclic of order a power of p and arbitrary. Actually, we can also construct a niveau spectral sequence, involving dimension instead of codimension. 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 it is well past time for an example. Hochschildserre spectral sequence of algebras converging as an algebra 10. We shall see in remark 6 that this fails in general. On hochschildserre spectral sequence of lie algebras.
Computing group cohomology rings from the lyndonhochschild. Milnor spectral sequence is another name for the bar spectral sequence. 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. Examples that we shall consider are another proof of the cechde rham theorem, the spectral sequence associated to an open cover cech spectral sequence, the leray spectral sequence, the localtoglobal spectral sequence, the frolicher spectral sequence of a complex manifold, the hochschildserre spectral sequence associated with an. The hochschildserre spectral sequence is a spectral sequence that expresses group cohomology by a special case of the grothendieck spectral sequence. But avoid asking for help, clarification, or responding to other answers. An introduction to spectral sequences matt booth december 4, 2016. The differential from odd rows to even rows involves a massey triple product, and we.
I have a hopefully simple question about the hochschildserre spectral sequence which may just be a simple question about general spectral sequences. Serre spectral sequence of a bration, lyndon hochschild serre spectral sequence of a short exact sequence of groups, atiyahhirzebruch spectral sequence of a bration. Using nonabelian exterior product and free presentation of a lie algebra the hochschildserre spectral sequence for cohomology of lie algebras will be extended a step further. The lyndon hochschild serre spectral sequences for sheaves with operators tatsuo suwa1 abstract. Pdf the relative hochschildserre spectral sequence and the. I will then demonstrate some neat applications of group cohomology to geometric, algebraic, and topological problems. For this spectral sequence the second and third differentials are known, and we. At the end i will discuss a powerful tool for computing group cohomology, the lyndon hochschild serre spectral sequence. A differential in the lyndonhochschildserre spectral. 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. May spectral sequence for calculating the tor or ext groups of an algebra. 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 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. 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. These are some notes for my talk in the bell show at the conference structured ring spectra tng, on august 4th, 2011, in hamburg, germany. We consider the lyndonhochschildserre spectral sequence with if, coefficients for a central extension with kernel cyclic of order a. Let ebe a spectral sequence, and suppose that for some r. At the end i will discuss a powerful tool for computing group cohomology, the lyndonhochschildserre spectral sequence.
Using nonabelian exterior product and free presentation of a lie algebra the hochschildserre spectral sequence for cohomology of lie algebras will b. The lyndonhochschildserre spectral sequences for sheaves with operators tatsuo suwa1 abstract. 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. In this section the ground ring is an arbitrary field k of characteristic zero. Associated to a fibration there is the serre spectral sequence 10. Vermani department of mathematics, university of exeter, exeter, department of mathematics, kurukshetra university, kurukshetra, india. If n is a normal subgroup of a group g, and m is an abelian group with an.
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. Transfer map and hochschildserre spectral sequences. G k\subset g a normal subgroup and a a a left g gmodule. With an eye towards applications in padic hodge theory, we extend this to some cases where h is not subnormal, assuming that the g. It is legitimate to ask whether the e 2page of our spectral sequence 1 coincides with that of 2. Let g be a pgroup with frattini subgroup z of order p, and let e r g be the hochschildserre spectral sequence corresponding to the central extension of z representing g. In the second chapter, we construct the serre spectral sequence of a.
Second, we use an implementation of the multiplicative structure of the lyndonhochschildserre spectral sequence for determining how much of a projective resolution needs to be computed in order to obtain a presentation of the cohomology ring. A generalization of the lyndon hochschildserre spectral sequence for polynomial cohomology a dissertation submitted to the faculty of purdue university by bobby william ramsey, jr. Kathryn hessbellwald supervisedby marc stephan academic year. A generalization of the lyndonhochschildserre spectral sequence for polynomial cohomology a dissertation submitted to the faculty of purdue university by bobby william ramsey, jr. Also, sketch of proofs of the two rst spectral sequences are included and the ltrations employed are. In many cases our cohomology agrees with that constructed by. The algebra is kept fairly simple here, and most of the discussion is about computing the homology of a double complex, and constructing the lyndonhochschildserre spectral sequence. Convergence proofs 12 references 15 this chapter contains those. It turns out that 5 is compatible with the hochschild serre filtrations on ck and.
A lyndon hochschild serre spectral sequence 3 arising from functoriality are maps of commutative s0algebras. Lyndonhochschildserre spectral sequence in group cohomology. A lyndonhochschildserre spectral sequence 3 arising from functoriality are maps of commutative s0algebras. For this spectral sequence the second and third differentials are known, and we give a description for the fourth differential.