Friday, may 26: lyndon-hochschild-serre spectral sequence. This note explains the following topics: introduction to the serre spectral sequence, with a number of applications, mostly fairly standard, the adams spectral sequence, eilenberg-moore spectral sequences. To read the original papers by lyndon, hochschild-serre, and serres thesis. The existence of such a sequence follows from the hochschildserre spectral sequence associated to the lie algebra extension. The hochschild-serre spectral sequence associated to a subalgebra of a lie algebra h,s and. Mology of groups and lie algebras, and the cohomology of associative algebras. 3 on the leray-serre spectral sequence in topology follows. Sheaf theory from harders book: basic language of sheaves, adjunction formula, sheaf cohomology and its functorial properties, cohomology with. The cohomology spectral sequence multiplicative structure. Spectral sequences are a key theoretical and computational tool in algebraic topology. Hs;djs2;dn;sd d 1si ?C n oc 2 how-to guide: we use the split short exact sequence1 f1g!C n!D n. Stating that the enveloping algebra of a torsion-free lie superalgebra is a do-. We can now explain the lyndon-serre-hochschild spectral sequence. Chapter 7 concems the homology and cohomology of lie algebras. 1015 Making it a spectral sequences of lie algebras by obtaining a result. Hochschild-serre spectral sequence for not normal subalgebra. Theory for calculating the cohomology of semi simple lie algebras, but more tools are needed for.
From the exact couple formulation of spectral sequences, we introduce a special case of the leray-serre spectral sequence and use it to compute h cpn;z. April 28th, 2014: lyndon-hochschild-serre spectral sequence. 326 As an application, we show that the hypercohomology of the atiyah algebroid of a line bundle has a natural hodge structure. This paper is a tutorial essay, intended for students and researchers as a quick introduction, providing a few nice examples, applications, and a short list of references for further inquiry. The computation of the mod p cohomology of compact lie groups was amply demonstrated in borels thesis. In mathematics, the serre spectral sequence sometimes lerayserre spectral sequence to acknowledge earlier work of jean leray in the leray spectral sequence is an important tool in algebraic topology. Consists of the one jets in wn- using the hochschild-serre spectral sequence associated to a subalgebra of a lie algebra hs and invariant theory w. Notation wn for this lie algebra following textbook 12. 5: lyndon-hochschild-serre; wild spectral sequences ep. Using the hochschild-serre spectral sequence to compute cohomology dvi ps. Absolutely irreducible lie algebra modules m,, with highest weight x. Chevalleyeilenberg cohomology and hochschild cohomology to address. This book describes some of the most important examples of spectral sequences and some of their most spectacular applications. We also use the hochschild-serre spectral sequence for the cohomology of alge-. A list of commutative algebra books for self study; a little intro to the jacobi symbol: part 1. The leray-serre spectral sequence arises when studying the homology and cohomology of fibrations with path-connected base spaces and connected fibers. 1 may then be interpreted as the hochschild-serre spectral se- quence for ne cn where ne.
A hochschildserre spectral sequence in the relative case. 349 The book covers four main spectral sequences that arise in algebraic topology: the leray-serre, eilenberg-moore, adams, and bockstein spectral sequences. Fibrations and the leray-serre spectral sequence 4 3. Friday, june 2: overview of lie algebra cohomology. Hochschild-serre spectral sequence for a lie algebra extension 0. In this section, we explain briefly the spectral sequence obtained by. When l is a restricted lie algebra over a field k and m is a module over the. We also prove a form of poincare duality for lie groups over. Hochschild-serre spectral sequence can be described as the cup product with. The last part of the book treats applications throughout mathematics, including. A users guide to spectral sequences cambridge studies in advanced mathematics book 58 - kindle edition by mccleary, john. Any of the standard texts on homological algebra or cohomology of groups contains an account of the theory of spectral sequences.
Download it once and read it on your kindle device, pc, phones or tablets. And ext, projective dimensions and spectral sequences. Spectral sequence for lie algebra extensions is discussed in 3 and 7. It expresses, in the language of homological algebra, the singular cohomology of the total space x of a serre fibration in terms of the cohomology of the base space b and the fiber f. The heart of the text is an exposition of the classical examples from homotopy theory, with chapters on the leray-serre spectral sequence, the eilenberg-moore spectral sequence, the adams spectral sequence, and, in this new edition, the bockstein spectral sequence. The homology of the lie algebra of polynomial vector fields on the line with. 295 Use features like bookmarks, note taking and highlighting while reading a users guide to spectral sequences cambridge studies in advanced mathematics book 58. The hochschild-serre spectral sequences relative to a suitable ideal. Computing hg beings in w 3, where we use botts lie-algebraic. Hochschild-serre spectral sequence of a split lie algebra extension, from which we derive.
Using the hochschild-serre spectral sequence to compute cohomology dvi ps pdf. Hochschild-serre spectral sequences implies main theorem 2. Theorems of appropriate hochschild-serre spectral sequences we avoid the need. Homology of group and lie algebras illustrate these topics. Sciendo provides publishing services and solutions to academic and professional organizations and individual authors. When the extension 8 splits, we propose a modi?Cation to the construction of the hochschild-serre spectral sequence. Stabilizer groups, which are the input for various spectral sequence. Bai and liu determined completely the betti numbers and associative superalgebra structures for trivial cohomology of heisenberg lie superalgebras by using the hochschild-serre spectral sequence. Introduction algebraic topology focuses on the properties of spaces and maps between. Serre spectral sequence of a short exact sequence of groups, atiyah-hirzebruch spectral. As above, let g be a group, n a normal subgroup, and a a g-module. Leray spectral sequence; base change spectral sequence for tor and ext; hochschild-serre spectral sequence. For every ideal h of a lie algebra g, there is a convergent first quadrant spectral sequence. Serre spectral sequences and applications david brown abstract. Shapiros lemma, the hochschild-serre spectral sequence and the classi-. The first third of the book treats the algebraic foundations for this sort of homological algebra, starting from informal calculations, to give. A spectral sequence is a sequence of complexes which, by taking successive cohomologies converges to the cohomology of a given complex or to some graded version thereof. De rham cohomology of g in terms of the lie algebra cartans theorem. 562
We also interpret our results in the settings of group and lie algebra extensions and prove some interesting corollaries concerning the collapse of the lyndon-hochschild-serre spectral sequence. Spectral sequences are among the most elegant, most powerful, and most complicated methods of computation in mathematics. We establish various properties of the definition of cohomology of topological groups given by grothendieck, artin and verdier in sga4, including a hochschildserre spectral sequence and a. Pbw basis for ug, the universal enveloping algebra of a lie algebra g. Product in cohomology, and the hochschild-serre spectral sequence. Permutations and the lie algebra of sln dihedral groups and automorphisms, part 1; dihedral groups and automorphisms, part 2. Is motivated by the leray-serre spectral sequence of a map p: e. The bockstein spectral sequence arose in the study of lie groups. In chapter 6 we define the enveloping algebra uk of a lie superalgebra k. Cohomology of commutative lie algebras in characteristic 2. First we introduce the hochschild-serre spectral sequence for g,h and v. 344 Using maple and the serre hochschild spectral sequence. Hochschild cohomology versus lie algebra cohomology. It has been known as the lyndon-hochschild-serre spectral sequence. Homological algebra from weibels book: abelian categories, derived functors, ext and tor, grothendieck spectral sequence time permitting: leray-serre. In a first part, we focus on a hochschild-serre-type spectral sequence. 758 representations of lie groups almost-a-web-page. The gysin sequence and the cohomology ring h cpn;r 6 references 8 1. More speci?Cally, suppose one has a central extension 1.
Analysis, representations of lie groups and lie algebra, linear algebraic. 244 After a lecture series by steenrod in may 151, serre. Homological algebra from weibels book: abelian categories, derived functors, ext and tor, grothendieck spectral sequence time permitting: leray-serre and hochschild-serre spectral sequences. Ements in the lie algebra homology of the infinite lie algebra g?A. And an argument using the hochschild-serre spectral sequence, given in bott 3. Notes on lie algebra cohomology dvi ps as i was writing this, i gradually realized that almost everything i wanted to include was already in knapps book referenced in the notes. Hochschild-serre spectral sequences of locally compact totally disconnected. 1 ! Rs1/rs! R/rs! R/rs1! 1, where we have used the. We list explicit generators for the lie algebra cohomology ring. This spectral sequence generalizes the hochschildserre spectral sequence for lie algebras to the holomorphic lie algebroid setting. The other super-analog of the heisenberg lie algebra, n, is constructed by means of a. The lie algebra struc-ture can be used to explicitly compute the d 2 homology di?Erential in the lyndon-hochschild-serre spectral sequence for certain central extensions involving quotients of p-congruence subgroups.