Journal ArticleDOI
On the Cohomology and K-Theory of the General Linear Groups Over a Finite Field
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This article is published in Annals of Mathematics.The article was published on 1972-11-01. It has received 546 citations till now. The article focuses on the topics: Group cohomology & Equivariant cohomology.read more
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Cohomology of Groups
TL;DR: In this paper, an advanced textbook introduces students to cohomology theory and no knowledge of homological algebra is assumed beyond what is normally taught in a first course in algebraic topology.
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E∞ ring spaces and E∞ ring spectra
TL;DR: In this article, the recognition principle for E? ring spaces was established for algebraic and topological K-theory, and pairings in infinite loop space theory were studied.
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Cohomology of finite group schemes over a field
Abstract: A finite group scheme G over a field k is equivalent to its coordinate algebra, a finite dimensional commutative Hopf algebra k[G] over k. In many contexts, it is natural to consider the rational (or Hochschild) cohomology of G with coefficients in a k[G]-comodule M . This is naturally isomorphic to the cohomology of the dual cocommutative Hopf algebra k[G] with coefficients in the k[G]-module M . In this latter formulation, we encounter familiar examples of the cohomology of group algebras kπ of a finite groups π and of restricted enveloping algebras V (g) of finite dimensional restricted Lie algebras g. In recent years, the representation theory of the algebras kπ and V (g) has been studied by considering the spectrum of the cohomology algebra with coefficients in the ground field k and the support in this spectrum of the cohomology with coefficients in various modules. This approach relies on the fact that H(π, k) and H(V (g), k) are finitely generated k-algebras as proved in [G], [E], [V], [FP2]. Rational representations of algebraic groups in positive characteristic correspond to representations of a hierarchy of finite group schemes. In order to begin the process of introducing geometric methods to the study of these other group schemes, finite generation must be proved. Such a proof has proved surprisingly elusive (though partial results can be found in [FP2]). The main theorem of this paper is the following:
References
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Journal ArticleDOI
The characters of the finite general linear groups
TL;DR: In this paper, the irreducible characters of the group GL(n, q) of all nonsingular matrices of degree n with coefficients in the finite field of q elements were calculated.
Book
Algebraic K-theory
TL;DR: The relation between algebraic and topological K theory has been investigated in this paper, where the relation between GO(R) and GO(RS) has also been studied in the context of topological k theory.
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The Mod 2 Cohomology Rings of Extra-special 2-groups and the Spinor Groups.
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The adams conjecture
TL;DR: In this paper, the Adams conjecture for real vector bundles has been shown to hold for vector bundles in the etale topology of algebraic varieties, and the proof of the Brauer induction theorem is given.