Donovan's conjecture, crossed products and algebraic group actions
type: Item
instance of: scholarly article
title: Donovan's conjecture, crossed products and algebraic group actions
zbMATH Open document ID: 0842.20011
DOI: 10.1007/BF02762084
published in: Israel Journal of Mathematics
publication date: 1996-02-13T00:00:00Z
review text: Donovan's conjecture is that for a given finite \(p\)-group \(D\), there are only finitely many Morita classes of \(p\)-blocks of group algebras of finite groups with defect group \(D\). Here, the group algebras are to be taken over an algebraically closed field of characteristic \(p\). The author reduces this conjecture to the case where the ambient finite group is generated by conjugates of \(D\). The proof uses Dade's theory of block extensions. Some of the steps of the proof are of independent interest. To wit, it is shown that if \(G\) is a finite group and \(C\) is a commutative \(G\)-algebra over an algebraically closed field \(F\) whose characteristic does not divide \(|G|\), then the cohomology groups \(H^i(G,U(C))\) are finite, where \(i\geq 1\) and \(U(C)\) is the group of units of \(C\). Also, if \(H\) is any linear algebraic group over \(F\), then there are only finitely many homomorphisms \(G\to H\), up to \(H\)-conjugacy.
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 20J06
Mathematics Subject Classification ID: 20G99
zbMATH DE Number: 836169
zbMATH Keywords: Donovan's conjecture
zbMATH Keywords: finite \(p\)-groups
zbMATH Keywords: Morita classes of \(p\)-blocks
zbMATH Keywords: group algebras of finite groups
zbMATH Keywords: defect groups
zbMATH Keywords: block extensions
zbMATH Keywords: cohomology groups
zbMATH Keywords: group of units
zbMATH Keywords: linear algebraic groups
Wikidata QID: Q123118198
author: Burkhard Külshammer
reviewed by: William H. Gustafson
MaRDI profile type: Publication
cites work: Q4724810
cites work: Blocks with cyclic defect groups
cites work: Q4050769
cites work: Block extensions
cites work: Blocks of tame representation type and related algebras
cites work: Indecomposable modules for finite groups
cites work: Q4722208
cites work: On p-blocks of p-solvable groups
cites work: Crossed products and blocks with normal defect groups
cites work: Q5203223
cites work: Q3957087
cites work: Cartan matrices and Morita equivalence for blocks of the symmetric groups
cites work: Remarks on the cohomology of groups
description: scientific article; zbMATH DE number 836169