Quillen stratification for block varieties.
type: Item
instance of: scholarly article
title: Quillen stratification for block varieties.
zbMATH Open document ID: 1062.20009
DOI: 10.1016/S0022-4049(01)00143-8
author: Markus Linckelmann
published in: Journal of Pure and Applied Algebra
publication date: 2002-08-22T00:00:00Z
review text: Let \(G\) be a finite group and let \(k\) be an algebraically closed field. By a classical result of Quillen the maximal ideal spectrum of \(H^*(G, k)\) -- which is an affine variety -- has a stratification labeled by the conjugacy classes of nontrivial elementary Abelian \(p\)-subgroups of \(G\). Later Carlson defined the cohomology variety \(V_G(M)\) of a finitely generated \(kG\)-module \(M\). Avrunin and Scott proved that \(V_G(M)\) has a stratification similar to the one in Quillen's theorem. Now suppose that \(M\) belongs to a \(p\)-block \(bkG\) where \(b\) is a primitive central idempotent in \(kG\). If \(b\) does not define the principal block then \(V_G(M)\) is, in general, not an invariant of \(M\) regarded as a \(bkG\)-module. Motivated by this lack the author defined [in J. Algebra 215, No. 2, 460-480 (1999; Zbl 0960.20008)] a variety \(V_G(M)\) related to the block cohomology \(H^*(G,b)\). This variety allows a surjective morphism onto \(V_G(M)\) which in case of the principal block turns out to be an isomorphism. The paper under review contains essentially two main theorems. In the first one the author describes a technique to compute \(V_{G,b}(M)\) via the truncation \(kP\)-module \(iM\) where \(i\) is a source idempotent and \(P\) a defect group of \(b\). In the other theorem he constructs a stratification of \(V_{G,b}(M)\) which in case of the principal block coincides with the Quillen stratification due to Avrunin and Scott. An illuminating example at the end of the paper clarifies the fact that one really has to work with the truncation \(iM\) and not only with the restriction \(M|_P\) of \(M\) to the defect group \(P\).
reviewed by: Wolfgang Willems
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20J06
Mathematics Subject Classification ID: 20C05
zbMATH DE Number: 1787474
zbMATH Keywords: block varieties
zbMATH Keywords: stratifications
zbMATH Keywords: cohomology varieties
zbMATH Keywords: finite groups
zbMATH Keywords: finitely-generated modules
zbMATH Keywords: block idempotents
zbMATH Keywords: central idempotents
zbMATH Keywords: defect groups
MaRDI profile type: Publication
cites work: Local methods in block theory
cites work: Representations, resolutions and Quillen's dimension theorem
cites work: Quillen stratification for modules
cites work: Q3999421
cites work: Q3997902
cites work: The homotopy theory of fusion systems
cites work: Characters and local structure in G-algebras
cites work: A Frobenius theorem for blocks
cites work: Q3950716
cites work: The varieties and the cohomology ring of a module
cites work: Q3240881
cites work: The Cohomology Ring of a Finite Group
cites work: The source algebras of blocks with a Klein four defect group
cites work: Q4208779
cites work: Transfer in Hochschild cohomology of blocks of finite groups
cites work: Varieties in block theory
cites work: Sur un théorème de Green
cites work: Pointed groups and construction of characters
cites work: The spectrum of an equivariant cohomology ring. I. II
cites work: Q4846455
cites work: Q3280631
full work available at URL: s0022-4049(01)00143-8
OpenAlex ID: W1991672270
description: scientific article; zbMATH DE number 1787474