The generating hypothesis for the stable module category of a \(p\)-group.
type: Item
instance of: scholarly article
title: The generating hypothesis for the stable module category of a \(p\)-group.
zbMATH Open document ID: 1120.20002
author: Sunil K. Chebolu
author: J. Daniel Christensen
author: David Benson
author: Ján Mináč
published in: Journal of Algebra
publication date: 2007-04-18T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0611403
review text: Let \(k\) be a field of characteristic \(p>0\), and let \(G\) be a nontrivial finite \(p\)-group. For finite-dimensional \(kG\)-modules \(M,N\), a map \(\varphi\in\Hom_{kG}(M,N)\) is called a ghost if the induced map in Tate cohomology \(\widehat H^i(G,M)\to\widehat H^i(G,N)\) is trivial for each \(i\). A ghost \(\varphi\) is called trivial if \(\varphi\) factors through a projective \(kG\)-module. The authors prove that all ghosts for \(G\) are trivial if and only if \(G\) is cyclic of order 2 or 3. This result is related to Freyd's generating hypothesis in stable homotopy theory.
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 16D90
Mathematics Subject Classification ID: 20J05
Mathematics Subject Classification ID: 55P42
Mathematics Subject Classification ID: 18E30
zbMATH DE Number: 5144401
zbMATH Keywords: stable module categories
zbMATH Keywords: generating hypothesis
zbMATH Keywords: Tate cohomology
zbMATH Keywords: ghost maps
zbMATH Keywords: finite \(p\)-groups
reviewed by: Burkhard Külshammer
MaRDI profile type: Publication
OpenAlex ID: W1967596497
arXiv ID: math/0611403
cites work: Products in negative cohomology
cites work: Groups which do not admit ghosts
cites work: Refinements for infinite direct decompositions of algebraic systems
cites work: K-Theory and the Generating Hypothesis
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cites work: Q3035669
cites work: The generating hypothesis in the derived category of \(R\)-modules
cites work: A Krull-Schmidt Theorem for Infinite Sums of Modules
DOI: 10.1016/J.JALGEBRA.2006.12.013
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description: scientific article