On equivalences between blocks of group algebras: Reduction to the simple components
type: Item
instance of: scholarly article
title: On equivalences between blocks of group algebras: Reduction to the simple components
zbMATH Open document ID: 0866.20007
author: Andrei Marcus
published in: Journal of Algebra
publication date: 1997-07-13T00:00:00Z
review text: Let \(p\) be a prime and let \(\mathcal O\) be a complete discrete valuation ring with quotient field \(K\) of characteristic zero and with residue field \(k={\mathcal O}/J({\mathcal O})\) of characteristic \(p\). All \(\mathcal O\)-algebras and \(\mathcal O\)-algebra modules will be assumed to be unitary, finitely generated and \(\mathcal O\)-free. Let \(G\) be a finite group and let \(R=\bigoplus_{g\in G}R_g\) and \(S=\bigoplus_{g\in G}S_g\) be \(G\)-crossed product \(\mathcal O\)-algebras. The opposite ring \(S^{\text{op}}\) is a \(G^{\text{op}}\)-crossed product \(\mathcal O\)-algebra and \(A=R\otimes_{\mathcal O}S^{\text{op}}\) is a \(G\times G^{\text{op}}\)-crossed product \(\mathcal O\)-algebra. Let \(\delta:G\to G\times G^{\text{op}}\) be defined by: \(g\mapsto(g,g^{-1})\) for all \(g\in G\). Then \(\Delta\equiv A_{\delta(G)}=\bigoplus_{g\in G}(R_g\otimes_{\mathcal O}S^{\text{op}}_{g^{-1}})\) is a \(\delta(G)\)-crossed product \(\mathcal O\)-subalgebra of \(A\). One of the main points of this paper is that there are three naturally isomorphic \(\mathcal O\)-linear functors \(A\otimes_\Delta-\), \(R\otimes_{R_1}-\), \(-\otimes_{S_1}S:\Delta\text{-mod}\to A\text{-mod}\) (Section 3). Let \(M\) be an \((R_1,S_1)\)-bimodule. Then its \(\mathcal O\)-dual \(M^*=\text{Hom}_{\mathcal O}(M,\mathcal O)\) is an \((S_1,R_1)\)-bimodule. One of the main results of this paper is: Theorem 3.4. (a) Assume that \((M,M^*)\) defines a Morita equivalence between \(R_1\) and \(S_1\) and that \(M\) is \(\delta(G)\)-invariant and extends to a \(\Delta\)-module. Then \(R\otimes_{R_1}M\) (resp. \(S\otimes_{S_1}M^*\)) is a naturally \(G\)-graded \((R,S)\)-bimodule (resp. \((S,R)\)-bimodule) and \((R\otimes_{R_1}M\), \(S\otimes_{S_1}M^*)\) defines a Morita equivalence between \(R\) and \(S\). (b) Assume that \(G\) is a \(p'\)-group and that \((C,C^*)\) defines a derived equivalence between \(R_1\) and \(S_1\). If \((C,d)\) extends to a complex of \(\Delta\)-modules, then \((R\otimes_{R_…
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 16D90
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 16S35
zbMATH DE Number: 930147
zbMATH Keywords: complete discrete valuation rings
zbMATH Keywords: finite groups
zbMATH Keywords: crossed product algebras
zbMATH Keywords: Morita equivalences
zbMATH Keywords: derived equivalences
zbMATH Keywords: block idempotents
zbMATH Keywords: Puig equivalences
zbMATH Keywords: Rickard tilting complexes
zbMATH Keywords: groups with cyclic Sylow \(p\)-subgroups
reviewed by: Morton E. Harris
MaRDI profile type: Publication
full work available at URL: jabr.1996.0265
OpenAlex ID: W1986863393
DOI: 10.1006/JABR.1996.0265
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description: scientific article; zbMATH DE number 930147