Higman's criterion revisited.
type: Item
instance of: scholarly article
title: Higman's criterion revisited.
zbMATH Open document ID: 1201.16014
author: Michel Broué
published in: Michigan Mathematical Journal
publication date: 2009-06-17T00:00:00Z
review text: The paper under review gives a self-contained introduction into certain aspects of the representation theory of symmetric algebras over a commutative ring \(R\). Recall that an associative \(R\)-algebra \(A\) is called symmetric if the \(A\)-\(A\)-bimodule \(A^*=\Hom_R(A,R)\) is isomorphic to \(A\). Important examples are the group algebras of finite groups and the Iwahori-Hecke algebras of finite Coxeter groups. In the representation theory of finite groups, Higman's criterion gives a useful characterization of projective and, more generally, relatively projective modules with respect to subgroups, in terms of a (relative) trace map. In the paper, this criterion is generalized to the setting of \(R\)-linear triangulated or Abelian categories \(\mathfrak A\) and \(\mathfrak B\). Suppose that \(M\colon\mathfrak B\to\mathfrak A\) and \(N\colon\mathfrak A\to\mathfrak B\) are exact functors forming a bi-adjoint pair. Then it is shown that an object \(X\) in \(\mathfrak A\) is isomorphic to a direct summand of \(MN(X)\) if and only if the identity morphism \(\text{id}_X\) is contained in the image of \(\text{Tr}^N_M(X)\). Here \(\text{Tr}^M_N(X)\colon\text{End}_{\mathfrak B}(NX)\to\text{End}_{\mathfrak A}(X)\) is a relative trace map defined in terms of the units and counits of the relevant adjunctions. The author illustrates his result by several applications, having to do with Casimir and Schur elements, with parabolic subalgebras and Heller operators, with blocks of finite groups and with stable equivalences of Morita type. Generalizations of Schanuel's Lemma and Shapiro's Lemma are also presented.
reviewed by: Burkhard Külshammer
Mathematics Subject Classification ID: 16G10
Mathematics Subject Classification ID: 18E30
Mathematics Subject Classification ID: 16L60
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 20C08
Mathematics Subject Classification ID: 16D90
zbMATH DE Number: 5566076
zbMATH Keywords: symmetric algebras
zbMATH Keywords: Abelian categories
zbMATH Keywords: triangulated categories
zbMATH Keywords: Higman criterion
zbMATH Keywords: relative projectivity
zbMATH Keywords: Harish-Chandra induction
zbMATH Keywords: Heller translate
zbMATH Keywords: Casimir elements
zbMATH Keywords: Schur elements
zbMATH Keywords: stable equivalences
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1242071686
OpenAlex ID: W1964554524
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DOI: 10.1307/MMJ/1242071686
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description: scientific article; zbMATH DE number 5566076