Static modules and Clifford theories
type: Item
instance of: scholarly article
title: Static modules and Clifford theories
zbMATH Open document ID: 0883.16004
published in: Journal of Algebra
publication date: 1998-01-25T00:00:00Z
review text: Let \(A\) be a ring and denote by Mod-\(A\) the category of right \(A\)-modules. Assume that \(M\) is a self-small \(A\)-module and let \(D=\text{End}_AM\). Then the category of \(A\)-modules which divide a direct sum of copies of \(M\) is equivalent to the category of projective \(D\)-modules. Let further \(\rho\colon A\to B\) be a ring homomorphism, \(K=M\otimes_AB\), \(E=\text{End}_B(K)\) and \(F=\text{End}_A(K)\). Denote by \(\text{Mod}(B|A\text{-split})\) the category of \(B\)-modules having a \(K\)-presentation which splits upon restriction to \(A\), and by \(\text{Mod}(E|\text{Res}_F)\) the category of \(E\)-modules \(W\) for which \(W\otimes_EF\) is a projective \(F\)-module. By a result of \textit{J. Alperin} [J. Aust. Math. Soc., Ser. A 49, No. 3, 347-353 (1990; Zbl 0739.16008)], the restrictions of the functors \(\text{Hom}_B(K,-)\) and \(-\otimes_E K\) determine an equivalence between these categories. The authors prove that the above mentioned theorems are in fact equivalent, since it turns out that \(\text{Mod}(B|A\text{-split})\) coincides with the category of \(B\)-modules which divide a direct sum of copies of \(K\), while \(\text{Mod}(E|\text{Res}_F)\) coincides with the category of projective \(E\)-modules.
Mathematics Subject Classification ID: 16D90
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 16S50
zbMATH DE Number: 1051582
zbMATH Keywords: static modules
zbMATH Keywords: endomorphism rings
zbMATH Keywords: Auslander equivalences
zbMATH Keywords: non-normal Clifford theories
zbMATH Keywords: categories of modules
zbMATH Keywords: equivalences of categories
zbMATH Keywords: projective modules
author: H. A. S. Abujabal
author: Syed Khalid Nauman
reviewed by: Andrei Marcus
MaRDI profile type: Publication
full work available at URL: jabr.1996.6961
OpenAlex ID: W2071503051
cites work: Q3971585
cites work: Modules with decompositions that complement direct summands
cites work: Representation Theory of Artin Algebras I
cites work: Stable Clifford theory
cites work: Group-graded rings and modules
cites work: Q4293469
cites work: Static modules and stable Clifford theory
cites work: Intersecting subcategories of static modules, stable Clifford theory and colocalization-localization
DOI: 10.1006/JABR.1996.6961
Recommended article: Q4865866
Recommended article: Some remarks on equivalences between categories of modules
Recommended article: LLI rings and modules of type LP
Recommended article: Q3971585
Recommended article: Some equivalences and dualities via static modules.
Recommended article: Q3531568
Recommended article: A Dundas-Goodwillie-McCarthy theorem for split square-zero extensions of exact categories
Recommended article: Q3110663
Recommended article: On the existence of left and right almost split morphisms.
Recommended article: Separable functors applied to graded rings
description: scientific article; zbMATH DE number 1051582