Endomorphism rings and category equivalences
type: Item
instance of: scholarly article
title: Endomorphism rings and category equivalences
zbMATH Open document ID: 0685.16023
DOI: 10.1016/0021-8693(89)90283-4
author: Yanyan Li
published in: Journal of Algebra
publication date: 1989-01-01T00:00:00Z
review text: If \({\mathcal C}\) is a locally finitely generated Grothendieck category and \(M\in Ob {\mathcal C}\), \(S=End_{{\mathcal C}}(M)\), the class of the M- distinguished objects of \({\mathcal C}\) is the torsion free class of a torsion theory (T,F) of \({\mathcal C}\) and the functor \(Hom_{{\mathcal C}}(M\),- ): \({\mathcal C}\to S\)-mod induces an equivalence between the quotient cateogory of \({\mathcal C} modulo T\) and a certain quotient category (S,\({\mathcal F})\)-mod of S-mod. This quotient category consists of all the S-modules iff M is a finitely generated quasiprojective object of \({\mathcal C}\) which is CQF-3. The properties of this construction are studied. If M is codivisible, then three different full subcategories of \({\mathcal C}\) exist and they are equivalent to (S,\({\mathcal F})\)-mod. These results are applied to characterize the modules M such that the endomorphism ring S of M is a left semihereditary ring, a left CS-ring or a left continuous ring.
Mathematics Subject Classification ID: 16D90
Mathematics Subject Classification ID: 16S50
Mathematics Subject Classification ID: 18E15
Mathematics Subject Classification ID: 18E40
zbMATH DE Number: 4122169
zbMATH Keywords: equivalence of module categories
zbMATH Keywords: locally finitely generated Grothendieck category
zbMATH Keywords: torsion free class
zbMATH Keywords: torsion theory
zbMATH Keywords: quotient cateogory
zbMATH Keywords: full subcategories
zbMATH Keywords: endomorphism ring
zbMATH Keywords: left semihereditary ring
zbMATH Keywords: left CS-ring
zbMATH Keywords: left continuous ring
reviewed by: Ketty Peeva
MaRDI profile type: Publication
full work available at URL: 0021-8693(89)90283-4
OpenAlex ID: W1963521603
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description: scientific article; zbMATH DE number 4122169