Categorical equivalences and realization theorems
type: Item
instance of: scholarly article
title: Categorical equivalences and realization theorems
zbMATH Open document ID: 0862.16002
DOI: 10.1016/0022-4049(96)00091-6
published in: Journal of Pure and Applied Algebra
publication date: 1997-04-07T00:00:00Z
review text: The main result of this paper is the following: Let \({\mathcal C}\) be a Grothendieck category having a set \((U_i)_{i\in I}\) of small projective generators, let \(A^0\) be a certain unital subring of the ring of endomorphisms of \(\prod_{i\in I} U_i\), and \(I\) a certain idempotent left ideal of \(A^0\). Then, for any subring \(A\) (not necessarily unital) of \(A^0\) containing \(I\), the category \({\mathcal C}\) is equivalent to the category \(\text{Gen}(_AI)\) of all unital left \(A\)-modules generated by \(I\). This theorem allows to provide a unified approach to various results concerning realization theorems of categories of graded objects as module categories, obtained by Cohen \& Montgomery, Beattie, Albu \& Năstăsescu, Albu and del Rio. The authors apply the theorem in the next section to the so-called ``rings graded by categories'', which include group-graded rings.
Mathematics Subject Classification ID: 16D90
Mathematics Subject Classification ID: 18E15
Mathematics Subject Classification ID: 16W50
zbMATH DE Number: 952351
zbMATH Keywords: rings graded by categories
zbMATH Keywords: Grothendieck categories
zbMATH Keywords: small projective generators
zbMATH Keywords: rings of endomorphisms
zbMATH Keywords: idempotent left ideals
zbMATH Keywords: realization theorems
zbMATH Keywords: categories of graded objects
zbMATH Keywords: module categories
zbMATH Keywords: group-graded rings
author: Gene Abrams
author: Claudia Menini
reviewed by: Toma Albu
MaRDI profile type: Publication
full work available at URL: 0022-4049(96)00091-6
OpenAlex ID: W2157908886
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description: scientific article; zbMATH DE number 952351