Gorenstein derived categories
type: Item
instance of: scholarly article
title: Gorenstein derived categories
zbMATH Open document ID: 1222.18005
author: Nan Gao
author: Pu Zhang
published in: Journal of Algebra
publication date: 2010-04-21T00:00:00Z
review text: Let \(\mathcal A\) be an abelian category with enough projective objects. An object \(G\) of \(\mathcal A\) is Gorenstein-projective if there is an exact sequence \(\dots \rightarrow P^1\rightarrow P^0\rightarrow P_0\rightarrow P_1\rightarrow\dots\) of projective objects of \(\mathcal A\) which stays exact under Hom\(_{\mathcal A}(-,P)\) for each projective objects \(P\) of \(\mathcal A\) and so that \(G\) is isomorphic to the image of \(P^0\rightarrow P_0\). The full subcategory of \(\mathcal A\) of Gorenstein-projective objects is denoted by \({\mathcal G}P\). The authors define the derived category \(D^*( {\mathcal G}P)\) in an appropriate way and study its properties, as well as related constructions such as homotopy category constructions. In particular a Morita theorem in the sense of Rickard is proven for equivalences between \(D^*( {\mathcal G}P)\) for two abelian categories \({\mathcal A}=A-{\mathcal M}od\) and \({\mathcal B}=B-{\mathcal M}od\). The relation to the usual derived category \(D^*({\mathcal A})\) is studied systematically.
reviewed by: Alexander Zimmermann
Mathematics Subject Classification ID: 18E30
zbMATH DE Number: 5696937
zbMATH Keywords: Gorenstein derived category
zbMATH Keywords: Gorenstein derived equivalences
zbMATH Keywords: relative homological algebra
zbMATH Keywords: Morita theorem
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2010.01.027
OpenAlex ID: W2043742780
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DOI: 10.1016/J.JALGEBRA.2010.01.027
description: scientific article; zbMATH DE number 5696937