Preservation of quasi-isomorphisms of complexes
type: Item
instance of: scholarly article
title: Preservation of quasi-isomorphisms of complexes
zbMATH Open document ID: 1260.13023
publication date: 2013-03-12T00:00:00Z
review text: Let \(A\) be a commutative ring, \(X \rightarrow Y\) a quasi-isomorphism of (unbounded) complexes of \(A\)-modules, and \(Z\) another complex. The author gives some conditions implying that \(\Hom(Z,X) \rightarrow \Hom(Z,Y)\), \(\Hom(Y,Z) \rightarrow \Hom(X,Z)\) or \(Z\otimes X \rightarrow Z \otimes Y\) are quasi-isomorphisms, improving some results existing in literature. For instance, the following theorem is proved: let \(C_\lrcorner (A)\) (resp. \(C_\llcorner (A)\)) be the category of complexes of \(A\)-modules \((T_i,d_i)\) (with differential of degree \(-1\)) such that \(d_i=0\) for an infinite decreasing (resp. increasing) sequence of integers \(i\). If \(Z\in C_\lrcorner (A)\), \(X,Y\in C_\llcorner (A)\) and \(\mathrm{Ext}_A^{\geq 1}(Z_j,X_k)=0=\mathrm{Ext}_A^{\geq 1}(Z_j,Y_k)\) for all \(j,k\), then any quasi-isomorphism \(X \rightarrow Y\) induces a quasi-isomorphism \(\Hom(Z,X) \rightarrow \Hom(Z,Y)\). Some applications are given.
Mathematics Subject Classification ID: 13D09
Mathematics Subject Classification ID: 18G10
zbMATH DE Number: 6143608
zbMATH Keywords: quasi-isomorphism
zbMATH Keywords: derived functor
zbMATH Keywords: Gorenstein homological dimension
zbMATH Keywords: hom evaluation morphism
zbMATH Keywords: unbounded complexes
author: Zhongkui Liu
reviewed by: Javier Majadas
MaRDI profile type: Publication
full work available at URL: s10114-012-0550-0
OpenAlex ID: W2125908648
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DOI: 10.1007/S10114-012-0550-0
published in: Acta Mathematica Sinica, English Series
description: scientific article; zbMATH DE number 6143608