Homology theories and Gorenstein dimensions for complexes
type: Item
instance of: scholarly article
title: Homology theories and Gorenstein dimensions for complexes
zbMATH Open document ID: 1505.16006
author: Yanyan Li
published in: Algebras and Representation Theory
publication date: 2021-12-08T00:00:00Z
full work available at URL: https://arxiv.org/abs/1808.07685
review text: In this paper, the author establishes various results about Tate homology \(\widehat{\mathrm{Tor}}\), stable homology \(\widetilde{\mathrm{Tor}}\), and so-called unbounded homology \(\overline{\mathrm{Tor}}\) (which can all be seen as variants of the usual \(\mathrm{Tor}\) functor) for certain types of complexes of modules over a ring. For example, for suitably nice complexes \(M\) and \(N\) it is shown that the Tate homology \(\widehat{\mathrm{Tor}}_*(M,N)\) can be computed from Tate projective/flat resolutions and that the unbounded homology is balanced, i.e.~one has \(\overline{\mathrm{Tor}}_*(M,N) \cong \overline{\mathrm{Tor}}_*(N,M)\). The paper also contains some results that relate the Gorenstein projective/flat dimension of a complex to the Gorenstein projective/flat dimensions of the modules it consists of.
reviewed by: Henrik Holm
Mathematics Subject Classification ID: 16E05
Mathematics Subject Classification ID: 16E30
Mathematics Subject Classification ID: 16E10
zbMATH DE Number: 7440310
zbMATH Keywords: Tate homology
zbMATH Keywords: unbounded homology
zbMATH Keywords: stable homology
zbMATH Keywords: Gorenstein projective dimension
zbMATH Keywords: Gorenstein flat dimension
MaRDI profile type: Publication
OpenAlex ID: W3088112666
arXiv ID: 1808.07685
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DOI: 10.1007/S10468-020-09999-1
description: scientific article