Generalized Tate cohomology.
type: Item
instance of: scholarly article
title: Generalized Tate cohomology.
zbMATH Open document ID: 1095.16003
published in: Tsukuba Journal of Mathematics
publication date: 2006-04-06T00:00:00Z
review text: Let \(R\) be an associative ring with identity. Given two classes of (left) \(R\)-modules, one contained in the other, we can construct the corresponding relative homology theories by viewing the two classes as relative projectives. If an \(R\)-module \(M\) has a relative projective resolution in each theory, then the identity map on \(M\) admits a lifting to the resolutions (in the direction from the smaller to the larger class). The standard cone-cylinder diagram gives rise to an exact sequence of complexes, going from the target resolution, via the mapping cone of the comparison, to the source resolution shifted by 1. Choosing an arbitrary coefficient module \(N\) and applying the functor \(\Hom(-,N)\), one obtains a long cohomology exact sequence. It is of interest because of the cohomology of the middle complex. The author calls it ``generalized Tate cohomology''. The motivation comes from one of the results in the paper which shows that, when the smaller class consists of absolute projectives, the larger class consists of Gorenstein projectives, and if \(M\) is of finite Gorenstein projective dimension, then this cohomology coincides with Tate cohomology in positive degrees. (In the opinion of the reviewer, such a name is not justified, as the new cohomology is only defined in nonnegative degrees, and thus misses the entire nonpositive part of Tate cohomology.) Under these assumptions, the long cohomology exact sequence becomes of the same type as the sequence obtained earlier by Avramov-Martsinkovsky, in the sense that it connects the same cohomology groups in the same order. In fact, the author claims that the two sequences are the same, but, in the paper, the reviewer could not find any proof of the functoriality (with respect to the maps in the sequences) for the cohomology theories in question. On the other hand, if the constructed map between generalized Tate cohomology and Tate cohomology is indeed a map of sequences (say, with respect to the identity m…
reviewed by: Alex Martsinkovsky
Mathematics Subject Classification ID: 16E30
Mathematics Subject Classification ID: 18G25
Mathematics Subject Classification ID: 16E05
zbMATH DE Number: 5017532
zbMATH Keywords: Avramov-Martsinkovsky exact sequences
zbMATH Keywords: complete injective resolutions
zbMATH Keywords: relative homology theories
zbMATH Keywords: relative projective resolutions
zbMATH Keywords: exact sequences of complexes
zbMATH Keywords: long cohomology exact sequences
zbMATH Keywords: generalized Tate cohomology
zbMATH Keywords: Gorenstein projectives
zbMATH Keywords: Gorenstein projective dimension
author: A. Iacob
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.21099/tkbjm/1496164963
OpenAlex ID: W272491648
DOI: 10.21099/TKBJM/1496164963
description: scientific article; zbMATH DE number 5017532