Generalized Tate cohomology.
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 maps on the other two cohomology theories), then there is no longer any need to prove that it is an isomorphism -- this would immediately follow from the 5-lemma. In the last section, the author proves dual results (replacing Gorenstein projectives by Gorenstein injectives). In particular, she establishes the balance property of Tate cohomology for arbitrary modules over an arbitrary Gorenstein ring.
scientific article; zbMATH DE number 5017532