Relative and Tate homology with respect to semidualizing modules (Q2878793)

Let \(R\) be a commutative noetherian ring (slightly weaker assumptions on \(R\) will also do). For every finitely generated \(R\)-module \(M\) of finite \(G\)-dimension and every \(R\)-module \(N\), Avramov and Martsinkovsky define and study in [\textit{L. L. Avramov} and \textit{A. Martsinkovsky}, Proc. Lond. Math. Soc. (3) 85, No. 2, 393--440 (2002; Zbl 1047.16002)] the Tate cohomology groups \(\widehat{\mathrm{Ext}}_R^*(M,N)\). One of their main results is the existence of a long exact sequence, NEWLINE\[NEWLINE 0 \to \mathrm{Ext}_{\mathcal{G}}^1(M,N) \to \mathrm{Ext}_R^1(M,N) \to \widehat{\mathrm{Ext}}_R^1(M,N) \to \mathrm{Ext}_{\mathcal{G}}^2(M,N) \to \cdots, NEWLINE\]NEWLINE where \(\mathrm{Ext}_{\mathcal{G}}^*(M,N)\) are the relative cohomology groups, that is, the right derived functors of \(\mathrm{Hom}_R(-,N)\) relative to the class \(\mathcal{G}\) of totally reflexive modules (= modules of G-dimension zero = finitely generated Gorenstein projective modules).NEWLINENEWLINEThe paper under review investigates the ``homology version'' of the work of [loc. cit.]. In fact, an entire family of such ``homology versions'' (parameterized by the set of semidualizing \(R\)-modules) is studied. The theory is nicely behaved, as expected.