A study of Tate homology via the approximation theory with applications to the depth formula (Q6043143)

Let \(R\) denote a commutative noetherian local ring and all \(R\)-modules be finitely generated. \textit{L. L. Avramov} and \textit{A. Martsinkovsky} [Proc. Lond. Math. Soc. (3) 85, No. 2, 393--440 (2002; Zbl 1047.16002)] show the following fact ``Let \(M\) and \(N\) be \(R\)-modules. Assume \(M\) has finite Gorenstein dimension \(n\) for some positive integer \(n\). Then there exists an exact sequence of the form: \[ 0 \rightarrow \widehat{\mathrm{Tor}}^{R}_{n}(M,N) \rightarrow \mathrm{Tor}_{n}(M,N) \rightarrow \mathcal{G}\mathrm{Tor}_{n}(M,N) \rightarrow \widehat{\mathrm{Tor}}^{R}_{n-1}(M,N)\rightarrow \cdots.'' \] The classical Tate Tor modules \(\widehat{\mathrm{Tor}}^{R}_{*}(M,N)\) are computed by using a complete resolution of \(M\), i.e., by using a diagram \(T \xrightarrow{\tau} P \rightarrow M\), where \(P \rightarrow M\) is a projective resolution of \(M\), \(T\) is a totally acyclic complex of free \(R\)-modules, and \(\tau_i\) is an isomorphism for all \(i \gg 0\). The other notation \(\mathcal{G}\mathrm{Tor}_{n}(M,N)\) as well as the following generalized notation are defined in section \(2\) of the paper in items 2.1-2.13. \begin{itemize} \item \(\widehat{\mathrm{Tor}}^{\mathcal{G}_{C}}_{*}(M,N)\), \item \(\mathcal{G_{C}}\mathrm{Tor}_{n}(M,N)\), \item \(\mathcal{G}_{C}\)-dim, Auslander class \(\mathcal{A}_{C}\) and \item \(\mathcal{I}_{C}\)-id\(_{R}(N)\). \end{itemize} The first main result of the paper is Theorem 1.2. ``Let \(M\) and \(N\) be \(R\)-modules, and let \(C\) be a semidualizing \(R\)-module. Assume \(M\) has finite \(\mathcal{G_{C}}\)-dimension \(n\) for some positive integer \(n\). Assume further \(N\) belongs to the Auslander class \(\mathcal{A}_{C}\). Then there is an exact sequence of the form: \[ 0 \rightarrow \widehat{\mathrm{Tor}}^{\mathcal{G}_{C}}_{n}(M,N) \rightarrow \mathrm{Tor}_{n}(M,N) \rightarrow \mathcal{G}_{C}\mathrm{Tor}_{n}(M,N) \rightarrow \widehat{\mathrm{Tor}}^{\mathcal{G}_{C}}_{n-1}(M,N)\rightarrow \cdots.'' \] In the second main result Theorem 1.3, the authors show that the Auslander depth formula holds in a wider category of \(R\)-modules, more precisely ``Let \(M\) and \(N\) be \(R-\)modules, and let \(C\) be a semidualizing \(R\)-module. Assume \(\mathcal{G_{C}}\)-\(\dim_{R}(M) <\infty\) and \(\mathcal{I}_{C}\)-id\(_{R}(N)<\infty\). Set \(q = \sup\{i |\mathrm{Tor}^{R}_{i} (M,N)\neq 0\}\). If \(q = 0\) or depth\(_{R}(\mathrm{Tor}^{R}_{q} (M,N))\leq 1\), then it follows that \[ \mathrm{depth}_{R}(M)+\mathrm{depth}_{R}(N)=\mathrm{depth}(R)+\mathrm{depth}_{R}(\mathrm{Tor}^{R}_{q} (M,N))-q.'' \]