Balance for Tate cohomology with respect to semidualizing modules (Q2873593)

Let \(R\) be a commutative noetherian ring, and denote by \(\mathcal{G}\) the class of finitely generated \(R\)-modules of \(G\)-dimension zero. Let \(M\) be a finitely generated \(R\)-module of finite \(G\)-dimension and let \(N\) be any \(R\)-module. \textit{L. L. Avramov} and \textit{A. Martsinkovsky} [Proc. Lond. Math. Soc., III. Ser. 85, No. 2, 393--440 (2002; Zbl 1047.16002)] defined relative cohomology groups \(\text{Ext}^{\ast}_\mathcal{G}(M,N)\), and Tate cohomology groups \({\widehat{\text{Ext}}}^{\ast}_R(M,N)\), and they showed that there exist an exact sequence connecting the absolute cohomology functor \(\text{Ext}^{\ast}_R(M,-)\), the relative cohomology functor \(\text{Ext}^{\ast}_\mathcal{G}(M,-)\), and the Tate cohomology functor \({\widehat{\text{Ext}}}^{\ast}_R(M,-)\).NEWLINENEWLINE\textit{S. Sather-Wagstaff} et al. [J. Algebra. 324 , No. 9, 2336--2368 (2010; Zbl 1207.13009)] constructed a theory of Tate cohomology in abelian categories and they proved same balance results.NEWLINENEWLINEThe authors further study balancedness of Tate cohomology in abelian categories in the present paper. Let \(\mathcal{A}\) be an abelian category, and let \(\mathcal{W}\subseteqq \mathcal{X}\subseteqq\mathcal{A}\) be full subcategories satisfying certain technical assumptions. For suitable types of objects \(M,N\) in \(\mathcal{A}\), the authors define relative cohomology groups \(\text{Ext}^{\ast}_{\mathcal{XA}}(M,N)\) and \(\text{Ext}^{\ast}_{\mathcal{WA}}(M,N)\), and Tate cohomology groups \({\widehat{\text{Ext}}}^{\ast}_{\mathcal{XA}}(M,N)\). One of the main results:NEWLINENEWLINETheorem. Assume that \(\mathcal{X}\) and \(\mathcal{Y}\) are exact, and \(\mathcal{X}\) is closed under kernels of epimorphisms and \(\mathcal{Y}\) is closed under cokernels of monomorphisms. Assume that \(\mathcal{W}\) is both an injective cogenerator and a projective generator for \(\mathcal{X}\) and \(\mathcal{V}\) is both an injective cogenerator and a projective generator for \(\mathcal{Y}\). Assume also that \(\mathcal{W}\) and \(\mathcal{V}\) are closed under direct summands and satisfy \(\mathcal{W}\bot\mathcal{Y}\), \(\mathcal{X}\bot\mathcal{V}\) and \(\text{Ext}^{\geq1}_{\mathcal{WA}}(\text{res}\widehat{\mathcal{W}},\mathcal{V})=0=\text{Ext}^{\geq1}_{\mathcal{AV}}(\mathcal{W},\text{cores}\widehat{\mathcal{V}})\). Then, for all \(M\in\text{res}\widehat{\mathcal{X}}\) and \(N\in\text{cores}\widehat{\mathcal{Y}}\), and all \(n\in \mathbb{Z}\), \({\widehat{\text{Ext}}}^n_{\mathcal{WA}}(M,N)\cong {\widehat{\text{Ext}}}^n_{\mathcal{AV}}(M,N)\).NEWLINENEWLINECorollary. Let \(R\) be a commutative ring, and let \(B\) and \(C\) be semidualizing \(R\)-modules such that \(B\in\mathcal{GP_C}(R)\). Set \(B^{\dag}=\text{Hom}_R(B,C)\). Let \(M\) and \(N\) be \(R\)-modules such that \(\mathcal{G}(\mathcal{P}_B)\)-\(\text{pd}_R(M)<\infty\) and \(\mathcal{G}(\mathcal{I}_{B^{\dag}})\)-\(\text{id}_R(N)<\infty\). Then, for each \(n\in\mathbb{Z}\), \({\widehat{\text{Ext}}}^n_{\mathcal{P}_B\mathcal{M}}(M,N)\cong{\widehat{\text{Ext}}}^n_{\mathcal{M}\mathcal{I}_{B^{\dag}}}(M,N)\).NEWLINENEWLINEThe last assertions was obtained for each \(n\geq1\) in [Zbl 1207.13009]