Tate cohomology with respect to semidualizing modules. (English) Zbl 1207.13009

Let \(R\) be a commutative noetherian ring, and denote by \(\mathcal{G}\) the class of finitely generated \(R\)-modules of G-dimension zero, in the sense of M. Auslander [Anneaux de Gorenstein et torsion en algèbre commutative. Séminaire d’algèbre commutative dirigé par P. Samuel, Secrétariat mathématique, Paris, (1967; Zbl 0157.08301)] (such modules are also called totally reflexive). Let \(M\) be a finitely generated \(R\)-module of finite G-dimension, and let \(N\) be any \(R\)-module. L. L. Avramov and A. Martsinkovsky [Proc. Lond. Math. Soc., III. Ser. 85, No. 2, 393-440 (2002; Zbl 1047.16002)] defined relative cohomology groups \(\mathrm{Ext}^*_\mathcal{G}(M,N)\), and Tate cohomology groups \(\widehat{\mathrm{Ext}}^*_R(M,N)\), and show that there is a long exact sequence: \[ \begin{split} 0 \to \mathrm{Ext}^1_\mathcal{G}(M,N) \to \mathrm{Ext}^1_R(M,N) \to \widehat{\mathrm{Ext}}^1_R(M,N) \to \mathrm{Ext}^2_\mathcal{G}(M,N) \to \\ \mathrm{Ext}^2_R(M,N) \to \cdots.\end{split} \] In the case where \(R\) has a dualizing complex, P. Jørgensen [J. Eur. Math. Soc. (JEMS) 9, No. 1, 59–76 (2007; Zbl 1114.13012)] extended the definitions of \(\mathrm{Ext}^*_\mathcal{G}(M,N)\) and \(\widehat{\mathrm{Ext}}^*_R(M,N)\), and proved that the sequence displayed above is exact for all \(R\)-modules \(M\) and \(N\).
The present paper generalizes the mentioned result from [Zbl 1047.16002] in another direction: Let \(\mathcal{A}\) be an abelian category, and let \(\mathcal{W} \subseteq \mathcal{X} \subseteq \mathcal{A}\) be full subcategories satisfying certain technical assumptions. For suitable types of objects \(M, N\) in \(\mathcal{A}\), the authors define relative cohomology groups \(\mathrm{Ext}^*_{\mathcal{X}\mathcal{A}}(M,N)\) and \(\mathrm{Ext}^*_{\mathcal{W}\mathcal{A}}(M,N)\), and Tate cohomology groups \(\widehat{\mathrm{Ext}}^*_{\mathcal{W}\mathcal{A}}(M,N)\), and prove the existence of a long exact sequence: \[ \begin{split} 0 \to \mathrm{Ext}^1_{\mathcal{X}\mathcal{A}}(M,N) \to \mathrm{Ext}^1_{\mathcal{W}\mathcal{A}}(M,N) \to \widehat{\mathrm{Ext}}^1_{\mathcal{W}\mathcal{A}}(M,N) \to \mathrm{Ext}^2_{\mathcal{X}\mathcal{A}}(M,N) \to\\ \mathrm{Ext}^2_{\mathcal{W}\mathcal{A}}(M,N) \to \cdots.\end{split} \] This long exact sequence specializes to that from [Zbl 1047.16002] in the situation where \(\mathcal{A}\) is the category of \(R\)-modules, \(\mathcal{X}\) is the category of projective \(R\)-modules, and \(\mathcal{W} = \mathcal{G}\) is the category of totally reflexive \(R\)-modules. More generally, the authors’ result applies to the situation where \(C\) is a semidualizing \(R\)-module, \(\mathcal{X}\) is the category of \(C\)-projective \(R\)-modules, and \(\mathcal{W}\) consists of those \(R\)-modules which are isomorphic to some cokernel in a totally \(\mathcal{X}\)-acyclic complex.
Under additional assumptions on \(\mathcal{W}\), and for a suitable type of object \(M\) in \(\mathcal{A}\), it is also proved that the following three conditions are equivalent:
(i) \(\widehat{\mathrm{Ext}}^n_{\mathcal{W}\mathcal{A}}(-,M)=0\) for all \(n \in \mathbb{Z}\);
(ii) \(\widehat{\mathrm{Ext}}^n_{\mathcal{W}\mathcal{A}}(M,-)=0\) for all \(n \in \mathbb{Z}\);
(iii) \(\widehat{\mathrm{Ext}}^0_{\mathcal{W}\mathcal{A}}(M,M)=0\).
Each of the results mentioned above has a “dual” counterpart. These dual results are also treated in the paper.


13D07 Homological functors on modules of commutative rings (Tor, Ext, etc.)
18G15 Ext and Tor, generalizations, Künneth formula (category-theoretic aspects)
18G25 Relative homological algebra, projective classes (category-theoretic aspects)
13D02 Syzygies, resolutions, complexes and commutative rings
13D05 Homological dimension and commutative rings
18G10 Resolutions; derived functors (category-theoretic aspects)
18G20 Homological dimension (category-theoretic aspects)
Full Text: DOI arXiv


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