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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.

MSC:

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)
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References:

[1] Asadollahi, J.; Salarian, S., Cohomology theories based on Gorenstein injective modules, Trans. Amer. Math. Soc., 358, 5, 2183-2203 (2006), MR 2197453 (2006k:13033) · Zbl 1093.13010
[2] Auslander, M.; Buchweitz, R.-O., The homological theory of maximal Cohen-Macaulay approximations, Colloque en l’honneur de Pierre Samuel. Colloque en l’honneur de Pierre Samuel, Orsay, 1987. Colloque en l’honneur de Pierre Samuel. Colloque en l’honneur de Pierre Samuel, Orsay, 1987, Mém. Soc. Math. Fr. (N.S.), 38, 5-37 (1989), MR 1044344 (91h:13010) · Zbl 0697.13005
[3] Avramov, L. L.; Martsinkovsky, A., Absolute, relative, and Tate cohomology of modules of finite Gorenstein dimension, Proc. Lond. Math. Soc. (3), 85, 393-440 (2002), MR 2003g:16009 · Zbl 1047.16002
[4] Bennis, D.; Mahdou, N., Strongly Gorenstein projective, injective, and flat modules, J. Pure Appl. Algebra, 210, 2, 437-445 (2007), MR 2320007 (2008d:13020) · Zbl 1118.13014
[5] Bruns, W.; Herzog, J., Cohen-Macaulay Rings, Stud. Adv. Math., vol. 39 (1998), Cambridge University Press: Cambridge University Press Cambridge, MR 1251956 (95h:13020) · Zbl 0909.13005
[6] Christensen, L. W., Semi-dualizing complexes and their Auslander categories, Trans. Amer. Math. Soc., 353, 5, 1839-1883 (2001), MR 2002a:13017 · Zbl 0969.13006
[7] Enochs, E. E.; Jenda, O. M.G., Gorenstein injective and projective modules, Math. Z., 220, 4, 611-633 (1995), MR 1363858 (97c:16011) · Zbl 0845.16005
[8] Foxby, H.-B., Gorenstein modules and related modules, Math. Scand., 31, 267-284 (1972), (1973). MR 48 #6094 · Zbl 0272.13009
[9] Foxby, H.-B., Gorenstein dimensions over Cohen-Macaulay rings, (Bruns, W., Proceedings of the International Conference on Commutative Algebra (1994), Universität Osnabrück), 59-63 · Zbl 0834.13014
[10] Gerko, A. A., On homological dimensions, Mat. Sb.. Mat. Sb., Sb. Math., 192, 7-8, 1165-1179 (2001), MR 2002h:13024 · Zbl 1029.13010
[11] Gerko, A. A., On the structure of the set of semidualizing complexes, Illinois J. Math., 48, 3, 965-976 (2004), MR 2114263 · Zbl 1080.13009
[12] Golod, E. S., \(G\)-dimension and generalized perfect ideals, Algebraic Geometry and Its Applications. Algebraic Geometry and Its Applications, Tr. Mat. Inst. Steklova, 165, 62-66 (1984), MR 85m:13011 · Zbl 0577.13008
[13] Holm, H., Gorenstein homological dimensions, J. Pure Appl. Algebra, 189, 1, 167-193 (2004), MR 2038564 (2004k:16013) · Zbl 1050.16003
[14] Holm, H.; Jørgensen, P., Semi-dualizing modules and related Gorenstein homological dimensions, J. Pure Appl. Algebra, 205, 2, 423-445 (2006), MR 2203625 · Zbl 1094.13021
[15] Holm, H.; White, D., Foxby equivalence over associative rings, J. Math. Kyoto Univ., 47, 4, 781-808 (2007), MR 2413065 · Zbl 1154.16007
[16] Iacob, A., Remarks on balance in generalized Tate cohomology, Arch. Math. (Basel), 85, 4, 335-344 (2005), MR 2174231 (2007b:16018) · Zbl 1080.18009
[17] Sather-Wagstaff, S.; Sharif, T.; White, D., Gorenstein cohomology in abelian categories, J. Math. Kyoto Univ., 48, 3, 571-596 (2008), MR 2511051 · Zbl 1172.18004
[18] Sather-Wagstaff, S.; Sharif, T.; White, D., Stability of Gorenstein categories, J. Lond. Math. Soc. (2), 77, 2, 481-502 (2008), MR 2400403 (2009f:13022) · Zbl 1140.18010
[19] Sather-Wagstaff, S.; Sharif, T.; White, D., Comparison of relative cohomology theories with respect to semidualizing modules, Math. Z., 264, 3, 571-600 (2010), MR 2591820 · Zbl 1190.13007
[20] Sather-Wagstaff, S.; Yassemi, S., Modules of finite homological dimension with respect to a semidualizing module, Arch. Math. (Basel), 93, 2, 111-121 (2009), MR 2534418 · Zbl 1173.13009
[21] Takahashi, R.; White, D., Homological aspects of semidualizing modules, Math. Scand., 106, 1, 5-22 (2010), MR 2603458 · Zbl 1193.13012
[22] Vasconcelos, W. V., Divisor Theory in Module Categories, North-Holland Math. Stud., vol. 14 (1974), North-Holland Publishing Co.: North-Holland Publishing Co. Amsterdam, MR 0498530 (58 #16637) · Zbl 0296.13005
[23] White, D., Gorenstein projective dimension with respect to a semidualizing module, J. Commut. Algebra, 2, 1, 111-137 (2010), MR 2607104 · Zbl 1237.13029
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