Relative Buchweitz-Happel theorem respect to a self-orthogonal class

Abstract

Let $R$ be a ring, $F$ a subbifunctor of the functor Ext$^{1}_{R}(-,-)$, $\mathcal{W}_{F}$ a self-orthogonal class of left $R$-modules respect to $F$. We introduce $\mathcal{W}_{F}$-Gorenstein modules $\mathcal{G}(\mathcal{W}_{F})$ as a generalization of $\mathcal{W}$-Gorenstein modules (Geng and Ding, 2011), $F$-Gorenstein projective and $F$-Gorenstein injective modules (Tang, 2014). We introduce the notion of relative singularity category $D_{\mathcal{W}_{F}} (R)$ with respect to $\mathcal{W}_{F}$. Moreover, we give a necessary and sufficient condition such that the stable category $\underline{\mathcal{G}(\mathcal{W}_{F})}$ and the relative singularity category $D_{\mathcal{W}_{F}} (R)$ are triangle-equivalence.

Keywords

• Buchweitz-Happel Theorem;
• self-orthogonal class;
• triangle-equivalence

References

1. [1] M. Auslander and $\phi$. Solberg, Relative homology and representation theory I. Relative homology and homologically finite subcategories, Comm. Algebra 21, 2995-3031, 1993.
2. [2] M. Auslander and $\phi$. Solberg, Relative homology and representation theory II. Relative cotilting theory, Comm. Algebra 21, 3033-3079, 1993.
3. [3] M. Auslander and $\phi$. Solberg, Relative homology and representation theory III, Cotilting modules and Wedderburn correspondence, Comm. Algebra 21, 3081-3097, 1993.
4. [4] A. Beligiannis, The homological theory of contravariantly finite subcategories: Auslander-Buchweitz contexts, Gorenstein categories and (co)stabilization, Comm. Algebra 28, 4547-4596, 2000.
5. [5] P. Bergh, S. Oppermann and D. Jorgensen, The Gorenstein defect category, Q. J. Math. 66 (2), 459-471, 2015.
6. [6] S. Bouchiba, Finiteness aspects of Gorenstein homological dimensions, Colloq. Math. 131 (2), 171-193, 2013.
7. [7] A. Buan,Closed subbifunctors of the extension functor, J. Algebra 244, 407-428, 2001.
8. [8] R. Buchweitz, Maximal Cohen-Macaulay modules and Tate cohomology over Gorenstein rings, unpublished manuscript, 1986.

9. [9] T. Bühler, Exact categories, Expo. Math. 28, 1-69, 2010.
10. [10] X. Chen, Relative singularity categories and Gorenstein-projective modules, Math. Nachr. 284, 199-212, 2011.
11. [11] L. Christensen, H. Foxby and H. Holm, Derived category methods in commutative algebra, preprint, 2012.
12. [12] L. Christensen, A. Frankild and H. Holm, On Gorenstein projective, injective and flat dimensions–a functorial description with applications, J. Algebra 302, 231-279, 2006.
13. [13] E. Enochs and O. Jenda, Gorenstein injective and projective modules, Math. Z. 220, 611-633, 1995.
14. [14] Y. Geng and N. Ding, W-Gorenstein modules, J. Algebra 325, 132-146, 2011.
15. [15] D. Happel, On Gorenstein algebras, in: Representation theory of finite groups and finite-dimensional algebras, Prog. Math. 95, 389-404, Birkhaüser, Basel, 1991.
16. [16] D. Happel, Triangulated Categories in the Representation Theory of Finite Dimensional Algebras, London Mathematical Society Lecture Notes Series Vol. 119, Cambridge University Press, Cambridge, 1988.
17. [17] H. Holm and D. White, Foxby equivalence over associative rings, J. Math. Kyoto Univ. 47, 781-808, 2007.
18. [18] M. Hoshino, Algebras of finite self-injective dimension, Proc. Amer. Math. Soc. 112 (3), 619-622, 1991.
19. [19] J. Hu, H. Li, J. Wei and N. Ding, Cotorsion Pairs, Gorenstein dimensions and tiangleequivalences, arXiv:1707.02678vl.
20. [20] M. Kashiwara and P. Schapira, Sheaves on manifolds, Springer-Verlag, 1990.
21. [21] B. Keller, Chain complexes and stable categories, Manuscripta Math. 67 (4), 379-417, 1990.
22. [22] B. Keller, Derived categories and universal problems, Comm. Algebra 19, 699-747, 1991.
23. [23] H. Li and Z. Huang, Relative singularity categories, J. Pure Appl. Algebra 219, 4090- 4104, 2015.
24. [24] H. Liu, R. Zhu and Y. Geng, Gorenstein global dimensions relative to balanced pairs, Electronic Res. Arch. 28 (4), 1563-1571, 2020.
25. [25] S. Pan, Relative derived equivalences and relative homological dimensions, Acta Mathematica Sinica, English Series 32, 439-456, 2016.
26. [26] S. Sather-Wagstaff, T. Sharif and D. White, Stability of Gorenstein categories, J. Lond. Math. Soc. 77, 481-502, 2008.
27. [27] X. Tang, On F-Gorenstein dimensions, J. Algebra Appl. 13 (6), 1450022, 2014.
28. [28] J. Verdier, Catégories dérivées, Etat 0, Lect. Notes Math. 569, 262-311, Springer- Verlag, Berlin, 1977.