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

Year 2023, Volume: 52 Issue: 2, 374 - 390, 31.03.2023

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.

Supporting Institution

National Natural Science Foundation of China

Project Number

No. 11771202

References

  • [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] M. Auslander and $\phi$. Solberg, Relative homology and representation theory II. Relative cotilting theory, Comm. Algebra 21, 3033-3079, 1993.
  • [3] M. Auslander and $\phi$. Solberg, Relative homology and representation theory III, Cotilting modules and Wedderburn correspondence, Comm. Algebra 21, 3081-3097, 1993.
  • [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] P. Bergh, S. Oppermann and D. Jorgensen, The Gorenstein defect category, Q. J. Math. 66 (2), 459-471, 2015.
  • [6] S. Bouchiba, Finiteness aspects of Gorenstein homological dimensions, Colloq. Math. 131 (2), 171-193, 2013.
  • [7] A. Buan,Closed subbifunctors of the extension functor, J. Algebra 244, 407-428, 2001.
  • [8] R. Buchweitz, Maximal Cohen-Macaulay modules and Tate cohomology over Gorenstein rings, unpublished manuscript, 1986.
  • [9] T. Bühler, Exact categories, Expo. Math. 28, 1-69, 2010.
  • [10] X. Chen, Relative singularity categories and Gorenstein-projective modules, Math. Nachr. 284, 199-212, 2011.
  • [11] L. Christensen, H. Foxby and H. Holm, Derived category methods in commutative algebra, preprint, 2012.
  • [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] E. Enochs and O. Jenda, Gorenstein injective and projective modules, Math. Z. 220, 611-633, 1995.
  • [14] Y. Geng and N. Ding, W-Gorenstein modules, J. Algebra 325, 132-146, 2011.
  • [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] 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] H. Holm and D. White, Foxby equivalence over associative rings, J. Math. Kyoto Univ. 47, 781-808, 2007.
  • [18] M. Hoshino, Algebras of finite self-injective dimension, Proc. Amer. Math. Soc. 112 (3), 619-622, 1991.
  • [19] J. Hu, H. Li, J. Wei and N. Ding, Cotorsion Pairs, Gorenstein dimensions and tiangleequivalences, arXiv:1707.02678vl.
  • [20] M. Kashiwara and P. Schapira, Sheaves on manifolds, Springer-Verlag, 1990.
  • [21] B. Keller, Chain complexes and stable categories, Manuscripta Math. 67 (4), 379-417, 1990.
  • [22] B. Keller, Derived categories and universal problems, Comm. Algebra 19, 699-747, 1991.
  • [23] H. Li and Z. Huang, Relative singularity categories, J. Pure Appl. Algebra 219, 4090- 4104, 2015.
  • [24] H. Liu, R. Zhu and Y. Geng, Gorenstein global dimensions relative to balanced pairs, Electronic Res. Arch. 28 (4), 1563-1571, 2020.
  • [25] S. Pan, Relative derived equivalences and relative homological dimensions, Acta Mathematica Sinica, English Series 32, 439-456, 2016.
  • [26] S. Sather-Wagstaff, T. Sharif and D. White, Stability of Gorenstein categories, J. Lond. Math. Soc. 77, 481-502, 2008.
  • [27] X. Tang, On F-Gorenstein dimensions, J. Algebra Appl. 13 (6), 1450022, 2014.
  • [28] J. Verdier, Catégories dérivées, Etat 0, Lect. Notes Math. 569, 262-311, Springer- Verlag, Berlin, 1977.
Year 2023, Volume: 52 Issue: 2, 374 - 390, 31.03.2023

Abstract

Project Number

No. 11771202

References

  • [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] M. Auslander and $\phi$. Solberg, Relative homology and representation theory II. Relative cotilting theory, Comm. Algebra 21, 3033-3079, 1993.
  • [3] M. Auslander and $\phi$. Solberg, Relative homology and representation theory III, Cotilting modules and Wedderburn correspondence, Comm. Algebra 21, 3081-3097, 1993.
  • [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] P. Bergh, S. Oppermann and D. Jorgensen, The Gorenstein defect category, Q. J. Math. 66 (2), 459-471, 2015.
  • [6] S. Bouchiba, Finiteness aspects of Gorenstein homological dimensions, Colloq. Math. 131 (2), 171-193, 2013.
  • [7] A. Buan,Closed subbifunctors of the extension functor, J. Algebra 244, 407-428, 2001.
  • [8] R. Buchweitz, Maximal Cohen-Macaulay modules and Tate cohomology over Gorenstein rings, unpublished manuscript, 1986.
  • [9] T. Bühler, Exact categories, Expo. Math. 28, 1-69, 2010.
  • [10] X. Chen, Relative singularity categories and Gorenstein-projective modules, Math. Nachr. 284, 199-212, 2011.
  • [11] L. Christensen, H. Foxby and H. Holm, Derived category methods in commutative algebra, preprint, 2012.
  • [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] E. Enochs and O. Jenda, Gorenstein injective and projective modules, Math. Z. 220, 611-633, 1995.
  • [14] Y. Geng and N. Ding, W-Gorenstein modules, J. Algebra 325, 132-146, 2011.
  • [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] 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] H. Holm and D. White, Foxby equivalence over associative rings, J. Math. Kyoto Univ. 47, 781-808, 2007.
  • [18] M. Hoshino, Algebras of finite self-injective dimension, Proc. Amer. Math. Soc. 112 (3), 619-622, 1991.
  • [19] J. Hu, H. Li, J. Wei and N. Ding, Cotorsion Pairs, Gorenstein dimensions and tiangleequivalences, arXiv:1707.02678vl.
  • [20] M. Kashiwara and P. Schapira, Sheaves on manifolds, Springer-Verlag, 1990.
  • [21] B. Keller, Chain complexes and stable categories, Manuscripta Math. 67 (4), 379-417, 1990.
  • [22] B. Keller, Derived categories and universal problems, Comm. Algebra 19, 699-747, 1991.
  • [23] H. Li and Z. Huang, Relative singularity categories, J. Pure Appl. Algebra 219, 4090- 4104, 2015.
  • [24] H. Liu, R. Zhu and Y. Geng, Gorenstein global dimensions relative to balanced pairs, Electronic Res. Arch. 28 (4), 1563-1571, 2020.
  • [25] S. Pan, Relative derived equivalences and relative homological dimensions, Acta Mathematica Sinica, English Series 32, 439-456, 2016.
  • [26] S. Sather-Wagstaff, T. Sharif and D. White, Stability of Gorenstein categories, J. Lond. Math. Soc. 77, 481-502, 2008.
  • [27] X. Tang, On F-Gorenstein dimensions, J. Algebra Appl. 13 (6), 1450022, 2014.
  • [28] J. Verdier, Catégories dérivées, Etat 0, Lect. Notes Math. 569, 262-311, Springer- Verlag, Berlin, 1977.
There are 28 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Haiyu Liu This is me 0000-0002-5612-397X

Yuxian Geng This is me 0000-0002-5431-5214

Rongmin Zhu 0000-0002-1815-5954

Project Number No. 11771202
Publication Date March 31, 2023
Published in Issue Year 2023 Volume: 52 Issue: 2

Cite

APA Liu, H., Geng, Y., & Zhu, R. (2023). Relative Buchweitz-Happel theorem respect to a self-orthogonal class. Hacettepe Journal of Mathematics and Statistics, 52(2), 374-390.
AMA Liu H, Geng Y, Zhu R. Relative Buchweitz-Happel theorem respect to a self-orthogonal class. Hacettepe Journal of Mathematics and Statistics. March 2023;52(2):374-390.
Chicago Liu, Haiyu, Yuxian Geng, and Rongmin Zhu. “Relative Buchweitz-Happel Theorem Respect to a Self-Orthogonal Class”. Hacettepe Journal of Mathematics and Statistics 52, no. 2 (March 2023): 374-90.
EndNote Liu H, Geng Y, Zhu R (March 1, 2023) Relative Buchweitz-Happel theorem respect to a self-orthogonal class. Hacettepe Journal of Mathematics and Statistics 52 2 374–390.
IEEE H. Liu, Y. Geng, and R. Zhu, “Relative Buchweitz-Happel theorem respect to a self-orthogonal class”, Hacettepe Journal of Mathematics and Statistics, vol. 52, no. 2, pp. 374–390, 2023.
ISNAD Liu, Haiyu et al. “Relative Buchweitz-Happel Theorem Respect to a Self-Orthogonal Class”. Hacettepe Journal of Mathematics and Statistics 52/2 (March 2023), 374-390.
JAMA Liu H, Geng Y, Zhu R. Relative Buchweitz-Happel theorem respect to a self-orthogonal class. Hacettepe Journal of Mathematics and Statistics. 2023;52:374–390.
MLA Liu, Haiyu et al. “Relative Buchweitz-Happel Theorem Respect to a Self-Orthogonal Class”. Hacettepe Journal of Mathematics and Statistics, vol. 52, no. 2, 2023, pp. 374-90.
Vancouver Liu H, Geng Y, Zhu R. Relative Buchweitz-Happel theorem respect to a self-orthogonal class. Hacettepe Journal of Mathematics and Statistics. 2023;52(2):374-90.