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Gorenstein projective modules over rings of Morita contexts

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Abstract

Under semi-weak and weak compatibility conditions of bimodules, we establish necessary and sufficient conditions of Gorenstein-projective modules over rings of Morita contexts with one bimodule homomorphism zero. This extends greatly the results on triangular matrix Artin algebras and on Artin algebras of Morita contexts with two bimodule homomorphisms zero in the literature, where only sufficient conditions are given under a strong assumption of compatibility of bimodules. An application is provided to describe Gorenstein-projective modules over noncommutative tensor products arising from Morita contexts. Our results are proved under a general setting of noetherian rings and modules instead of Artin algebras and modules.

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References

  1. Anderson F W, Fuller K R. Rings and Categories of Modules. Graduate Texts in Mathematics, vol. 13. New York: Springer, 1974

    MATH  Google Scholar 

  2. Auslander M, Bridger M. Stable Module Theory. Memoirs of the American Mathematical Society, vol. 94. Providence: Amer Math Soc, 1969

    MATH  Google Scholar 

  3. Bass H. Algebraic K-Theory. New York-Amsterdam: W. A. Benjamin, 1968

    MATH  Google Scholar 

  4. Buchweitz R O. Morita contexts, idempotents, and Hochschild cohomology-with applications to invariant rings. Contemp Math, 2003, 331: 25–53

    Article  MathSciNet  MATH  Google Scholar 

  5. Chen H X, Xi C C. Exact contexts, noncommutative tensor products and universal localizations. Trans Amer Math Soc, 2019, 371: 3647–3672

    Article  MathSciNet  MATH  Google Scholar 

  6. Chen H X, Xi C C. Recollements of derived categories I: Construction from exact contexts. J Pure Appl Algebra, 2021, 225: 106648

    Article  MathSciNet  MATH  Google Scholar 

  7. Enochs E E, Jenda O M G. Gorenstein injective and projective modules. Math Z, 1995, 220: 611–633

    Article  MathSciNet  MATH  Google Scholar 

  8. Enochs E E, Jenda O M G, Torrecillas B. Gorenstein flat modules. Nanjing Daxue Xuebao Shuxue Bannian Kan, 1993, 10: 1–9

    MathSciNet  MATH  Google Scholar 

  9. Franjou V, Pirashvili T. Comparison of abelian categories recollements. Doc Math, 2004, 9: 41–56

    Article  MathSciNet  MATH  Google Scholar 

  10. Gao N, Psaroudakis C. Gorenstein homological aspects of monomorphism categories via Morita rings. Algebr Represent Theory, 2017, 20: 487–529

    Article  MathSciNet  MATH  Google Scholar 

  11. Green E L. On the representation theory of rings in matrix form. Pacific J Math, 1982, 100: 123–138

    Article  MathSciNet  MATH  Google Scholar 

  12. Green E L, Psaroudakis C. On Artin algebras arising from Morita contexts. Algebr Represent Theory, 2014, 17: 1485–1525

    Article  MathSciNet  MATH  Google Scholar 

  13. Holm H. Gorenstein homological dimensions. J Pure Appl Algebra, 2004, 189: 167–193

    Article  MathSciNet  MATH  Google Scholar 

  14. Hu J S, Zhu H Y. Special precovering classes in comma categories. Sci China Math, 2022, 65: 933–950

    Article  MathSciNet  MATH  Google Scholar 

  15. Kerner O, Yamagata K. Morita algebras. J Algebra, 2013, 382: 185–202

    Article  MathSciNet  MATH  Google Scholar 

  16. Krylov P A, Tuganbaev A A. Modules over formal matrix rings. J Math Sci, 2010, 171: 248–295

    Article  MathSciNet  MATH  Google Scholar 

  17. Li Z-W, Zhang P. A construction of Gorenstein-projective modules. J Algebra, 2010, 323: 1802–1812

    Article  MathSciNet  MATH  Google Scholar 

  18. McConnell J C, Robson J C. Noncommutative Noetherian Rings. Chichester: John Wiley & Sons, 1987

    MATH  Google Scholar 

  19. Morita K. Duality for modules and its applications to the theory of rings with minimum condition. Sci Rep Tokyo Kyoiku Daigaku Sect A, 1958, 6: 83–142

    MathSciNet  MATH  Google Scholar 

  20. Müller M. Rings of quotients of generalized matrix rings. Comm Algebra, 1987, 15: 1991–2015

    Article  MathSciNet  MATH  Google Scholar 

  21. Psaroudakis C. Homological theory of recollements of abelian categories. J Algebra, 2014, 398: 63–110

    Article  MathSciNet  MATH  Google Scholar 

  22. Xiong B L, Zhang P. Gorenstein-projective modules over triangular matrix Artin algebras. J Algebra Appl, 2012, 11: 1250066

    Article  MathSciNet  MATH  Google Scholar 

  23. Zhang P. Gorenstein-pro jective modules and symmetric recollements. J Algebra, 2013, 388: 65–80

    Article  MathSciNet  Google Scholar 

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Acknowledgements

Changchang Xi was supported by National Natural Science Foundation of China (Grant Nos. 12031014 and 12226314).

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Authors and Affiliations

  1. School of Mathematical Sciences, Capital Normal University, Beijing, 100048, China

    Qianqian Guo & Changchang Xi

  2. School of Mathematical Sciences, Laboratory of Mathematics and Complex Systems (Ministry of Education of China), Beijing Normal University, Beijing, 100875, China

    Qianqian Guo

  3. School of Mathematics and Statistics, Shaanxi Normal University, Xi’an, 710062, China

    Changchang Xi

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  1. Qianqian Guo
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  2. Changchang Xi
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Correspondence to Changchang Xi.

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Guo, Q., Xi, C. Gorenstein projective modules over rings of Morita contexts. Sci. China Math. (2023). https://doi.org/10.1007/s11425-022-2206-8

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  • DOI: https://doi.org/10.1007/s11425-022-2206-8

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