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The stable category of monomorphisms between (Gorenstein) projective modules with applications

  • Abdolnaser Bahlekeh , Fahimeh Sadat Fotouhi , Mohammad Amin Hamlehdari and Shokrollah Salarian EMAIL logo
Published/Copyright: September 3, 2024
Forum Mathematicum
From the journal Forum Mathematicum Volume 37 Issue 4

Abstract

Let ( S , 𝔫 ) be a commutative noetherian local ring and let Ο‰ ∈ 𝔫 be non-zerodivisor. This paper is concerned with the two categories of monomorphisms between finitely generated (Gorenstein) projective S-modules, such that their cokernels are annihilated by Ο‰. It is shown that these categories, which will be denoted by π–¬π—ˆπ—‡ ⁒ ( Ο‰ , 𝒫 ) and π–¬π—ˆπ—‡ ⁒ ( Ο‰ , 𝒒 ) , are both Frobenius categories with the same projective objects. It is also proved that the stable category π–¬π—ˆπ—‡ Β― ⁒ ( Ο‰ , 𝒫 ) is triangle equivalent to the category of D-branes of type B, 𝖣𝖑 ⁒ ( Ο‰ ) , which has been introduced by Kontsevich and studied by Orlov. Moreover, it will be observed that the stable categories π–¬π—ˆπ—‡ Β― ⁒ ( Ο‰ , 𝒫 ) and π–¬π—ˆπ—‡ Β― ⁒ ( Ο‰ , 𝒒 ) are closely related to the singularity category of the factor ring R = S / ( Ο‰ ) . Precisely, there is a fully faithful triangle functor from the stable category π–¬π—ˆπ—‡ Β― ⁒ ( Ο‰ , 𝒒 ) to 𝖣 π—Œπ—€ ⁑ ( R ) , which is dense if and only if R (and so S) are Gorenstein rings. Particularly, it is proved that the density of the restriction of this functor to π–¬π—ˆπ—‡ Β― ⁒ ( Ο‰ , 𝒫 ) , guarantees the regularity of the ring S.

Keywords: Monomorphism category; stable category; triangulated category; singularity category; the category of D-branes of type B; matrix factorization
MSC 2020: 13D09; 18G80; 13H10; 13C60

Communicated by Freydoon Shahidi

Funding source: Iran National Science Foundation

Award Identifier / Grant number: 4001480

Funding statement: This work is based upon research funded by Iran National Science Foundation (INSF) under project No. 4001480.

Acknowledgements

The authors would like to thank the referee for reading the paper very carefully and giving a lot of valuable suggestions kindly and patiently.

References

[1] J. Asadollahi, R. Hafezi and S. Sadeghi, On the monomorphism category of n-cluster tilting subcategories, Sci. China Math. 65 (2022), no. 7, 1343–1362. 10.1007/s11425-020-1879-9Search in Google Scholar

[2] A. Bahlekeh, F. S. Fotouhi, A. Nateghi and S. Salarian, The homotopy category of monomorphisms between projective modules, Bull. Malays. Math. Sci. Soc. 46 (2023), no. 3, Paper No. 91. 10.1007/s40840-023-01483-5Search in Google Scholar

[3] A. Bahlekeh, F. S. Fotouhi, A. Nateghi and S. Salarian, Gorenstein D-branes of type B, preprint (2024), https://arxiv.org/abs/2402.13833v1. Search in Google Scholar

[4] P. A. Bergh and D. A. Jorgensen, Complete intersections and equivalences with categories of matrix factorizations, Homology Homotopy Appl. 18 (2016), no. 2, 377–390. 10.4310/HHA.2016.v18.n2.a21Search in Google Scholar

[5] P. A. Bergh, S. Oppermann and D. A. Jorgensen, The Gorenstein defect category, Q. J. Math. 66 (2015), no. 2, 459–471. 10.1093/qmath/hav001Search in Google Scholar

[6] G. Birkhoff, Subgroups of Abelian Groups, Proc. Lond. Math. Soc. (2) 38 (1935), 385–401. 10.1112/plms/s2-38.1.385Search in Google Scholar

[7] R.-O. Buchweitz, Maximal Cohen–Macaulay Modules and Tate Cohomology, Math. Surveys Monogr. 262, American Mathematical Society, Providence, 2021. 10.1090/surv/262Search in Google Scholar

[8] T. BΓΌhler, Exact categories, Expo. Math. 28 (2010), no. 1, 1–69. 10.1016/j.exmath.2009.04.004Search in Google Scholar

[9] X.-W. Chen, The stable monomorphism category of a Frobenius category, Math. Res. Lett. 18 (2011), no. 1, 125–137. 10.4310/MRL.2011.v18.n1.a9Search in Google Scholar

[10] X.-W. Chen, Three results on Frobenius categories, Math. Z. 270 (2012), no. 1–2, 43–58. 10.1007/s00209-010-0785-3Search in Google Scholar

[11] L. W. Christensen, Gorenstein Dimensions, Lecture Notes in Math. 1747, Springer, Berlin, 2000. 10.1007/BFb0103980Search in Google Scholar

[12] D. Eisenbud, Homological algebra on a complete intersection, with an application to group representations, Trans. Amer. Math. Soc. 260 (1980), no. 1, 35–64. 10.1090/S0002-9947-1980-0570778-7Search in Google Scholar

[13] E. E. Enochs and O. M. G. Jenda, Relative Homological Algebra. Volume 2, De Gruyter Exp. Math. 54, Walter de Gruyter, Berlin, 2011. 10.1515/9783110215212Search in Google Scholar

[14] R. Hafezi, From subcategories to the entire module categories, Forum Math. 33 (2021), no. 1, 245–270. 10.1515/forum-2019-0276Search in Google Scholar

[15] R. Hafezi and Y. Zhang, The stable Auslander–Reiten components of certain monomorphism categories, Sci. China Math. 67 (2024), no. 3, 505–526. 10.1007/s11425-022-2095-1Search in Google Scholar

[16] D. Happel, Triangulated Categories in the Representation Theory of Finite-Dimensional Algebras, London Math. Soc. Lecture Note Ser. 119, Cambridge University, Cambridge, 1988. 10.1017/CBO9780511629228Search in Google Scholar

[17] B. Keller, Chain complexes and stable categories, Manuscripta Math. 67 (1990), no. 4, 379–417. 10.1007/BF02568439Search in Google Scholar

[18] B. Keller, On differential graded categories, International Congress of Mathematicians. Vol. II, European Mathematical Society, ZΓΌrich (2006), 151–190. 10.4171/022-2/8Search in Google Scholar

[19] M. Kontsevich, Homological algebra of mirror symmetry, Proceedings of the International Congress of Mathematicians. Vol. 1, 2 (ZΓΌrich 1994), BirkhΓ€user, Basel (1995), 120–139. 10.1007/978-3-0348-9078-6_11Search in Google Scholar

[20] J. Kosakowska, M. Schmidmeier and M. Schreiner, Abelian p-groups with a p-bounded factor or a p-bounded subgroup, preprint (2023), https://arxiv.org/abs/2312.01451. Search in Google Scholar

[21] D. Kussin, H. Lenzing and H. Meltzer, Nilpotent operators and weighted projective lines, J. Reine Angew. Math. 685 (2013), 33–71. 10.1515/crelle-2012-0014Search in Google Scholar

[22] H. Matsumura, Commutative Ring Theory, Cambridge Stud. Adv. Math. 8, Cambridge University, Cambridge, 1989. Search in Google Scholar

[23] B. Mitchell, Theory of Categories, Pure Appl. Math. 17, Academic Press, New York, 1965. Search in Google Scholar

[24] D. O. Orlov, Triangulated categories of singularities and D-branes in Landau–Ginzburg models, Tr. Mat. Inst. Steklova 246 (2004), no. 3, 240–262. Search in Google Scholar

[25] D. Quillen, Higher algebraic K-theory. I, Algebraic K-Theory, I: Higher K-Theories, Lecture Notes in Math. 341, Springer, Berlin (1973), 85–147. 10.1007/BFb0067053Search in Google Scholar

[26] C. M. Ringel and M. Schmidmeier, Invariant subspaces of nilpotent linear operators. I, J. Reine Angew. Math. 614 (2008), 1–52. 10.1515/CRELLE.2008.001Search in Google Scholar

[27] C. M. Ringel and P. Zhang, From submodule categories to preprojective algebras, Math. Z. 278 (2014), no. 1–2, 55–73. 10.1007/s00209-014-1305-7Search in Google Scholar

[28] O. Veliche, Gorenstein projective dimension for complexes, Trans. Amer. Math. Soc. 358 (2006), no. 3, 1257–1283. 10.1090/S0002-9947-05-03771-2Search in Google Scholar

[29] B.-L. Xiong, P. Zhang and Y.-H. Zhang, Auslander–Reiten translations in monomorphism categories, Forum Math. 26 (2014), no. 3, 863–912. 10.1515/forum-2011-0003Search in Google Scholar

[30] Y. Yoshino, Cohen–Macaulay Modules Over Cohen–Macaulay Rings, London Math. Soc. Lecture Note Ser. 146, Cambridge University, Cambridge, 1990. 10.1017/CBO9780511600685Search in Google Scholar
Received: 2023-09-03
Revised: 2024-04-06
Published Online: 2024-09-03
Published in Print: 2025-06-01

Β© 2024 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Rational points on a class of cubic hypersurfaces
  3. On fractional inequalities on metric measure spaces with polar decomposition
  4. Quantitative weighted estimates for the multilinear pseudo-differential operators in function spaces
  5. The 𝐿𝑝 restriction bounds for Neumann data on surface
  6. Quasi-triangular, factorizable Leibniz bialgebras and relative Rota–Baxter operators
  7. Big pure projective modules over commutative noetherian rings: Comparison with the completion
  8. Any Sasakian structure is approximated by embeddings into spheres
  9. GIT quotient of holomorphic foliations on β„‚β„™2 of degree 2 and quartic plane curves
  10. The stable category of monomorphisms between (Gorenstein) projective modules with applications
  11. Gradings and graded linear maps on algebras
  12. A note on conjugacy of supplements in soluble periodic linear groups
  13. Discrete Ξ©-results for the Riemann zeta function
  14. Is addition definable from multiplication and successor?
  15. Hausdorff dimension of removable sets for elliptic and canceling homogeneous differential operators in the class of bounded functions
  16. Statistics of ranks, determinants and characteristic polynomials of rational matrices
  17. Controllability and diffeomorphism groups on manifolds with boundary
https://doi.org/10.1515/forum-2023-0317
Keywords for this article
Monomorphism category; stable category; triangulated category; singularity category; the category of D-branes of type B; matrix factorization

Articles in the same Issue

  1. Frontmatter
  2. Rational points on a class of cubic hypersurfaces
  3. On fractional inequalities on metric measure spaces with polar decomposition
  4. Quantitative weighted estimates for the multilinear pseudo-differential operators in function spaces
  5. The 𝐿𝑝 restriction bounds for Neumann data on surface
  6. Quasi-triangular, factorizable Leibniz bialgebras and relative Rota–Baxter operators
  7. Big pure projective modules over commutative noetherian rings: Comparison with the completion
  8. Any Sasakian structure is approximated by embeddings into spheres
  9. GIT quotient of holomorphic foliations on β„‚β„™2 of degree 2 and quartic plane curves
  10. The stable category of monomorphisms between (Gorenstein) projective modules with applications
  11. Gradings and graded linear maps on algebras
  12. A note on conjugacy of supplements in soluble periodic linear groups
  13. Discrete Ξ©-results for the Riemann zeta function
  14. Is addition definable from multiplication and successor?
  15. Hausdorff dimension of removable sets for elliptic and canceling homogeneous differential operators in the class of bounded functions
  16. Statistics of ranks, determinants and characteristic polynomials of rational matrices
  17. Controllability and diffeomorphism groups on manifolds with boundary
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