Abstract
Let A be an Artin algebra and let Gprj-Λ denote the class of all the finitely generated Gorenstein projective Λ-modules. In this paper, we study the components of the stable Auslander-Reiten quiver of a certain subcategory of the monomorphism category \({\cal{S}}(\text{Gprj-}\Lambda)\) containing boundary vertices. We describe the shape of such components. It is shown that certain components are linked to the orbits of an auto-equivalence on the stable category Gprj-Λ. In particular, for the finite components, we show that under certain mild conditions, their cardinalities are divisible by 3. We see that this three-periodicity phenomenon reoccurs several times in the paper.
Similar content being viewed by others
References
Amiot C. On the structure of triangulated categories with finitely many indécomposables. Bull Soc Math France, 2007, 135: 435–474
Ariki S, Kase R, Miyamoto K. On components of stable Auslander-Reiten quivers that contain Heller lattices: The case of truncated polynomial rings. Nagoya Math J, 2017, 228: 72–113
Assem I, Simson D, Skowroński A. Elements of the Representation Theory of Associative Algebras I: Techniques of Representation Theory. Cambridge: Cambridge University Press, 2006
Auslander M, Bridger M. Stable Module Category. Memoirs of the American Mathematical Society, vol. 94. Providence: Amer Math Soc, 1969
Auslander M, Reiten I. Stable equivalence of dualizing R-varieties. Adv Math, 1974, 12: 306–366
Auslander M, Reiten I. Representation theory of Artin algebras. III. Almost split sequences. Comm Algebra, 1975, 3: 239–294
Auslander M, Reiten I. Representation theory of Artin algebras. IV. Invariants given by almost split sequences. Comm Algebra, 1977, 5: 443–518
Auslander M, Reiten I, Smalø S O. Representation Theory of Artin Algebras. Cambridge: Cambridge University Press, 1995
Auslander M, Smalø S O. Almost split sequences in subcategories. J Algebra, 1981, 69: 426–454
Beligiannis A. On algebras of finite Cohen-Macaulay type. Adv Math, 2011, 226: 1973–2019
Bergh P A, Erdmann K. The stable Auslander-Reiten quiver of a quantum complete intersection. Bull Lond Math Soc, 2011, 43: 79–90
Birkhoff G. Subgroups of abelian groups. Proc Lond Math Soc, 1935, 38: 385–401
Bühler T. Exact categories. Expo Math, 2010, 28: 1–69
Chen X W. The stable monomorphism category of a Frobenius category. Math Res Lett, 2011, 18: 125–137
Chen X W. Gorenstein homological algebra of Artin algebras. arXiv:1712.04587, 2017
Enochs E E, Jenda O M G. Gorenstein injective and projective modules. Math Z, 1995, 220: 611–633
Eshraghi H, Hafezi R, Salarian S. Total acyclicity for complexes of representations of quivers. Comm Algebra, 2013, 41: 4425–4441
Gabriel P, Roiter A V. Representations of Finite-Dimensional Algebras. Encyclopedia of Mathematical Sciences. Berlin: Springer, 1992
Gao N, Külshammer J, Kvamme S, et al. A functorial approach to monomorphism categories for species I. Commun Contemp Math, 2022, 24: 2150069
Hafezi R. From subcategories to the entire module categories. Forum Math, 2021, 33: 245–270
Hafezi R. When stable Cohen-Macaulay Auslander algebra is semisimple. arXiv:2109.00467, 2021
Hafezi R, Muchtadi-Alamsyah I. Different exact structures on the monomorphism categories. Appl Categ Structures, 2021, 29: 31–68
Happel D. Triangulated Categories in the Representation Theory of Finite Dimensional Algebras. Cambridge: Cambridge University Press, 1988
Hu W, Luo X H, Xiong B L, et al. Gorenstein projective bimodules via monomorphism categories and filtration categories. J Pure Appl Algebra, 2019, 223: 1014–1039
Iyama O, Nakaoka H, Palu Y. Auslander-Reiten theory in extriangulated categories. arXiv:1805.03776, 2018
Keller B. Chain complexes and stable categories. Manuscripta Math, 1990, 67: 379–417
Krause H, Solberg Ø. Applications of cotorsion pairs. J Lond Math Soc (2), 2003, 68: 631–650
Liu S P. Auslander-Reiten theory in a Krull-Schmidt category. São Paulo J Math Sci, 2010, 4: 425–472
Liu S P, Ng P M, Paquette C. Almost split sequences and approximations. Algebr Represent Theory, 2013, 16: 1809–1827
Lu M. Gorenstein properties of simple gluing algebras. Algebr Represent Theory, 2019, 22: 517–543
Luo X H, Zhang P. Monic representations and Gorenstein-projective modules. Pacific J Math, 2013, 264: 163–194
Quillen D. Higher algebraic K-theory. I. In: Algebraic K-theory, I: Higher K-theories. Lecture Notes in Mathematics, vol. 341. Berlin: Springer, 1973, 85–147
Riedtmann C. Algebren, Darstellungsköcher, Überlagerungen und zurück. Comment Math Helv, 1980, 55: 199–224
Riedtmann C. Many algebras with the same Auslander-Reiten quiver. Bull Lond Math Soc, 1983, 15: 43–47
Riedtmann C. Representation-finite selfinjective algebras of class Dn Compos Math, 1983, 49: 231–282
Ringel C M. Finite-dimensional hereditary algebras of wild representation type. Math Z, 1978, 161: 235–255
Ringel C M, Schmidmeier M. Submodule categories of wild representation type. J Pure Appl Algebra, 2006, 205: 412–422
Ringel C M, Schmidmeier M. Invariant subspaces of nilpotent linear operators, I. J Reine Angew Math, 2008, 614:1–52
Ringel C M, Schmidmeier M. The Auslander-Reiten translation in submodule categories. Trans Amer Math Soc, 2008, 60: 691–716
Ringel C M, Zhang P. From submodule categories to preprojective algebras. Math Z, 2014, 278: 55–73
Simson D. Representation types of the category of subprojective representations of a finite poset over K[t]/(tm) and a solution of a Birkhoff type problem. J Algebra, 2007, 311: 1–30
Skowroński A, Yamagata K. Frobenius Algebras I: Basic Representation Theory. EMS Textbooks in Mathematics. Zürich: Eur Math Soc, 2011
Thiel J N. On Euclidean components of Auslander-Reiten quivers of finite group schemes. Algebr Represent Theory, 2023, 26: 1217–1230
Wang Z P, Liu Z K. Recollements induced by monomorphism categories. J Algebra, 2022, 594: 614–635
Xiao J, Zhu B. Locally finite triangulated categories. J Algebra, 2005, 290: 473–490
Xiong B L, Zhang P, Zhang Y H. Auslander-Reiten translations in monomorphism categories. Forum Math, 2014, 26: 863–912
Xu J. Flat Covers of Modules. Lecture Notes in Mathematics, vol. 1634. Berlin: Springer-Verlag, 1996
Zhang P. Monomorphism categories, cotilting theory, and Gorenstein-projective modules. J Algebra, 2011, 339: 181–202
Zhang P, Xiong B L. Separated monic representations II: Frobenius subcategories and RSS equivalences. Trans Amer Math Soc, 2019, 372: 981–1021
Acknowledgements
This work was supported by National Natural Science Foundation of China (Grant No. 12101316). The authors thank the referees, whose many useful comments significantly improved the paper.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Hafezi, R., Zhang, Y. The stable Auslander-Reiten components of certain monomorphism categories. Sci. China Math. 67, 505–526 (2024). https://doi.org/10.1007/s11425-022-2095-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11425-022-2095-1