Abstract
Let 𝒯 be a Hom-finite triangulated Krull-Schmidt category over a field k. Inspired by a definition of Koenig and Liu (Q. J. Math 63(3), 653-674, 2012), we say that a family 𝒮 ⊆ 𝒯 of pairwise orthogonal bricks is a simple-minded system if its closure under extensions is all of 𝒯. We construct torsion pairs in 𝒯 associated to any subset 𝒳 of a simple-minded system 𝒮, and use these to define left and right mutations of 𝒮 relative to 𝒳. When 𝒯 has a Serre functor ν and 𝒮 and 𝒳 are invariant under ν ∘ [1], we show that these mutations are again simple-minded systems. We are particularly interested in the case where 𝒯 = mod-Λ for a self-injective algebra Λ. In this case, our mutation procedure parallels that introduced by Koenig and Yang for simple-minded collections in D b(mod-Λ) (Koenig and Yang, 2013). It follows that the mutation of the set of simple Λ-modules relative to 𝒳 yields the images of the simple Γ-modules under a stable equivalence mod-Γ → mod-Λ, where Γ is the tilting mutation of Λ relative to 𝒳.
References
Aihara, T., Iyama, O.: Silting mutation in triangulated categories. J. Lond. Math. Soc. (2) 85(3), 633–668 (2012)
Al-Nofayee, S.: Equivalences of derived categories for self-injective algebras. J. Algebra 313, 897–904 (2007)
Auslander, M., Reiten, I., Smalø, S.: Representation theory of Artin algebras. Cambridge Studies in Advanced Mathematics, vol. 36. Cambridge Univ. Press (1995)
Beilinson, A., Bernstein, J., Deligne, P.: Faisceaux pervers. Astérisque 100, Socit Mathmatique de France, Paris (1982)
Beligiannis, A., Reiten, I.: Homological and homotopical aspects of torsion theories. Mem. Amer. Math. Soc. 188(883), (2007)
Carlson, J., Thévenaz, J.: Torsion endo-trivial modules. Algebr. Represent. Theor 3, 303–335 (2000)
Dugas, A.: Tilting mutation of weakly symmetric algebras and stable equivalence. Algebr. Represent. Theor. doi:10.1007/s10468-013-9422-2. arXiv:1110.1679v1 [math.RT]
Dugas, A., Martínez Villa, R.: Stable equivalences of graded algebras. J. Algebra 320(12), 4215–4241 (2008)
Happel, D.: Triangulated categories in the representation theory of finite-dimensional algebras. London Mathematical Society Lecture Note Series, vol. 119. Cambridge University Press, Cambridge (1988)
Hu, W., Xi, C.: Derived equivalences and stable equivalences of Morita type. I. Nagoya Math. J 200, 107–152 (2010)
Iyama, O., Yoshino, Y.: Mutation in triangulated categories and rigid Cohen-Macaulay modules. Invent. Math. 172, 117–168 (2008)
Keller, B.: Calabi-Yau triangulated categories. Trends in Representation Theory of Algebras and Related Topics, pp. 467–489. EMS Ser. Congr. Rep., Eur. Math, Soc., Zurich (2008)
Keller, B., Yang, D.: Derived equivalences from mutations of quivers with potential. Adv. Math. 226, 2118–2168 (2011)
Koenig, S., Liu, Y.: Simple-minded systems in stable module categories. Q. J. Math 63(3), 653–674 (2012)
Koenig, S., Yang, D.: Silting objects, simple-minded collections, t-structures and co-t-structures for finite-dimensional algebras. Preprint (2013) arXiv:1203.5657v4
Linckelmann, M.: Stable equivalences of Morita type for self-injective algebras and p-groups. Math. Z. 223(1), 87–100 (1996)
Martínez Villa, R.: Properties that are left invariant under stable equivalence. Comm. Algebra 18(12), 4141–4169 (1990)
Okuyama, T.: Some examples of derived equivalent blocks of finite groups. Preprint (1998)
Pogorzały, Z.: Algebras stably equivalent to self-injective special biserial algebras. Comm. Algebra 22(4), 1127–1160 (1994)
Rickard, J.: Derived categories and stable equivalence. J. Pure Appl. Algebra 61(3), 303–317 (1989)
Rickard, J.: Derived equivalences as derived functors. J. London Math. Soc. (2) 43(1), 37–48 (1991)
Rickard, J.: Equivalences of derived categories for symmetric algebras. J. Algebra 257(2), 460–481 (2002)
Yoshino, Y.: Cohen-Macaulay modules over Cohen-Macaulay rings. London Mathematical Society Lecture Note Series, vol. 146. Cambridge University Press, Cambridge (1990)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Dugas, A. Torsion Pairs and Simple-Minded Systems in Triangulated Categories. Appl Categor Struct 23, 507–526 (2015). https://doi.org/10.1007/s10485-014-9365-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10485-014-9365-8