Nagoya Mathematical Journal

Derived equivalences and stable equivalences of Morita type, I

Wei Hu and Changchang Xi

Full-text: Open access


For self-injective algebras, Rickard proved that each derived equivalence induces a stable equivalence of Morita type. For general algebras, it is unknown when a derived equivalence implies a stable equivalence of Morita type. In this article, we first show that each derived equivalence F between the derived categories of Artin algebras A and B arises naturally as a functor F¯ between their stable module categories, which can be used to compare certain homological dimensions of A with that of B. We then give a sufficient condition for the functor F¯ to be an equivalence. Moreover, if we work with finite-dimensional algebras over a field, then the sufficient condition guarantees the existence of a stable equivalence of Morita type. In this way, we extend the classical result of Rickard. Furthermore, we provide several inductive methods for constructing those derived equivalences that induce stable equivalences of Morita type. It turns out that we may produce a lot of (usually not self-injective) finite-dimensional algebras that are both derived-equivalent and stably equivalent of Morita type; thus, they share many common invariants.

Article information

Nagoya Math. J., Volume 200 (2010), 107-152.

First available in Project Euclid: 28 December 2010

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 18E30: Derived categories, triangulated categories 16G10: Representations of Artinian rings
Secondary: 18G20: Homological dimension [See also 13D05, 16E10] 16D90: Module categories [See also 16Gxx, 16S90]; module theory in a category-theoretic context; Morita equivalence and duality


Hu, Wei; Xi, Changchang. Derived equivalences and stable equivalences of Morita type, I. Nagoya Math. J. 200 (2010), 107--152. doi:10.1215/00277630-2010-014.

Export citation


  • [1] M. Auslander, Representation Dimension of Artin Algebras, Queen Mary Coll. Math. Notes, Queen Mary College, London, 1971.
  • [2] M. Barot and H. Lenzing, One-point extensions and derived equivalences, J. Algebra 264 (2003), 1–5.
  • [3] M. Broué, “Equivalences of blocks of group algebras” in Finite Dimensional Algebras and Related Topics, Kluwer, Dordrecht, 1994, 1–26.
  • [4] E. Cline, B. Parshall, and L. Scott, Derived categories and Morita theory, J. Algebra 104 (1986), 397–409.
  • [5] D. Happel, Triangulated Categories in the Representation Theory of Finite Dimensional Algebras, Cambridge Univ. Press, Cambridge, 1988.
  • [6] W. Hu and C. C. Xi, Almost D-split sequences and derived equivalences, preprint, 2007, arXiv:0810.4757v1[math.RT]
  • [7] W. Hu and C. C. Xi, Derived equivalences for Φ-Auslander-Yoneda algebras, preprint, 2009, arXiv:9012.0647v2[math.RT]
  • [8] H. Krause, Stable equivalence preserves representation type, Comment. Math. Helv. 72 (1997), 266–284.
  • [9] Y. M. Liu and C. C. Xi, Constructions of stable equivalences of Morita type for finite dimensional algebras, III, J. Lond. Math. Soc. (2) 76 (2007), 567–585.
  • [10] R. Martinez-Villa, Properties that are left invariant under stable equivalence, Comm. Algebra 18 (1990), 4141–4169.
  • [11] A. Neeman, Triangulated Categories, Princeton University Press, Princeton, 2001.
  • [12] S. Y. Pan and C. C. Xi, Finiteness of finitistic dimension is invariant under derived equivalences, J. Algebra 322 (2009), 21–24.
  • [13] J. Rickard, Derived categories and stable equivalences, J. Pure Appl. Algebra 64 (1989), 303–317.
  • [14] J. Rickard, Morita theory for derived categories, J. Lond. Math. Soc. (2) 39 (1989), 436–456.
  • [15] J. Rickard, Derived equivalence as derived functors, J. Lond. Math. Soc. (2) 43 (1991), 37–48.
  • [16] J. Rickard, “The abelian defect group conjecture” in Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998), Doc. Math. 1998, Extra Vol. II, Documenta Mathematica, Bielefeld, 1998, 121–128.
  • [17] R. Rouquier, Representation dimension of exterior algebras, Invent. Math. 165 (2006), 357–367.
  • [18] C. A. Weibel, An Introduction to Homological Algebra, Cambridge University Press, Cambridge, 1994.
  • [19] C. C. Xi, Representation dimension and quasi-hereditary algebras, Adv. Math. 168 (2002), 193–212.
  • [20] C. C. Xi, Stable equivalences of adjoint type, Forum Math. 20 (2008), 81–97.
  • [21] C. C. Xi and D. M. Xu, The finitistic dimension conjecture and relatively projective modules, preprint, 2007.