Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Derived equivalences induced by nonclassical tilting objects

Authors: Luisa Fiorot, Francesco Mattiello and Manuel Saorín
Journal: Proc. Amer. Math. Soc. 145 (2017), 1505-1514
MSC (2010): Primary 18E30; Secondary 18E10, 18G55
Published electronically: November 21, 2016
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Suppose that $ \mathcal {A}$ is an abelian category whose derived category $ \mathcal {D}(\mathcal {A})$ has $ Hom$ sets and arbitrary (small) coproducts, let $ T$ be a (not necessarily classical) ($ n$-)tilting object of $ \mathcal {A}$ and let $ \mathcal {H}$ be the heart of the associated t-structure on $ \mathcal {D}(\mathcal {A})$. We show that there is a triangulated equivalence of unbounded derived categories $ \mathcal {D}(\mathcal {H})\stackrel {\cong }{\longrightarrow }\mathcal {D}(\mathcal {A})$ which is compatible with the inclusion functor $ \mathcal {H}\hookrightarrow \mathcal {D}(\mathcal {A})$. The result admits a straightforward dualization to cotilting objects in abelian categories whose derived category has $ Hom$ sets and arbitrary products.

References [Enhancements On Off] (What's this?)

  • [AC01] Lidia Angeleri-Hügel and Flávio Ulhoa Coelho, Infinitely generated tilting modules of finite projective dimension, Forum Math. 13 (2001), no. 2, 239-250. MR 1813669,
  • [AHK07] Lidia Angeleri-Hügel, Dieter Happel, and Henning Krause (eds.), Handbook of tilting theory, London Mathematical Society Lecture Note Series, vol. 332, Cambridge University Press, Cambridge, 2007. MR 2385175
  • [BMT11] Silvana Bazzoni, Francesca Mantese, and Alberto Tonolo, Derived equivalence induced by infinitely generated $ n$-tilting modules, Proc. Amer. Math. Soc. 139 (2011), no. 12, 4225-4234. MR 2823068,
  • [BR07] Apostolos Beligiannis and Idun Reiten, Homological and homotopical aspects of torsion theories, Mem. Amer. Math. Soc. 188 (2007), no. 883, viii+207. MR 2327478,
  • [Bo81] Klaus Bongartz, Tilted algebras, Representations of algebras (Puebla, 1980) Lecture Notes in Math., vol. 903, Springer, Berlin-New York, 1981, pp. 26-38. MR 654701
  • [BB80] Sheila Brenner and M. C. R. Butler, Generalizations of the Bernstein-Gelfand-Ponomarev reflection functors, Representation theory, II (Proc. Second Internat. Conf., Carleton Univ., Ottawa, Ont., 1979) Lecture Notes in Math., vol. 832, Springer, Berlin-New York, 1980, pp. 103-169. MR 607151
  • [BBD82] A. A. Beĭlinson, J. Bernstein, and P. Deligne, Faisceaux pervers, Analysis and topology on singular spaces, I (Luminy, 1981) Astérisque, vol. 100, Soc. Math. France, Paris, 1982, pp. 5-171 (French). MR 751966
  • [Ch10] Xiao-Wu Chen, A short proof of HRS-tilting, Proc. Amer. Math. Soc. 138 (2010), no. 2, 455-459. MR 2557163,
  • [CF90] R. R. Colby and K. R. Fuller, Tilting, cotilting, and serially tilted rings, Comm. Algebra 18 (1990), no. 5, 1585-1615. MR 1059750,
  • [CT95] Riccardo Colpi and Jan Trlifaj, Tilting modules and tilting torsion theories, J. Algebra 178 (1995), no. 2, 614-634. MR 1359905,
  • [CPS86] E. Cline, B. Parshall, and L. Scott, Derived categories and Morita theory, J. Algebra 104 (1986), no. 2, 397-409. MR 866784,
  • [FMT14] Luisa Fiorot, Francesco Mattiello, and Alberto Tonolo, A classification theorem for $ t$-structures, J. Algebra 465 (2016), 214-258. MR 3537822,
  • [H87] Dieter Happel, On the derived category of a finite-dimensional algebra, Comment. Math. Helv. 62 (1987), no. 3, 339-389. MR 910167,
  • [H88] Dieter Happel, Triangulated categories in the representation theory of finite-dimensional algebras, London Mathematical Society Lecture Note Series, vol. 119, Cambridge University Press, Cambridge, 1988. MR 935124
  • [HRS96] Dieter Happel, Idun Reiten, and Sverre O. Smalø, Tilting in abelian categories and quasitilted algebras, Mem. Amer. Math. Soc. 120 (1996), no. 575, viii+ 88. MR 1327209,
  • [HR82] Dieter Happel and Claus Michael Ringel, Tilted algebras, Trans. Amer. Math. Soc. 274 (1982), no. 2, 399-443. MR 675063,
  • [KS05] Masaki Kashiwara and Pierre Schapira, Categories and sheaves, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 332, Springer-Verlag, Berlin, 2006. MR 2182076
  • [Ke94] Bernhard Keller, Deriving DG categories, Ann. Sci. École Norm. Sup. (4) 27 (1994), no. 1, 63-102. MR 1258406
  • [M86] Yoichi Miyashita, Tilting modules of finite projective dimension, Math. Z. 193 (1986), no. 1, 113-146. MR 852914,
  • [NSZ15] P. Nicolás, M. Saorín, and A. Zvonareva, Silting objects in arbitrary triangulated categories, Preprint available at
  • [PV15] C. Psaroudakis and J. Vitória, Realisation functors in tilting theory, Preprint available at
  • [Ri89] Jeremy Rickard, Morita theory for derived categories, J. London Math. Soc. (2) 39 (1989), no. 3, 436-456. MR 1002456,
  • [Sto14] Jan Šťovíček, Derived equivalences induced by big cotilting modules, Adv. Math. 263 (2014), 45-87. MR 3239134,

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 18E30, 18E10, 18G55

Retrieve articles in all journals with MSC (2010): 18E30, 18E10, 18G55

Additional Information

Luisa Fiorot
Affiliation: Dipartimento di Matematica, Università degli studi di Padova, Via Trieste 63, 35121 Padova, Italy

Francesco Mattiello
Affiliation: Dipartimento di Matematica, Università degli studi di Padova, Via Trieste 63, 35121 Padova, Italy

Manuel Saorín
Affiliation: Departamento de Matemáticas, Universidad de Murcia, Aptdo. 4021, 30100 Espinardo, Murcia, Spain

Received by editor(s): November 21, 2015
Received by editor(s) in revised form: December 8, 2015, May 2, 2016, and June 27, 2016
Published electronically: November 21, 2016
Additional Notes: The second author was supported by Assegno di ricerca “Tilting theory in triagulated categories” del Dipartimento di Matematica dell’Università degli Studi di Padova and by Progetto di Eccellenza della fondazione Cariparo.
The third author was supported by research projects from the Spanish Ministerio de Economía y Competitividad (MTM2013-46837-P) and from the Fundación ‘Séneca’ of Murcia (19880/GERM/15), with a part of FEDER funds.
The authors thank their institutions for their help. They also thank Jorge Vitória and Alexandra Zvonareva for pointing out some omissions in a first draft of the paper.
Communicated by: Pham Huu Tiep
Article copyright: © Copyright 2016 American Mathematical Society

American Mathematical Society