Abstract
Extriangulated category was introduced by H. Nakaoka and Y. Palu to give a unification of properties in exact categories and triangulated categories. A notion of tilting (resp., cotilting) subcategories in an extriangulated category is defined in this paper. We give a Bazzoni characterization of tilting (resp., cotilting) subcategories and obtain an Auslander-Reiten correspondence between tilting (resp., cotilting) subcategories and coresolving covariantly (resp., resolving contravariantly) finite subcatgories which are closed under direct summands and satisfy some cogenerating (resp., generating) conditions. Applications of the results are given: we show that tilting (resp., cotilting) subcategories defined here unify many previous works about tilting modules (subcategories) in module categories of Artin algebras and in abelian categories admitting a cotorsion triples; we also show that the results work for the triangulated categories with a proper class of triangles introduced by A. Beligiannis.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Angeleri Hügel L, Happel D, Krause H. Handbook of Tilting Theory. London Math Soc Lecture Note Ser, Vol 332. Cambridge: Cambridge Univ Press, 2007
Assem I, Simson D, Skowronski A. Elements of the Representation Theory of Associative Algebras. Vol 1: Techniques of Representation Theory. London Math Soc Stud Texts, Vol 65. Cambridge: Cambridge Univ Press, 2006
Auslander M. Coherent functors. In: Eilenberg S, Harrison D K, MacLane S, Röhrl H, eds. Proceedings of the Conference on Categorical Algebra. Berlin: Springer, 1965, 189–231
Auslander M, Buchweitz R O. The homological theory of maximal Cohen-Macaulay approximations. Mém Soc Math Fr, 1989, 38: 5–37
Auslander M, Platzeck M I, Reiten I. Coxeter functors without diagrams. Trans Amer Math Soc, 1979, 250: 1–46
Auslander M, Reiten I. Applications of contravariantly finite subcategories. Adv Math, 1991, 86(1): 111–152
Auslander M, Reiten I, Smalø S O. Representation Theory of Artin Algebras. Cambridge Stud Adv Math, Vol 36. Cambridge: Cambridge Univ Press, 1997
Auslander M, Solberg O. Relative homology and representation theory II: Relative cotilting theory. Comm Algebra, 1993, 21(9): 3033–3079
Bazzoni S. A characterization of n-cotilting and n-tilting modules. J Algebra, 2004, 273(1): 359–372
Beligiannis A. Relative homological algebra and purity in triangulated categories. J Algebra, 2000, 227(1): 268–361
Bernstein I N, Gelfand I M, Ponomarev V A. Coxeter functors and Gabriel’s theorem. Uspichi Mat Nauk, 1973, 28: 19–33 (in Russian); Russian Math Surveys, 1973, 28: 17-32
Brenner S, Butler M C R. Generalizations of the Bernstein-Gelfand-Ponomarev reflection functors. In: Dlab V, Gabriel P, eds. Representation Theory II. Lecture Notes in Math, Vol 832. Berlin: Springer, 1980, 103–169
Chen X W. Homotopy equivalences induced by balanced pairs. J Algebra, 2010, 324(10): 2718–2731
Colpi R, Fuller K. Tilting objects in abelian categories and quasitilted rings. Trans Amer Math Soc, 2007, 359(2): 741–765
Colpi R, Trlifaj J. Tilting modules and tilting torsion theories. J Algebra, 1995, 178(2): 614–634
Di Z, Wei J, Zhang X, Chen J. Tilting subcategories with respect to cotorsion triples in abelian categories. Proc Roy Soc Edinburgh Sect A, 2017, 147(4): 703–726
Dr¨axler P, Reiten I, Smalø S, Keller B. Exact categories and vector space categories. Trans Amer Math Soc, 1999, 351(2): 647–682
Enomoto H. Classifying exact categories via Wakamatsu tilting. J Algebra, 2017, 485: 1–44
Happel D. Triangulated Categories in the Representation of Finite Dimensional Algebras. London Math Soc Lecture Note Ser, Vol 119. Cambridge: Cambridge Univ Press, 1988
Happel D, Ringel C M. Tilted algebras. Trans Amer Math Soc, 1982, 274(2): 399–443
Happel D, Unger L. On a partial order of tilting modules. Algebra Represent Theory, 2005, 8(2): 147–156
Hu J, Zhang D, Zhou P. Proper classes and Gorensteinness in extriangulated categories. arXiv: 1906.10989
Iyama O, Nakaoka H, Palu Y. Auslander-Reiten theory in extriangulated categories. arXiv: 1805.03776
Liu Y, Nakaoka H. Hearts of twin cotorsion pairs on extriangulated categories. J Algebra, 2019, 528: 96–149
Miyashita Y. Tilting modules of finite projective dimension. Math Z, 1986, 193(1): 113–146
Nakaoka H, Palu Y. Extriangulated categories, Hovey twin cotorsion pairs and model structures. Cah Topol Geom Differ Categ, 2019, 60(2): 117–193
Rickard J. Morita theory for derived categories. J Lond Math Soc, 1989, 2(3): 436–456
Zhao T, Huang Z. Phantom ideals and cotorsion pairs in extriangulated categories. Taiwanese J Math, 2019, 23(1): 29–61
Zhou P, Zhu B. Triangulated quotient categories revisited. J Algebra, 2018, 502: 196–232
Acknowledgements
The authors would like to thank Dr. Tiwei Zhao for helpful comments on the paper, and they also would like to thank the referees for reading the paper carefully and for many suggestions on mathematical and English expressions. This work was supported by the National Natural Science Foundation of China (Grant No. 11671221).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Zhu, B., Zhuang, X. Tilting subcategories in extriangulated categories. Front. Math. China 15, 225–253 (2020). https://doi.org/10.1007/s11464-020-0811-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11464-020-0811-7