Derived equivalences induced by nonclassical tilting objects

Fiorot, Luisa; Mattiello, Francesco; Saorín, Manuel

doi:10.1090/proc/13368

Derived equivalences induced by nonclassical tilting objects
HTML articles powered by AMS MathViewer

by Luisa Fiorot, Francesco Mattiello and Manuel Saorín PDF

Proc. Amer. Math. Soc. 145 (2017), 1505-1514 Request permission

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

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, DOI 10.1515/form.2001.006
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, DOI 10.1017/CBO9780511735134
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, DOI 10.1090/S0002-9939-2011-10900-6
Apostolos Beligiannis and Idun Reiten, Homological and homotopical aspects of torsion theories, Mem. Amer. Math. Soc. 188 (2007), no. 883, viii+207. MR 2327478, DOI 10.1090/memo/0883
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
Sheila Brenner and M. C. R. Butler, Generalizations of the Bernstein-Gel′fand-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
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
Xiao-Wu Chen, A short proof of HRS-tilting, Proc. Amer. Math. Soc. 138 (2010), no. 2, 455–459. MR 2557163, DOI 10.1090/S0002-9939-09-10101-6
R. R. Colby and K. R. Fuller, Tilting, cotilting, and serially tilted rings, Comm. Algebra 18 (1990), no. 5, 1585–1615. MR 1059750, DOI 10.1080/00927879008823985
Riccardo Colpi and Jan Trlifaj, Tilting modules and tilting torsion theories, J. Algebra 178 (1995), no. 2, 614–634. MR 1359905, DOI 10.1006/jabr.1995.1368
E. Cline, B. Parshall, and L. Scott, Derived categories and Morita theory, J. Algebra 104 (1986), no. 2, 397–409. MR 866784, DOI 10.1016/0021-8693(86)90224-3
Luisa Fiorot, Francesco Mattiello, and Alberto Tonolo, A classification theorem for $t$-structures, J. Algebra 465 (2016), 214–258. MR 3537822, DOI 10.1016/j.jalgebra.2016.07.008
Dieter Happel, On the derived category of a finite-dimensional algebra, Comment. Math. Helv. 62 (1987), no. 3, 339–389. MR 910167, DOI 10.1007/BF02564452
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, DOI 10.1017/CBO9780511629228
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, DOI 10.1090/memo/0575
Dieter Happel and Claus Michael Ringel, Tilted algebras, Trans. Amer. Math. Soc. 274 (1982), no. 2, 399–443. MR 675063, DOI 10.1090/S0002-9947-1982-0675063-2
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, DOI 10.1007/3-540-27950-4
Bernhard Keller, Deriving DG categories, Ann. Sci. École Norm. Sup. (4) 27 (1994), no. 1, 63–102. MR 1258406
Yoichi Miyashita, Tilting modules of finite projective dimension, Math. Z. 193 (1986), no. 1, 113–146. MR 852914, DOI 10.1007/BF01163359
P. Nicolás, M. Saorín, and A. Zvonareva, Silting objects in arbitrary triangulated categories, Preprint available at http://arxiv.org/pdf/1512.04700v3.pdf.
C. Psaroudakis and J. Vitória, Realisation functors in tilting theory, Preprint available at http://arxiv.org/pdf/1511.02677v1.pdf
Jeremy Rickard, Morita theory for derived categories, J. London Math. Soc. (2) 39 (1989), no. 3, 436–456. MR 1002456, DOI 10.1112/jlms/s2-39.3.436
Jan Šťovíček, Derived equivalences induced by big cotilting modules, Adv. Math. 263 (2014), 45–87. MR 3239134, DOI 10.1016/j.aim.2014.06.007