Derived equivalence induced by infinitely generated 𝑛-tilting modules

Bazzoni, Silvana; Mantese, Francesca; Tonolo, Alberto

doi:10.1090/S0002-9939-2011-10900-6

Derived equivalence induced by infinitely generated $n$-tilting modules
HTML articles powered by AMS MathViewer

by Silvana Bazzoni, Francesca Mantese and Alberto Tonolo

Proc. Amer. Math. Soc. 139 (2011), 4225-4234

DOI: https://doi.org/10.1090/S0002-9939-2011-10900-6

Published electronically: April 28, 2011

PDF | Request permission

Abstract:

Let $T_R$ be a right $n$-tilting module over an arbitrary associative ring $R$. In this paper we prove that there exists an $n$-tilting module $T’_R$ equivalent to $T_R$ which induces a derived equivalence between the unbounded derived category $\mathcal {D}(R)$ and a triangulated subcategory $\mathcal E_{\perp }$ of $\mathcal {D}(\operatorname {End}(T’))$ equivalent to the quotient category of $\mathcal {D}(\operatorname {End}(T’))$ modulo the kernel of the total left derived functor $-\otimes ^{\mathbb L}_{S’}T’$. If $T_R$ is a classical $n$-tilting module, we have that $T=T’$ and the subcategory $\mathcal E_{\perp }$ coincides with $\mathcal {D}(\operatorname {End}|(T))$, recovering the classical case.

References

Leovigildo Alonso Tarrío, Ana Jeremías López, and María José Souto Salorio, Localization in categories of complexes and unbounded resolutions, Canad. J. Math. 52 (2000), no. 2, 225–247. MR 1755776, DOI 10.4153/CJM-2000-010-4
L. Angeleri Hügel and J. Sánchez. Tilting modules arising from ring epimorphisms. Algebr. Represent. Theory, 14:217–246, 2011.
Lidia Angeleri-Hügel and Jan Trlifaj, Tilting theory and the finitistic dimension conjectures, Trans. Amer. Math. Soc. 354 (2002), no. 11, 4345–4358. MR 1926879, DOI 10.1090/S0002-9947-02-03066-0
Silvana Bazzoni, A characterization of $n$-cotilting and $n$-tilting modules, J. Algebra 273 (2004), no. 1, 359–372. MR 2032465, DOI 10.1016/S0021-8693(03)00432-0
Silvana Bazzoni, Equivalences induced by infinitely generated tilting modules, Proc. Amer. Math. Soc. 138 (2010), no. 2, 533–544. MR 2557170, DOI 10.1090/S0002-9939-09-10120-X
I. N. Bernšteĭn, I. M. Gel′fand, and V. A. Ponomarev, Coxeter functors, and Gabriel’s theorem, Uspehi Mat. Nauk 28 (1973), no. 2(170), 19–33 (Russian). MR 0393065
Marcel Bökstedt and Amnon Neeman, Homotopy limits in triangulated categories, Compositio Math. 86 (1993), no. 2, 209–234. MR 1214458
A. K. Bousfield, The localization of spectra with respect to homology, Topology 18 (1979), no. 4, 257–281. MR 551009, DOI 10.1016/0040-9383(79)90018-1
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, 1980, pp. 103–169. MR 607151
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
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
Alberto Facchini, A tilting module over commutative integral domains, Comm. Algebra 15 (1987), no. 11, 2235–2250. MR 912770, DOI 10.1080/00927878708823535
Alberto Facchini, Divisible modules over integral domains, Ark. Mat. 26 (1988), no. 1, 67–85. MR 948281, DOI 10.1007/BF02386109
L. Fuchs, On divisible modules over domains, Abelian groups and modules (Udine, 1984) CISM Courses and Lect., vol. 287, Springer, Vienna, 1984, pp. 341–356. MR 789830, DOI 10.1007/978-3-7091-2814-5_{2}5
P. Gabriel and M. Zisman, Calculus of fractions and homotopy theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 35, Springer-Verlag New York, Inc., New York, 1967. MR 0210125, DOI 10.1007/978-3-642-85844-4
Rüdiger Göbel and Jan Trlifaj, Approximations and endomorphism algebras of modules, De Gruyter Expositions in Mathematics, vol. 41, Walter de Gruyter GmbH & Co. KG, Berlin, 2006. MR 2251271, DOI 10.1515/9783110199727
Enrico Gregorio and Alberto Tonolo, Weakly tilting bimodules, Forum Math. 13 (2001), no. 5, 589–614. MR 1858490, DOI 10.1515/form.2001.024
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 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
D. K. Harrison, Infinite abelian groups and homological methods, Ann. of Math. (2) 69 (1959), 366–391. MR 104728, DOI 10.2307/1970188
Robin Hartshorne, Residues and duality, Lecture Notes in Mathematics, No. 20, Springer-Verlag, Berlin-New York, 1966. Lecture notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/64; With an appendix by P. Deligne. MR 0222093, DOI 10.1007/BFb0080482
B. Keller. On the construction of triangle equivalences. Lectures Workshop Pappenheim 1994, 1994.
H. Krause. Localization theory for triangulated categories. Proceedings of “Workshop on Triangulated Categories” (Leeds, 2006), vol. 375 of London Math. Soc. Lecture Note Ser., pages 161–235. Cambridge Univ. Press, Cambridge, 2010.
Frank Lukas, Infinite-dimensional modules over wild hereditary algebras, J. London Math. Soc. (2) 44 (1991), no. 3, 401–419. MR 1149004, DOI 10.1112/jlms/s2-44.3.401
F. Mantese and A. Tonolo. Reflexivity in derived categories. Forum Math., 22(6):1161–1191, 2010.
Eben Matlis, Cotorsion modules, Mem. Amer. Math. Soc. 49 (1964), 66. MR 178025
Yoichi Miyashita, Tilting modules of finite projective dimension, Math. Z. 193 (1986), no. 1, 113–146. MR 852914, DOI 10.1007/BF01163359
Idun Reiten and Claus Michael Ringel, Infinite dimensional representations of canonical algebras, Canad. J. Math. 58 (2006), no. 1, 180–224. MR 2195596, DOI 10.4153/CJM-2006-008-1
Charles A. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge University Press, Cambridge, 1994. MR 1269324, DOI 10.1017/CBO9781139644136