Abstract
We use recollement and HRS-tilt to describe bounded t-structures on the bounded derived category \(\mathcal {D}^{b}(\mathbb {X})\) of coherent sheaves over a weighted projective line \(\mathbb {X}\) of domestic or tubular type. We will see from our description that the combinatorics in the classification of bounded t-structures on \(\mathcal {D}^{b}(\mathbb {X})\) can be reduced to that in the classification of bounded t-structures on the bounded derived categories of finite dimensional right modules over representation-finite finite dimensional hereditary algebras.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Aihara, T., Iyama, O.: Silting mutation in triangulated categories. J. Lond. Math. Soc. (2) 85(3), 633–668 (2012)
Assem, I., Salorio, M.J.S., Trepode, S.: Ext-projectives in suspended subcategories. J. Pure Appl. Algebra 212(2), 423–434 (2008)
Auslander, M.: Functors and morphisms determined by objects. In: Representation Theory of Algebras Proc. Conf. Lecture Notes in Pure Appl. Math., 37, Dekker, New York, 1978, pp 1–244. Temple Univ., Philadelphia (1976)
Auslander, M., Reiten, I., Smalø, S.O.: Representation Theory of Artin Algebras, vol. 36. Cambridge University Press (1997)
Auslander, M., Smalø, S.O.: Almost split sequences in subcategories. J. Algebra 71(2), 592–594 (1981)
Beilinson, A.A.: On the Derived Category of Perverse Sheaves. K-theory, arithmetic and geometry (Moscow, 1984-1986), pp. 27–41, Lecture Notes in Math. 1289. Springer, Berlin (1987)
Beilinson, A.A., Bernstein, J., Deligne, P.: Faisceaux pervers. In: Analysis and Topology on Singular Spaces, I (Luminy, 1981) Asteŕisque, vol. 100, pp 5–171. Soc. Math. France, Paris (1982)
Beilinson, A., Ginzburg, V., Soergel, W.: Koszul duality patterns in representation theory. J. Amer. Math. Soc. 9 (♯2), 473–527 (1996)
Bondal, A.I.: Representation of associative algebras and coherent sheaves. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 53(1), 25–44 (1989). translation in Math. USSR-Izv. 34 (1989), no. 1, 23–42
Bondal, A.I., Kapranov, M.M.: Representable functors, Serre functors, and reconstructions. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 53(6), 1183–1205, 1337 (1989). translation in Math. USSR-Izv. 35 (1989), no. 3, 519–541
Bondal, A.I., van den Bergh, M.: Generators and representability of functors in commutative and noncommutative geometry. Mosc. Math. J. 3(1), 1–36,258 (2003)
Bridgeland, T.: Stability conditions on triangulated categories. Ann. Math. (2) 166(2), 317–345 (2007)
Chen, X.-W., Han, Z., Zhou, Y.: Derived equivalences via HRS-tilting. Adv. Math. 354(106749), 26 (2019)
Chen, X.-W., Krause, H.: Introduction to coherent sheaves on weighted projective lines. arXiv:0911.4473 (2009)
Chen, X.-W.: A short proof of HRS-tilting. Proc. Amer. Math. Soc. 138(2), 455–459 (2010)
Chen, X.-W., Ringel, C.M.: Hereditary triangulated categories. J. Noncommut. Geom. 12(4), 1425–1444 (2018)
Geigle, W., Lenzing, H.: A class of weighted projective curves arising in representation theory of finite-dimensional algebras. In: Singularities, Representation of Algebras, and Vector Bundles (Lambrecht, 1985). Lecture Notes in Math., vol. 1273, pp 265–297. Springer, Berlin (1987)
Geigle, W., Lenzing, H.: Perpendicular categories with applications to representations and sheaves. J. Algebra 144(2), 273–343 (1991)
Gorodentscev, A., Kuleshov, S., Rudakov, A.: Stability data and t-structures on a triangulated category. arXiv:math/0312442 (2003)
Happel, D.: A characterization of hereditary categories with tilting object. Invent. Math. 144(2), 381–398 (2001)
Happel, D.: Triangulated categories in the representation of finite dimensional algebras, vol. 119. Cambridge University Press (1988)
Happel, D., Reiten, I., Smalø, S.O.: Tilting in Abelian categories and quasitilted algebras. Mem. Amer. Math. Soc. 120(575), viii+ 88 (1996)
Hübner, T.: Classification of Indecomposable Vector Bundles on Weighted Curves. Diplomarbeit, Paderborn (1989)
Hübner, T.: Exzeptionelle Vektorbündel und Reflektionen an Kippgarben über projectiven gewichteten Kurven. Dissertion, Universität Paderborn (1996)
Hübner, T., Lenzing, H.: Categories Perpendicular to Exceptional Bundles. Preprint, Paderborn (1993)
Keller, B.: On triangulated orbit categories. Doc. Math 10.551–581, 21–56 (2005)
Keller, B., Vossieck, D.: Aisles in derived categories. Bull. Soc. Math. Belg. Sér. A 40(2), 239–253 (1988)
Koenig, S., Yang, D.: Silting objects, simple-minded collections, t-structures and co-t-structures for finite-dimensional algebras. Doc. Math. 19, 403–438 (2014)
Kussin, D., Lenzing, H., Meltzer, H.: Triangle singularities, ADE-chains, and weighted projective lines. Adv. Math. 237, 194–251 (2013)
Lenzing, H.: Weighted projective lines and applications. In: Representations of Algebras and Related Topics, pp 153–187. European Mathematical Society (2011)
Lenzing, H.: Hereditary categories. In: Handbook of Tilting Theory, London Math. Soc., vol. 332, pp 105–146. Lecture Note Ser (2007)
Lenzing, H., Meltzer, H.: Tilting sheaves and concealed-canonical algebras. Representation theory of algebras (Cocoyoc 1994) 18, 455–473 (1996)
Lenzing, H., Meltzer, H.: Sheaves on a weighted projective line of genus one, and representations of a tubular algebra. Representations of algebras (Ottawa, ON), 313–337. CMS Conf. Proc. vol. 14 (1992)
Lenzing, H., de la Peña, J.A.: Wild canonical algebras. Math. Z. 224(3), 403–425 (1997)
Liu, Q.-H., Vitória, J.: T-structures via recollements for piecewise hereditary algebras. J. Pure Appl. Algebra 216(4), 837–849 (2012)
Liu, Q.-H., Vitória, J., Yang, D.: Gluing silting objects. Nagoya Math. J. 216, 117–151 (2014)
Meltzer, H.: Exceptional vector bundles, tilting sheaves and tilting complexes for weighted projective lines. Mem. Amer. Math. Soc., 171(808) (2004)
Meltzer, H.: Tubular mutations. Colloq. Math., 74(2) (1997)
Okonek, C., Schneider, M., Spindler, H.: Vector Bundles on Complex Projective Spaces, with an Appendix by S. I. Gelfand. Corrected reprint of the 1988 Edition. Birkhäuser, Boston (2011)
Polishchuk, A.: Constant families of t-structures on derived categories of coherent sheaves. Mosc. Math. J. 7.1, 109–134 (2007)
Reiten, I., Van den Bergh, M.: Noetherian hereditary abelian categories satisfying Serre duality. J. Am. Math. Soc. 15.2, 295–366 (2002)
Rickard, J.: Equivalences of derived categories for symmetric algebras. J. Algebra 257(2), 460–481 (2002)
Ringel, C.M.: Tame Algebras and Integral Quadratic Forms. In: Lecture Notes in Mathematics, vol. 1099. Springer, Berlin (1984)
Stanley, D., van Roosmalen, A.C.: Derived equivalences for hereditary Artin algebras. Adv. Math. 303, 415–463 (2016)
Niven, I.M.: Diophantine Approximations. Reprint of the 1963 Original. Dover Publications, Inc., Mineola (2008). ix+ 68 pp.
Wei, J.Q.: Semi-tilting complexes. Israel J. Math. 194(2), 871–893 (2013)
Acknowledgements
The question of this article originated from a seminar on Bridgeland’s stability conditions organized by Prof. Xiao-Wu Chen, Prof. Mao Sheng and Prof. Bin Xu. I thank these organizers who gave me the opportunity to report. I am grateful to the participants for their patience and critical questions. Thanks are once again due to Prof. Xiao-Wu Chen, my supervisor, for his guidance and kindness.
I thank Peng-Jie Jiao for discussion, thank Prof. Helmut Lenzing for carefully answering my question on stable bundles over a tubular weighted projective line, thank Prof. Zeng-Qiang Lin for communication on realization functors, thank Prof. Hagen Meltzer for his lectures on weighted projective lines, thank Prof. Dong Yang for explaning the results in [28] and for stimulating conversations, and thank Prof. Pu Zhang for a series of lectures on triangulated categories based on his newly-written book titled “triangulated categories and derived categories” (in Chinese).
Moreover, I would like to express my deep gratitude to two anonymous referees. The referee of the first version of this article gave me a long list of suggestions, which pointed out many mistakes and inaccuracies and helped in improving the exposition and in reshaping some parts of this article. Another referee of the second version of this article also poinited out several mistakes, many typos and grammar problems.
This work is supported by the National Science Foundation of China (No. 11522113 and No. 115771329) and also by the Fundamental Research Funds for the Central Unviersities.
Author information
Authors and Affiliations
Corresponding author
Additional information
Presented by: Henning Krause.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Sun, C. Bounded t-Structures on the Bounded Derived Category of Coherent Sheaves over a Weighted Projective Line. Algebr Represent Theor 23, 2167–2235 (2020). https://doi.org/10.1007/s10468-019-09929-w
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10468-019-09929-w