Abstract
We study possible values of the global dimension of endomorphism algebras of 2-term silting complexes. We show that for any algebra A whose global dimension gl.dim A ≤ 2 and any 2-term silting complex P in the bounded derived category D b(A) of A, the global dimension of \(\text {End}_{{D^b(A)}}(\mathbf {P})\) is at most 7. We also show that for each n > 2, there is an algebra A with gl.dim A = n such that D b(A) admits a 2-term silting complex P with \(\mathrm {gl. dim~}\text {End}_{{D^b(A)}}(\mathbf {P})\) infinite.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Adachi, T., Iyama, O., Reiten, I.: τ −tilting theory. Compos. Math. 150(3), 415–452 (2015)
Aihara, T., Iyama, O.: Silting mutation in triangulated categories. J. Lond. Math. Soc. (2) 85(3), 633–668 (2012)
Assem, I., Simson, D., Skowronski, A.: Elements of the representation theory of associative algebras. In: Techniques of Representation Theory, London Mathematical Society Student Texts, vol. 1, p 65. Cambridge University Press, Cambridge (2006)
Brüstle, T., Yang, D.: Ordered Exchange Graphs, Advances in representation theory of algebras, pp 135-193. EMS Series of Congress Report, European Mathematical Society, Zürich (2013)
Buan, A.B., Zhou, Y.: A silting theorem. J. Pure Appl. Algebra 220(7), 2748–2770 (2016)
Buan, A.B., Zhou, Y.: Silted algebras. Adv. Math. 303, 859–887 (2016)
Coelho, F.U., Lanzilotta, M.A.: Algebras with small homological dimensions. Manuscripta Math. 100, 1–11 (1999)
Happel, D.: Triangulated categories in the representation theory of finite dimensional algebras, London Mathematical Society Lecture Note Series, p 119. Cambridge University Press, Cambridge (1988)
Happel, D., Reiten, I., Smalø, S.O.: Tilting in abelian categories and quasitilted algebra. Mem. Amer. Math. Soc. 120, 575 (1996)
Iyama, O., Yoshino, Y.: Mutation in triangulated categories and rigid Cohen-Macaulay modules. Invent. Math. 172(1), 117–168 (2008)
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)
Rickard, J.: Morita theory for derived categories. J. London Math. Soc. (2) 39 (3), 436–456 (1989)
Acknowledgments
This work was supported by FRINAT grant number 231000, from the Norwegian Research Council. Support by the Institut Mittag-Leffler (Djursholm, Sweden) is gratefully acknowledged.
Author information
Authors and Affiliations
Corresponding author
Additional information
Presented by Steffen Koenig.
Rights and permissions
About this article
Cite this article
Buan, A., Zhou, Y. Endomorphism Algebras of 2-term Silting Complexes. Algebr Represent Theor 21, 181–194 (2018). https://doi.org/10.1007/s10468-017-9709-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10468-017-9709-9