Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-26T06:43:17.489Z Has data issue: false hasContentIssue false
  • English
  • Français

Report on the finiteness of silting objects

Part of: Representation theory of rings and algebras

Published online by Cambridge University Press:  07 May 2021

Takuma Aihara ,
Takahiro Honma ,
Kengo Miyamoto  and
Qi Wang
Takuma Aihara
Affiliation:
Department of Mathematics, Tokyo Gakugei University, 4-1-1 Nukuikita-machi, Koganei, Tokyo184-8501, Japan (aihara@u-gakugei.ac.jp)
Takahiro Honma
Affiliation:
Graduate School of Mathematics, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku, Tokyo162-8601, Japan (1119704@ed.tus.ac.jp)
Kengo Miyamoto
Affiliation:
Department of Computer and Information Science, Ibaraki University, 4-12-1, Nakanarusawa-cho, Hitachi, Ibaraki 316-8511, Japan (kengo.miyamoto.uz63@vc.ibaraki.ac.jp)
Qi Wang
Affiliation:
Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Suita, Osaka565-0871, Japan (q.wang@ist.osaka-u.ac.jp)
Get access Rights & Permissions [Opens in a new window]

Abstract

We discuss the finiteness of (two-term) silting objects. First, we investigate new triangulated categories without silting object. Second, we study two classes of $\tau$-tilting-finite algebras and give the numbers of their two-term silting objects. Finally, we explore when $\tau$-tilting-finiteness implies representation-finiteness and obtain several classes of algebras in which a $\tau$-tilting-finite algebra is representation-finite.

Keywords

silting objectsupport τ-tilting moduleτ-tilting-finite

MSC classification

Primary: 16G20: Representations of quivers and partially ordered sets
Secondary: 16G60: Representation type (finite, tame, wild, etc.)
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 64 , Issue 2 , May 2021 , pp. 217 - 233
Check for updates
Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Adachi, T., Characterizing $\tau$-tilting finite algebras with radical square zero, Proc. Amer. Math. Soc. 144(11) (2016), 4673–4685.CrossRefGoogle Scholar
Adachi, T., The classification of $\tau$-tilting modules over Nakayama algebras, J. Algebra 452 (2016), 227–262.CrossRefGoogle Scholar
Adachi, T., Iyama, O. and Reiten, I., $\tau$-tilting theory, Compos. Math. 150(3) (2014), 415–452.CrossRefGoogle Scholar
Adachi, T., Aihara, T. and Chan, A., Classification of two-term tilting complexes over Brauer graph algebras, Math. Z. 290(1–2) (2018), 1–36.CrossRefGoogle Scholar
Aihara, T. and Iyama, O., Silting mutation in triangulated categories, J. Lond. Math. Soc. (2) 85(3) (2012), 633–668.CrossRefGoogle Scholar
Aihara, T. and Mizuno, Y., Classifying tilting complexes over preprojective algebras of Dynkin type, Algeb. Num. Theory 11(6) (2017), 1287–1315.CrossRefGoogle Scholar
Assem, I. and Skowroński, A., Quadratic forms and iterated tilted algebras, J. Algebra 128(1) (1990), 55–85.CrossRefGoogle Scholar
Assem, I., Happel, D. and Roldán, O., Representation-finite trivial extension algebras, J. Pure Appl. Algebra 33(3) (1984), 235–242.CrossRefGoogle Scholar
Assem, I., Simson, D. and Skowroński, A., Elements of the representation theory of associative algebras. Vol. 1. Techniques of representation theory, London Mathematical Society Student Texts, Volume 65 (Cambridge University Press, Cambridge, 2006).CrossRefGoogle Scholar
Auslander, M., Reiten, I. and Smalø, S. O., Representation theory of Artin algebras, Cambridge Studies in Advanced Mathematics, Volume 36 (Cambridge University Press, Cambridge, 1995).CrossRefGoogle Scholar
Bautista, R., Classification of certain algebras of finite representation type, An. Int. Mat. Univ. Nac. Autónoma México 22 (1982), 1–82.Google Scholar
Białkowski, J. and Skowroński, A., On tame weakly symmetric algebras having only periodic modules, Arch. Math. (Basel) 81(2) (2003), 142–154.CrossRefGoogle Scholar
Białkowski, J. and Skowroński, A., Socle deformations of self-injective algebras of tubular type, J. Math. Soc. Japan 56(3) (2004), 687–716.CrossRefGoogle Scholar
Białkowski, J., Holm, T. and Skowroński, A., Derived equivalences for tame weakly symmetric algebras having only periodic modules, J. Algebra 269(2) (2003), 652–668.CrossRefGoogle Scholar
Coelho, F. U. and Happel, D., Quasitilted algebras admit a preprojective component, Proc. Am. Math. Soc. 125(5) (1997), 1283–1291.CrossRefGoogle Scholar
Demonet, L., Iyama, O., Reading, N., Reiten, I. and Thomas, H., Lattice theory of torsion classes. Preprint (2017), arXiv:1711.01785.Google Scholar
Eisele, F., Janssens, G. and Raedschelders, T., A reduction theorem for $\tau$-rigid modules, Math. Z. 290(3–4) (2018), 1377–1413.CrossRefGoogle Scholar
Escolar, E. G. and Hiraoka, Y., Persistence modules on commutative ladders of finite type, Discrete Comput. Geom. 55(1) (2016), 100–157.CrossRefGoogle Scholar
Happel, D., Tilting sets on cylinders, Proc. London Math. Soc. (3) 51(1) (1985), 21–55.CrossRefGoogle Scholar
Happel, D. and Vossieck, D., Minimal algebras of infinite representation type with preprojective component, Manuscripta Math. 42(2–3) (1983), 221–243.CrossRefGoogle Scholar
Iyama, O. and Yang, D., Silting reduction and Calabi-Yau reduction of triangulated categories, Trans. Amer. Math. Soc. 370(11) (2018), 7861–7898.CrossRefGoogle Scholar
Leszczyński, Z., The completely separating incidence algebras of tame representation type, Colloq. Math. 94(2) (2002), 243–262.CrossRefGoogle Scholar
Leszczyński, Z. and Simson, D., On triangular matrix rings of finite representation type, J. Lond. Math. Soc. (2) 20(3) (1979), 396–402.CrossRefGoogle Scholar
Malicki, P. and Skowroński, A., Cycle-finite algebras with finitely many $\tau$-rigid indecomposable modules, Comm. Algebra 44(5) (2016), 2048–2057.CrossRefGoogle Scholar
Martínez-Villa, R., Algebras stably equivalent to $\ell$-hereditary, in Representation theory, II (Proc. Second Internat. Conf., Carleton Univ., Ottawa, Ont., 1979), pp. 396–431, Lecture Notes in Math., Volume 832 (Springer, Berlin, 1980).CrossRefGoogle Scholar
Mizuno, Y., Classifying $\tau$-tilting modules over preprojective algebras of Dynkin type, Math. Z. 277(3–4) (2014), 665–690.CrossRefGoogle Scholar
Mousavand, K., $\tau$-tilting finiteness of biserial algebras. Preprint (2019), arXiv:1904.11514.Google Scholar
Plamondon, P.-G., $\tau$-tilting finite gentle algebras are representation-finite, Pacific J. Math. 302(2) (2019), 709–716.CrossRefGoogle Scholar
Ringel, C. M., Tame algebras and integral quadratic forms, Lecture Notes in Mathematics, Volume 1099 (Springer-Verlag, Berlin, 1984).CrossRefGoogle Scholar
Simson, D. and Skowroński, A., Elements of the representation theory of associative algebras. Vol. 2. Tubes and concealed algebras of Euclidean type, London Mathematical Society Student Texts, Volume 71 (Cambridge University Press, Cambridge, 2007).Google Scholar
Skowroński, A., Minimal representation-infinite Artin algebras, Math. Proc. Cambridge Philos. Soc. 116(2) (1994), 229–243.CrossRefGoogle Scholar
Skowroński, A., Selfinjective algebras: Finite and tame type, in Trends in representation theory of Algebras and related topics, pp. 69–238, Contemp. Math., Volume 406 (American Mathematical Society, 2006).CrossRefGoogle Scholar
Wang, Q., On $\tau$-tilting finite simply connected algebras. Preprint (2019), arXiv:1910.01937.Google Scholar
Zito, S., $\tau$-tilting finite tilted and cluster-tilted algebras, Proc. Edinb. Math. Soc. (2) 63(4) (2020), 950–955.10.1017/S0013091520000255CrossRefGoogle Scholar