Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-10T08:36:34.205Z Has data issue: false hasContentIssue false
  • English
  • Français

Characterizations of right rejective chains

Part of: Representation theory of rings and algebras

Published online by Cambridge University Press:  14 July 2021

Mayu Tsukamoto
Mayu Tsukamoto*
Affiliation:
Graduate school of Sciences and Technology for Innovation, Yamaguchi University, 1677-1 Yoshida, Yamaguchi753-8512, Japan
*
e-mail: tsukamot@yamaguchi-u.ac.jp
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we give characterizations of the category of finitely generated projective modules having a right rejective chain. By focusing on the characterizations, we give sufficient conditions for right rejective chains to be total right rejective chains. Moreover, we prove that Nakayama algebras with heredity ideals, locally hereditary algebras and algebras of global dimension at most two satisfy the sufficient conditions. As an application, we show that these algebras are right-strongly quasi-hereditary algebras.

Keywords

Right rejective subcategoriescoreflective subcategoriesright-strongly quasi-hereditary algebrasneat algebras

MSC classification

Primary: 16G10: Representations of Artinian rings
Type
Article
Information
Canadian Mathematical Bulletin , Volume 65 , Issue 3 , September 2022 , pp. 557 - 570
Check for updates
Copyright
© Canadian Mathematical Society 2021

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

Ágoston, I., Dlab, V., and Wakamatsu, T., Neat algebras. Comm. Algebra 19(1991), no. 2, 433–442.Google Scholar
Anderson, F. W. and Fuller, K. R., Rings and categories of modules . Graduate Texts in Mathematics, 13, Springer-Verlag, Berlin, 1992.Google Scholar
Auslander, M., Platzeck, M. I., and Todorov, G., Homological theory of idempotent ideals. Trans. Amer. Math. Soc. 332(1992), no. 2, 667–692.CrossRefGoogle Scholar
Burgess, W. D. and Fuller, K. R., On quasihereditary rings. Proc. Amer. Math. Soc. 106(1989), no. 2, 321–328.CrossRefGoogle Scholar
Cline, E., Parshall, B., and Scott, L., Finite-dimensional algebras and highest weight categories, J. Reine Angew. Math. 391(1988), 85–99.Google Scholar
Cline, E., Parshall, B., and Scott, L., Stratifying endomorphism algebras. Mem. Amer. Math. Soc. 124(1996), no. 591, viii+119pp.Google Scholar
Conde, T., The quasihereditary structure of the Auslander-Dlab-Ringel algebra. J. Algebra 460(2016), 181–202.CrossRefGoogle Scholar
Coulembier, K., Ringel duality and Auslander-Dlab-Ringel algebras, J. Pure Appl. Algebra 222(2018), no. 12, 3831–3848.CrossRefGoogle Scholar
Dlab, V. and Ringel, C. M., Quasi-hereditary algebras. Illinois J. Math. 33(1989), no. 2, 280–291.CrossRefGoogle Scholar
Geiss, C., Leclerc, B., and Schröer, J., Cluster algebra structures and semicanonical bases for unipotentgroups. arXiv:math/0703039v4 Google Scholar
Gustafson, W. H., Global dimension in serial rings. J. Algebra 97(1985), no. 1, 14–16.CrossRefGoogle Scholar
Iyama, O., A generalization of rejection lemma of Drozd-Kirichenko. J. Math. Soc. Japan 50(1998), no. 3, 697–718.Google Scholar
Iyama, O., Finiteness of representation dimension. Proc. Amer. Math. Soc. 131(2003), no. 4, 1011–1014.CrossRefGoogle Scholar
Iyama, O., $\tau$ -categories. II. Nakayama pairs and rejective subcategories. Algebr. Represent. Theory 8(2005), no. 4, 449–477.10.1007/s10468-005-0969-4CrossRefGoogle Scholar
Iyama, O., Rejective subcategories of artin algebras and orders. arXiv:math/0311281 Google Scholar
Kalck, M. and Karmazyn, J., Ringel duality for certain strongly quasi-hereditary algebras. Eur. J. Math. 4(2018), no. 3, 1100–1140.CrossRefGoogle Scholar
Martinez-Villa, R., Algebras stably equivalent to ℓ-hereditary. Representation theory, II (Proc. Second Internat. Conf., Carleton Univ., Ottawa, Ont., 1979), Lecture Notes in Math., no. 832, Springer, Berlin, 1980, 396–431.CrossRefGoogle Scholar
Ringel, C. M., Iyama’s finiteness theorem via strongly quasi-hereditary algebras. J. Pure Appl. Algebra 214(2010), no. 9, 1687–1692.10.1016/j.jpaa.2009.12.012CrossRefGoogle Scholar
Rump, W., The category of lattices over a lattice-finite ring. Algebr. Represent. Theory 8(2005), no. 3, 323–345.CrossRefGoogle Scholar
Scott, L., Simulating algebraic geometry with algebra. I. The algebraic theory of derived categories. In: The Arcata Conference on Representations of Finite Groups (Arcata, Calif., 1986), Proc. Sympos. Pure Math., 47, Part 2, Amer. Math. Soc., Providence, RI, 1987, pp. 271–281.CrossRefGoogle Scholar
Tsukamoto, M., On an upper bound for the global dimension of Auslander–Dlab–Ringel algebras. Arch. Math. (Basel) 112(2019), no. 1, 41–51.CrossRefGoogle Scholar
Tsukamoto, M., Strongly quasi-hereditary algebras and rejective subcategories. Nagoya Math. J. 237(2020), 10–38.CrossRefGoogle Scholar