Abstract
Lin and Xi introduced Auslander–Dlab–Ringel (ADR) algebras of semilocal modules as a generalization of original ADR algebras and showed that they are quasi-hereditary. In this paper, we prove that such algebras are always left-strongly quasi-hereditary. As an application, we give a better upper bound for the global dimension of ADR algebras of semilocal modules. Moreover, we describe characterizations of original ADR algebras to be strongly quasi-hereditary.
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References
Auslander, M.: Representation Dimension of Artin Algebras. Queen Mary College, London (1971)
Cline, E., Parshall, B., Scott, L.: Finite-dimensional algebras and highest weight categories. J. Reine Angew. Math. 391, 85–99 (1988)
Conde, T.: The quasihereditary structure of the Auslander–Dlab–Ringel algebra. J. Algebra 460, 181–202 (2016)
Conde, T.: \(\Delta \)-filtrations and projective resolutions for the Auslander–Dlab–Ringel algebra. Algebr. Represent. Theory 21(3), 605–625 (2018)
Conde, T., Erdmann, K.: The Ringel dual of the Auslander–Dlab–Ringel algebra. J. Algebra 504, 506–535 (2018)
Coulembier, K.: Ringel duality and Auslander-Dlab-Ringel algebras. J. Pure Appl. Algebra 222(12), 3831–3848 (2018)
Dlab, V., Ringel, C .M.: Auslander algebras as quasi-hereditary algebras. J. Lond. Math. Soc. (2) 39(3), 457–466 (1989)
Dlab, V., Ringel, C.M.: Every semiprimary ring is the endomorphism ring of a projective module over a quasihereditary ring. Proc. Am. Math. Soc. 107(1), 1–5 (1989)
Dlab, V., Ringel, C.M.: Quasi-hereditary algebras. Ill. J. Math. 33(2), 280–291 (1989)
Dlab, V., Ringel, C.M.: The module theoretical approach to quasi-hereditary algebras. In: Representations of Algebras and Related Topics (Kyoto, 1990), London Math. Soc. Lecture Note Ser., vol. 168, pp. 200–224. Cambridge University Press, Cambridge (1992)
Iyama, O.: Finiteness of representation dimension. Proc. Am. Math. Soc. 131(4), 1011–1014 (2003)
Iyama, O.: Rejective subcategories of Artin algebras and orders. arXiv:math/0311281
Kalck, M., Karmazyn, J.: Ringel duality for certain strongly quasi-hereditary algebras. Eur. J. Math. (to appear). arXiv:1711.00416
Lin, Y.N., Xi, C.C.: Semilocal modules and quasi-hereditary algebras. Arch. Math. (Basel) 60(6), 512–516 (1993)
Ringel, C.M.: Iyama’s finiteness theorem via strongly quasi-hereditary algebras. J. Pure Appl. Algebra 214(9), 1687–1692 (2010)
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, pp. 271–281. Amer. Math. Soc., Providence (1987)
Smalø, S.O.: Global dimension of special endomorphism rings over Artin algebras. Ill. J. Math. 22(3), 414–427 (1978)
Tsukamoto, M.: Strongly quasi-hereditary algebras and rejective subcategories. Nagoya Math. J. (to appear). arXiv:1705.03279
Acknowledgements
The author wishes to express her sincere gratitude to Takahide Adachi and Professor Osamu Iyama. The author thanks Teresa Conde and Aaron Chan for informing her about the reference [17, Proposition 2], which greatly shortens her original proof.
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Tsukamoto, M. On an upper bound for the global dimension of Auslander–Dlab–Ringel algebras. Arch. Math. 112, 41–51 (2019). https://doi.org/10.1007/s00013-018-1226-5
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DOI: https://doi.org/10.1007/s00013-018-1226-5