Hostname: page-component-8448b6f56d-cfpbc Total loading time: 0 Render date: 2024-04-24T11:31:46.647Z Has data issue: false hasContentIssue false
  • English
  • Français

THE FINITISTIC DIMENSION AND CHAIN CONDITIONS ON IDEALS

Part of: Homological methods

Published online by Cambridge University Press:  03 November 2020

JUNLING ZHENG  and
ZHAOYONG HUANG
JUNLING ZHENG
Affiliation:
Department of Mathematics, China Jiliang University, Hangzhou310018, Zhejiang Province, P.R. China Department of Mathematics, Nanjing University, Nanjing210093, Jiangsu Province, P.R. China e-mail: zjlshuxue@163.com
ZHAOYONG HUANG
Affiliation:
Department of Mathematics, Nanjing University, Nanjing210093, Jiangsu Province, P.R. China e-mail: huangzy@nju.edu.cn
Get access Rights & Permissions [Opens in a new window]

Abstract

Let Λ be an artin algebra and $0=I_{0}\subseteq I_{1} \subseteq I_{2}\subseteq\cdots \subseteq I_{n}$ a chain of ideals of Λ such that $(I_{i+1}/I_{i})\rad(\Lambda/I_{i})=0$ for any $0\leq i\leq n-1$ and $\Lambda/I_{n}$ is semisimple. If either none or the direct sum of exactly two consecutive ideals has infinite projective dimension, then the finitistic dimension conjecture holds for Λ. As a consequence, we have that if either none or the direct sum of exactly two consecutive terms in the radical series of Λ has infinite projective dimension, then the finitistic dimension conjecture holds for Λ. Some known results are obtained as corollaries.

MSC classification

Primary: 16E10: Homological dimension
Secondary: 16E05: Syzygies, resolutions, complexes
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 64 , Issue 1 , January 2022 , pp. 37 - 44
Copyright
© The Author(s), 2020. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

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

REFERENCES

Bass, H., Finitistic dimension and a homological generalization of semiprimary rings, Trans. Amer. Math. Soc. 95 (1960), 466–488.CrossRefGoogle Scholar
Beligiannis, A. and Reiten, I., Homological and homotopical aspects of torsion theories , Memoirs of the American Mathematical Society, vol. 188 (American Mathematical Society, Providence, RI, 2007).Google Scholar
Fernandes, S. M., Lanzilotta, M. and Mendoza, O., The Φ-dimension: a new homological measure, Algebr. Represent. Theory 18 (2015), 463–476.CrossRefGoogle Scholar
Gatica, M. A., Lanzilotta, M. and Platzeck, M. I., Idempotent ideals and the Igusa-Todorov functions, Algebr. Represent. Theory 20 (2017), 275–287.CrossRefGoogle Scholar
Green, E. L. and Zimmermann-Huisgen, B., Finitistic dimension of artinian rings with vanishing radical cube, Math. Z. 206 (1991), 505–526.CrossRefGoogle Scholar
Huang, Z. Y., Proper resolutions and Gorenstein categories, J. Algebra 393 (2013), 142–169.CrossRefGoogle Scholar
Huang, Z. Y. and Sun, J. X., Endomorphism algebras and Igusa-Todorov algebras, Acta Math. Hungar. 140 (2013), 60–70.CrossRefGoogle Scholar
Huard, F. and Lanzilotta, M., Self-injective right artinian rings and Igusa-Todorov functions, Algebr. Reprenent Theory 16 (2013), 765–770.CrossRefGoogle Scholar
Huard, F., Lanzilotta, M. and Mendoza, O., An approach to the finitistic dimension conjecture, J. Algebra 319 (2008), 3918–3934.CrossRefGoogle Scholar
Huard, F., Lanzilotta, M. and Mendoza, O., Finitistic dimension through infinite projective dimension, Bull. London Math. Soc. 41 (2009), 367–376.CrossRefGoogle Scholar
Igusa, K. and Todorov, G., On the finitistic global dimension conjecture for artin algebras, Representations of Algebras and Related Topics, Fields Institute Communications, vol. 45 (American Mathematical Society, Providence, RI, 2005), 201–204.Google Scholar
Lanzilotta, M., Marcos, E. N. and Mata, G., Igusa-Todorov functions for radical square zero algebras, J. Algebra 487 (2017), 357–385.CrossRefGoogle Scholar
Lanzilotta, M. and Mata, G., Igusa-Todorov functions for artin algebras, J. Pure Appl. Algebra 222 (2018), 202–212.CrossRefGoogle Scholar
Lanzilotta, M. and Mendoza, O., Relative Igusa-Todorov functions and relative homological dimensions, Algebr. Represent. Theory 20 (2017), 765–802.CrossRefGoogle Scholar
Wang, C. X and Xi, C. C., Finitistic dimension conjecture and radical-power extensions, J. Pure Appl. Algebra 221 (2016), 832–846.CrossRefGoogle Scholar
Wang, Y., A note on the finitistic dimension conjecture, Comm. Algebra 22 (1994), 2525–2528.CrossRefGoogle Scholar
Wei, J. Q., Finitistic dimension and Igusa-Todorov algebras, Adv. Math. 222 (2009), 2215–2226.CrossRefGoogle Scholar
Wei, J. Q., Finitistic dimension conjecture and conditions on ideals, Forum Math. 23 (2011), 549–564.CrossRefGoogle Scholar
Xi, C. C., On the finitistic dimension conjecture I: related to representation-finite algebras, J. Pure Appl. Algebra 193 (2004), 287–305; and Erratum, J. Pure Appl. Algebra 202 (2005), 325–328.CrossRefGoogle Scholar
Xi, C. C., On the finitistic dimension conjecture II: related to finite global dimension, Adv. Math. 201 (2006), 116–142.CrossRefGoogle Scholar
Xi, C. C., Some new advances in finitistic dimension conjecture (in Chinese), Adv. Math. (China) 36 (2007), 13–17.Google Scholar
Xi, C. C., On the finitistic dimension conjecture III: related to $eAe\subseteq A$ , J. Algebra 319 (2008), 3666–3688.CrossRefGoogle Scholar
Xu, D. M., Generalized Igusa-Todorov function and finitistic dimensions, Arch. Math. 100 (2013), 309–322.CrossRefGoogle Scholar
Yamagata, K., Frobenius algebras, in Handbook of algebra, Handbook of Algebra, vol. 1, (Elsevier/North-Holland, Amsterdam, 1996), 841–887.CrossRefGoogle Scholar
Zhang, A. P. and Zhang, S. H., Subalgebras and finitistic dimensions of artin algebras, Acta Math. Sin. English. Ser. 31 (2011), 2033–2040.Google Scholar
Zimmerman-Huisgen, B., The finitistic dimension conjectures–a tale of 3.5 decades, in Abelian groups and modules (Padova, 1994), Mathematics and Its Applications, vol. 343 (Kluwer Academic Publishers, Dordrecht, 1995), 501–517.CrossRefGoogle Scholar