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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
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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

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