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Proceedings of the American Mathematical Society

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Idempotent reduction for the finitistic dimension conjecture


Authors: Diego Bravo and Charles Paquette
Journal: Proc. Amer. Math. Soc. 148 (2020), 1891-1900
MSC (2010): Primary 16E10, 16G20
DOI: https://doi.org/10.1090/proc/14945
Published electronically: January 21, 2020
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Abstract: In this note, we prove that if $ \Lambda $ is an Artin algebra with a simple module $ S$ of finite projective dimension, then the finiteness of the finitistic dimension of $ \Lambda $ implies that of $ (1-e)\Lambda (1-e)$ where $ e$ is the primitive idempotent supporting $ S$. We derive some consequences of this. In particular, we recover a result of Green-Solberg-Psaroudakis: if $ \Lambda $ is the quotient of a path algebra by an admissible ideal $ I$ whose defining relations do not involve a certain arrow $ \alpha $, then the finitistic dimension of $ \Lambda $ is finite if and only if the finitistic dimension of $ \Lambda /\Lambda \alpha \Lambda $ is finite.


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Additional Information

Diego Bravo
Affiliation: Instituto de Matemática y Estadística “Rafael Laguardia”, Universidad de la República, Julio Herrera y Reissig 565, Montevideo, Uruguay
Email: dbravo@fing.edu.uy

Charles Paquette
Affiliation: Department of Mathematics and Computer Science, Royal Military College of Canada, Kingston, Ontario K7K 7B4, Canada
Email: charles.paquette.math@gmail.com

DOI: https://doi.org/10.1090/proc/14945
Keywords: Finitistic dimension, primitive idempotent, reduction technique, projective dimension, projective ideal
Received by editor(s): February 13, 2019
Received by editor(s) in revised form: September 10, 2019
Published electronically: January 21, 2020
Additional Notes: The second author was supported by the Natural Sciences and Engineering Research Council of Canada and by CDARP.
Communicated by: Jerzy Weyman
Article copyright: © Copyright 2020 American Mathematical Society