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Homological dimensions for co-rank one idempotent subalgebras


Authors: Colin Ingalls and Charles Paquette
Journal: Trans. Amer. Math. Soc. 369 (2017), 5317-5340
MSC (2010): Primary 16E10, 16G20
DOI: https://doi.org/10.1090/tran/6815
Published electronically: November 28, 2016
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Abstract: Let $ k$ be an algebraically closed field and $ A$ a (left and right) Noetherian associative $ k$-algebra. Assume further that $ A$ is either positively graded or semiperfect (this includes the class of finite dimensional $ k$-algebras and $ k$-algebras that are finitely generated modules over a Noetherian central Henselian ring). Let $ e$ be a primitive idempotent of $ A$, which we assume is of degree 0 if $ A$ is positively graded. We consider the idempotent subalgebra $ \Gamma = (1-e)A(1-e)$ and $ S_e$ the simple right $ A$-module $ S_e = eA/e{\rm rad}A$, where $ {\rm rad}A$ is the Jacobson radical of $ A$ or the graded Jacobson radical of $ A$ if $ A$ is positively graded. In this paper, we relate the homological dimensions of $ A$ and $ \Gamma $, using the homological properties of $ S_e$. First, if $ S_e$ has no self-extensions of any degree, then the global dimension of $ A$ is finite if and only if that of $ \Gamma $ is. On the other hand, if the global dimensions of both $ A$ and $ \Gamma $ are finite, then $ S_e$ cannot have self-extensions of degree greater than one, provided $ A/{\rm rad}A$ is finite dimensional.


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Additional Information

Colin Ingalls
Affiliation: Department of Mathematics & Statistics, University of New Brunswick-Fredericton, P.O. Box 4400, Fredericton, New Brunswick E3B 5A3, Canada
Email: cingalls@unb.ca

Charles Paquette
Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269-1009
Email: charles.paquette@usherbrooke.ca

DOI: https://doi.org/10.1090/tran/6815
Received by editor(s): June 4, 2014
Received by editor(s) in revised form: August 21, 2015
Published electronically: November 28, 2016
Article copyright: © Copyright 2016 American Mathematical Society