Homological dimensions for co-rank one idempotent subalgebras
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- by Colin Ingalls and Charles Paquette PDF
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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 $S_e = eA/e\textrm {rad}A$, where $\textrm {rad}A$ 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 $A/\textrm {rad}A$ is finite dimensional.References
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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
- MR Author ID: 787027
- Email: charles.paquette@usherbrooke.ca
- Received by editor(s): June 4, 2014
- Received by editor(s) in revised form: August 21, 2015
- Published electronically: November 28, 2016
- © Copyright 2016 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 369 (2017), 5317-5340
- MSC (2010): Primary 16E10, 16G20
- DOI: https://doi.org/10.1090/tran/6815
- MathSciNet review: 3646764