Global Dimensions of Some Artinian Algebras
M\"uge Kanuni, Atabey Kaygun
TL;DR
This paper develops bounds and an algorithm to determine the global dimension of certain artinian algebras using a directed graph, aiding in understanding their homological properties.
Contribution
It introduces a novel algorithm based on a directed graph to estimate and decide finiteness of global dimensions of artinian algebras.
Findings
Provided bounds for global dimensions in terms of subalgebras.
Developed an explicit algorithm for global dimension estimation.
Enabled decision of finite global dimension in many cases.
Abstract
In this article we obtain lower and upper bounds for global dimensions of a class of artinian algebras in terms of global dimensions of a finite subset of their artinian subalgebras. Finding these bounds for the global dimension of an artinian algebra is realized via an explicit algorithm we develop. This algorithm is based on a directed graph (not the Auslander-Reiten quiver) we construct, and it allows us to decide whether an artinian algebra has finite global dimension in good number of cases.
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Taxonomy
TopicsAlgebraic structures and combinatorial models · Advanced Topics in Algebra · Rings, Modules, and Algebras