Truncated path algebras and Betti numbers of polynomial growth

Coopergard, Ryan; Purin, Marju

doi:10.2140/involve.2019.12.919

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Truncated path algebras and Betti numbers of polynomial growth

Ryan Coopergard and Marju Purin

Vol. 12 (2019), No. 6, 919–940

DOI: 10.2140/involve.2019.12.919

Abstract

We investigate a class of truncated path algebras in which the Betti numbers of a simple module satisfy a polynomial of arbitrarily large degree. We produce truncated path algebras where the $i$ -th Betti number of a simple module $S$ is $β_{i} (S) = i^{k}$ for $2 \leq k \leq 4$ and provide a result of the existence of algebras where $β_{i} (S)$ is a polynomial of degree 4 or less with nonnegative integer coefficients. In particular, we prove that this class of truncated path algebras produces Betti numbers corresponding to any polynomial in a certain family.

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Keywords

finite-dimensional algebra, Betti number, path algebra, quiver

Mathematical Subject Classification 2010

Primary: 16P90

Secondary: 16P10, 16G20

Milestones

Received: 23 December 2016

Revised: 24 May 2018

Accepted: 31 January 2019

Published: 3 August 2019

Communicated by Kenneth S. Berenhaut

Authors

Ryan Coopergard
Department of Mathematics University of Minnesota - Twin Cities Minneapolis, MN United States

Marju Purin
Department of Mathematics, Statistics, and Computer Science St. Olaf College Northfield, MN United States

Vol. 12, No. 6, 2019