Vol. 12, No. 6, 2019

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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
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 βi(S) = ik for 2 k 4 and provide a result of the existence of algebras where βi(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.

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