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 ${\beta }_{i}\left(S\right)={i}^{k}$ for $2\le k\le 4$ and provide a result of the existence of algebras where ${\beta }_{i}\left(S\right)$ 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