Truncated path algebras and Betti numbers of polynomial growth

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.