Koszul modules (and the $\Omega$-growth of modules) over short local algebras

TL;DR

This paper investigates the growth patterns of Betti numbers of modules over short local algebras, generalizing known results from commutative algebra and revealing limited growth possibilities.

Contribution

It extends the understanding of Betti number growth to non-commutative short local algebras, providing new insights and generalizations of existing commutative algebra results.

Findings

Limited possibilities for Betti number growth over short local algebras

Generalization of known commutative algebra results to non-commutative case

Some results are new even in the commutative setting

Abstract

Following the well-established terminology in commutative algebra, any (not necessarily commutative) finite-dimensional local algebra $A$ with radical $J$ will be said to be short provided $J^3 = 0$. As in the commutative case, also in general, the asymptotic behavior of the Betti numbers of modules seems to be of interest. As we will see, there are only few possibilities for the growth of the Betti numbers of modules. We generalize results which are known for commutative algebras, but some of our results seem to be new also in the commutative case.