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

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.