Cohomology of finite modules over short Gorenstein rings

Abstract

Let $R$ be a Gorenstein local ring with maximal ideal $\mathfrak{m}$ satisfying $\mathfrak{m}^3=0\ne\mathfrak{m}^2$. Set $k=R/\mathfrak{m}$ and $e=\text{rank}_{k}(\mathfrak{m}/\mathfrak{m}^2)$. If $e>2$ and $M$, $N$ are finitely generated $R$-modules, we show that the formal power series $\sum_{i=0}^\infty\text{rank}_{k}\left(\text{Ext}^i_R(M,N)\otimes_R k \right)t^i$ and $\sum_{i=0}^\infty\text{rank}_{k}\left(\text{Tor}_i^R(M,N)\otimes_R k \right)t^i$ are rational, with denominator $1-et+t^2$.