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 $\mathfrak{k} =R/\mathfrak{m} $ and $e=rank _{\mathfrak{k} }(\mathfrak{m} /\mathfrak{m} ^2)$. If $e>2$ and $M$, $N$ are finitely generated $R$-modules, we show that the formal power series \[ \sum _{i=0}^\infty rank _{\mathfrak{k} }\left (Ext ^i_R(M,N)\otimes _R\mathfrak{k} \right )t^i \] and \[ \sum _{i=0}^\infty rank _{\mathfrak{k} }\left (Tor _i^R(M,N)\otimes _R \mathfrak{k} \right )t^i \] are rational, with denominator $1-et+t^2$.