Cohomology of finite modules over short Gorenstein rings

TL;DR

This paper proves that for certain Gorenstein rings with specific nilpotent maximal ideals, the generating functions for Ext and Tor modules are rational with a specific quadratic denominator.

Contribution

It establishes the rationality of Ext and Tor series over short Gorenstein rings with nilpotent maximal ideals, extending understanding of their homological properties.

Findings

The Ext series is rational with denominator 1 - e t + t^2.

The Tor series is rational with denominator 1 - e t + t^2.

Results apply when the maximal ideal cubed is zero but not squared.

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 $1-et+t^2$.