Free resolutions over short Gorenstein local rings

TL;DR

This paper investigates minimal free resolutions over certain Gorenstein local rings with nilpotent maximal ideals, revealing rational Poincaré series with specific denominators in the case when m^4=0 and e≥3.

Contribution

It establishes the rationality of Poincaré series for modules over Gorenstein rings with m^4=0 and e≥3, extending understanding of their free resolutions.

Findings

Poincaré series are rational with a specific denominator

Results apply to generic Gorenstein algebras of socle degree 3

Provides explicit formulas for minimal free resolutions

Abstract

Let R be a local ring with maximal ideal m admitting a non-zero element a\in\fm for which the ideal (0:a) is isomorphic to R/aR. We study minimal free resolutions of finitely generated R-modules M, with particular attention to the case when m^4=0. Let e denote the minimal number of generators of m. If R is Gorenstein with m^4=0 and e\ge 3, we show that \Poi MRt is rational with denominator \HH R{-t} =1-et+et^2-t^3, for each finitely generated R-module M. In particular, this conclusion applies to generic Gorenstein algebras of socle degree 3.