Free resolutions over short Gorenstein local rings

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.