Free resolutions over short Gorenstein local rings

Let R be a local ring with maximal ideal m admitting a non-zero element a∈m 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 m4=0. Let e denote the minimal number of generators of m. If R is Gorenstein with m4=0 and e ≥ 3, we show that PRM(t) is rational with denominator H 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.