Free resolutions over short Gorenstein local rings

Let R be a local ring with maximal ideal \({\mathfrak{m}}\) admitting a non-zero element \({a\in\mathfrak{m}}\) 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 \({\mathfrak{m}^4=0}\). Let e denote the minimal number of generators of \({\mathfrak{m}}\). If R is Gorenstein with \({\mathfrak{m}^4=0}\) and e ≥ 3, we show that \({{\rm P}_{M}^{R}(t)}\) 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.