2 00 9 Exact Zero Divisors and Free Resolutions over Short Local Rings

Let R be a local ring with maximal ideal m such that there exists a pair of elements a, b with (0 : a) = bR and (0 : b) = aR; we say that a pair a, b as above is an exact pair of zero divisors. We study minimal free resolutions of finitely generated R-modules M , with particular attention to the case when m = 0. Let e denote the minimal number of generators of m. If R is Gorenstein with m = 0 and e ≥ 3, we show that P M (t) is rational with denominator HR(−t) = 1− et+ et 2 − t, for each finitely generated R-module M . In particular, this conclusion applies to generic Gorenstein algebras of socle degree 3.