Free resolutions over short local rings

Abstract

Abstract. 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 m 4 = 0. Let e denote the minimal number of generators of m. If R is Gorenstein with m 4 = 0 and e ≥ 3, we show that P R M (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.

Free resolutions over short Gorenstein local rings

“…6 Conca generators. Following [5], we say that an element c ∈ m is a Conca generator modulo a R if it satisfies m 2 + a R = cm + a R, c / ∈ a R and c 2 ∈ a R.…”

“…In the terminology of [5], the image in R/a R of a Conca generator modulo a R is exactly a Conca generator of the maximal ideal m/a R. 3.7 Golod homomorphisms. The notion of Golod homomorphism was introduced by Levin [16].…”

“…All Gorenstein rings (R, m, k) with m 3 = 0 are known to be good, due toSjödin [22]; see also Avramov, Iyengar,Şega [5]. However, Gorenstein local rings with m 4 = 0 can exhibit bad behavior, as there exist examples with μ(m) ≥ 12 for which P R k (t) is irrational (cf.…”

On the Regularity and Koszulness of Modules Over Local Rings

“…In the following we give some information about the Koszul modules and the regularity of modules over these kind of rings. The first statement was already proved in [3], but we insert here a proof as a consequence of Corollary 2.11. Theorem 2.13.…”

“…Avramov, Iyengar and Şega proved that over a local ring whose maximal ideal has a Conca generator, every module has a Koszul syzygy module (see [3]). In the following we give some information about the Koszul modules and the regularity of modules over these kind of rings.…”