Koszul Modules

Abstract

This paper studies a new class of modules over noetherian local rings, called Koszul modules. It is proved that when a Koszul local ring R is a complete intersection, all high syzygies of finitely generated R-modules are Koszul. A stronger result is obtained when R is Golod and of embedding dimension d: the 2dth syzygy of every R-module is Koszul. In addition, results are established that demonstrate that Koszul modules possess good homological properties; for instance, their Poincaré series is a rational function. IntroductionConsider a noetherian local ring R with maximal ideal m and residue field k. Given a finitely generated R-module M and F its minimal free resolution, denote lin R (F ) the associated graded complex of F arising from the standard m-adic filtration; the details are given in Section 1. This is a complex of free modules over the associated graded ring gr m (R) of R and its differentials depend only on the linear part-in a sense made precise in Section 4-of the corresponding differentials on F; whence the notation.We say that M is Koszul if lin R (F ) is acyclic. Such modules are module theoretic analogues of Koszul algebras introduced by Priddy [19].

Linearity defect and regularity over a Koszul algebra

“…Clearly, if A itself is complete intersection or Golod, then it comes from a complete intersection by a Golod map. On the other hand, even if A is Koszul and commutative, ld A (M) can be infinite for some M ∈ * mod A, as pointed out in [11].…”

“…The linear part lin(P • ) of P • is the chain complex such that lin(P • ) i = P i for all i and its differential maps are given by erasing all the entries of degree ≥ 2 from the matrices representing the differentials of P • . According to Herzog-Iyengar [11], we call ld A (M) := sup{ i | H i (lin(P • )) = 0 } the linearity defect of M. This invariant is related to the regularity via Koszul duality (see Theorem 3.9 below). In §4, we mainly treat a Koszul commutative algebra A and its dual A !…”

“…Even in this case, it can occur that ld A (M) = ∞ for some M ∈ * mod A ( [11]), while Avramov-Eisenbud [1] showed that reg A (M) < ∞ for all M ∈ * mod A. On the other hand, Herzog-Iyengar [11] proved that if A is complete intersection or Golod then ld A (M) < ∞ for all M ∈ * mod A. Initiated by these results, we will show the following.…”

“…If A is a Koszul commutative algebra and S := Sym K A 1 is the polynomial ring, then we have A = S/I for a graded ideal I of S. In this situation, A is Golod if and only if I has a 2-linear resolution as an S-module (i.e., β i, j (I ) = 0 implies j = i + 2), see [11,Proposition 5.8]. We say A comes from a complete intersection by a Golod map (see [2], [11], although they do not use this terminology), if there is an intermediate graded ring R with S → → R → → A satisfying the following conditions:…”

“…We say A comes from a complete intersection by a Golod map (see [2], [11], although they do not use this terminology), if there is an intermediate graded ring R with S → → R → → A satisfying the following conditions:…”