Ordinary and symbolic powers are Golod

Abstract

Let S be a positively graded polynomial ring over a field of characteristic 0, and I ⊂ S a proper graded ideal. In this note it is shown that S/I is Golod if ∂(I) 2 ⊂ I. Here ∂(I) denotes the ideal generated by all the partial derivatives of elements of I. We apply this result to find large classes of Golod ideals, including powers, symbolic powers, and saturations of ideals.2000 Mathematics Subject Classification. 13A02, 13D40.

Powers of sums and their homological invariants

“…, x m ] is a polynomial ring over k (where m ≥ 0) and (0) = I ⊆ R is a proper homogeneous ideal. Following Herzog and Huneke [31], if char k = 0 we say that a homogeneous (but possibly non-monomial) ideal I is strongly Golod if ∂(I) 2 ⊆ I. Independently of char k, we say that an ideal I is * strongly Golod if I is a monomial ideal and ∂ * (I) 2 ⊆ I.…”

Koszul cycles and Golod rings

“…As a first application we prove a result analogue to [8,Theorem 2.3], whose proof follows very much the line of arguments given there. The new and important fact is that no assumptions on the characteristic of the base field are made.…”

“…On the other hand, if char(K) = 0 and I is a monomial ideal, then I is strongly Golod in the sense of [8] if and only if it is strongly d-Golod. This follows from the remarks at the begin of Section 3 in [8], where it is observed that a monomial ideal I is strongly Golod if and only if for all monomial generators u, v ∈ I and all integers i and j with x i |u and x j |v it follows that uv/x i x j ∈ I. Indeed, it is obvious that strongly Golod implies strongly d-Golod.…”

The absolutely Koszul property of Veronese subrings and Segre products

“…, x n ] and s ≥ 2. Herzog and Huneke [22] proved recently that any such R is a Golod ring, in particular any finitely generated graded module over R has rational Poincaré series.…”