A∞–resolutions and the Golod property for monomial rings

Abstract

Let R = S/I be a monomial ring whose minimal free resolution F is rooted. We describe an A∞-algebra structure on F . Using this structure, we show that R is Golod if and only if the product on Tor S (R, k) vanishes. Furthermore, we give a necessary and sufficient combinatorial condition for R to be Golod.

On Monomial Golod Ideals

“…The notion is defined and studied extensively in the local setting, but in this paper we shall restrict ourselves to the graded situation. Golod rings and ideals have attracted increasing attention recently (see [6,8,9,12,15]), but they remain mysterious even when n = 3. For instance, we do not know if the product of any two homogeneous ideals in Q = k[x, y, z] is Golod.…”

Golodness and polyhedral products of simplicial complexes with minimal Taylor resolutions

“…We will see in the next section that minimality of the Taylor resolution of k[K] does not depend on k, so in fact, Theorem 1.3 does not depend on k. 2. Recently, Frankhuizen [5] proved the equivalence between 1 and 2 in a more general setting by a purely algebraic manner.…”

Massey products and the Golod property for simplicially resolvable rings

“…In [11], we developed an approach to study Massey products on Tor S (R, k) using A ∞ -algebras and applied this to give necessary and sufficient conditions for Golodness for rooted rings. The purpose of this paper is to extend the methods developed in [11] to monomial rings whose minimal free resolution is simplicial in the sense of [4].…”

“…The main idea is the following. As a consequence of the result in [11], we can study the Golod property in terms of A ∞ -structures on the minimal free resolution of R. In this paper, we construct such A ∞ -structures by applying algebraic Morse theory to the Taylor resolution of a monomial ring R. When R is simplicially resolvable (see Definition 7.3), it turns out that this structure is a comparatively simple description which is given in Lemma 7.5. By using this description, we obtain the first main result of this paper.…”