Free resolution of powers of monomial ideals and Golod rings

Let $S = \mathbb{K}[x_1, \dots, x_n]$ be the polynomial ring over a field $\mathbb{K}$. In this paper we present a criterion for componentwise linearity of powers of monomial ideals. In particular, we prove that if a square-free monomial ideal $I$ contains no variable and some power of $I$ is componentwise linear, then $I$ satisfies the gcd condition. For a square-free monomial ideal $I$ which contains no variable, we show that $S/I$ is a Golod ring provided that for some integer $s\geq 1$, the ideal $I^s$ has linear quotients with respect to a monomial order.

Shellability and the Strong gcd-Condition

Shellability is a well-known combinatorial criterion on a simplicial complex $\Delta$ for verifying that the associated Stanley-Reisner ring $k[\Delta]$ is Cohen-Macaulay. A notion familiar to commutative algebraists, but which has not received as much attention from combinatorialists as the Cohen-Macaulay property, is the notion of a Golod ring. Recently, Jöllenbeck introduced a criterion on simplicial complexes reminiscent of shellability, called the strong gcd-condition, and he together with the author proved that it implies Golodness of the associated Stanley-Reisner ring. The two algebraic notions were earlier tied together by Herzog, Reiner and Welker, who showed that if $k[\Delta^\vee]$ is sequentially Cohen-Macaulay, where $\Delta^\vee$ is the Alexander dual of $\Delta$, then $k[\Delta]$ is Golod. In this paper, we present a combinatorial companion of this result, namely that if $\Delta^\vee$ is (non-pure) shellable then $\Delta$ satisfies the strong gcd-condition. Moreover, we show that all implications just mentioned are strict in general but that they are equivalences if $\Delta$ is a flag complex.