$A_{\infty}$–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 $A_{\infty }$–algebra structure on $F$. Using this structure, we show that $R$ is Golod if and only if the product on ${Tor}^S\left(R,k\right)$ vanishes. Furthermore, we give a necessary and sufficient combinatorial condition for $R$ to be Golod.