A non-Golod ring with a trivial product on its Koszul homology

Abstract

We present a monomial ideal $\mathfrak{a}\subset S$ such that $S/\mathfrak{a}$ is not Golod, even though the product on its Koszul homology is trivial. This constitutes a counterexample to a well-known result by Berglund and Jöllenbeck (the error can be traced to a mistake in an earlier article by Jöllenbeck). On the positive side, we show that if $R$ is a monomial ring such that the $r$-ary Massey product vanish for all $r\le \max (2,\mathrm{reg}R-2)$, then $R$ is Golod. In particular, if $R$ is the Stanley-Reisner ring of a simplicial complex of dimension at most $3$, then $R$ is Golod if and only if the product on its Koszul homology is trivial. Moreover, we show that if $\Delta$ is a triangulation of a $\mathbb{k}$-orientable manifold whose Stanley-Reisner ring is Golod, then $\Delta$ is $2$-neighborly. This extends a recent result of Iriye and Kishimoto.

arXiv comment17 pages, simplified the main example and improved exposition