On the golod property of Stanley-Reisner rings

Recently in [M. Jollenbeck, On the multigraded Hilbert and Poincare series of monomial rings, J. Pure Appl. Algebra 207 (2) (2006) 261-298] the second author made a conjecture about the structure of Ext*(A) (k, k) as an N x N-n-graded vector space, where A is a monomial ring over a field k, that is, the A quotient of a polynomial ring P = k[x(1),..., x(n)] by a monomial ideal, and he verified this conjecture for several classes of such rings. Using the results of [A. Berglund, Poincare series and homotopy Lie algebras of monomial rings, Licentiate thesis, Stockholm University, http://www.math.su.se/reports/2005/6/, 2005] by the first author, we are able to prove this conjecture in general. In particular we get a new explicit formula for the multigraded Hilbert series of Ext*(A) (k, k). A surprising consequence of our results is that a monomial ring A is Golod if and only if the product on Tor(*)(p) (A, k) is trivial. For Stanley-Reisner rings of flag complexes we get a complete combinatorial characterization of Golodness. We introduce the concept of minimally non-Golod complexes,' and show that boundary complexes of stacked polytopes are minimally non-Golod. Finally we discuss the relation between minimal non-Golodness and the Gorenstein* property for simplicial complexes. (C) 2007 Elsevier Inc. All rights reserved.