On the Golod property of Stanley–Reisner rings

Abstract

Recently in [M. Jöllenbeck, On the multigraded Hilbert and Poincaré 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 × N n -graded vector space, where A is a monomial ring over a field k, that is, the 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, Poincaré 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.