G-Elements of matroid complexes

Abstract

A g-element for a graded R-module is a one-form with properties similar to a Lefschetz class in the cohomology ring of a compact complex projective manifold, except that the induced multiplication maps are injections instead of bijections. We show that if k( Δ) is the face ring of the independence complex of a matroid and the characteristic of k is zero, then there is a non-empty Zariski open subset of pairs ( Θ, ω) such that Θ is a linear set of parameters for k( Δ) and ω is a g-element for k( Δ)/〈 Θ〉. This leads to an inequality on the first half of the h-vector of the complex similar to the g-theorem for simplicial polytopes.