Betti numbers of skeletons

Abstract

We demonstrate that the Betti numbers associated to an N-graded minimal free resolution of the Stanley-Reisner ring of the (d-1)-skeleton of a simplicial complex of dimension d can be expressed as a Z-linear combination of the corresponding Betti numbers of the complex itself. An immediate implication of our main result is that the projective dimension of the Stanley-Reisner ring of the (d-1)-skeleton is at most one greater than the projective dimension of the Stanley-Reisner ring of the original complex, and it thus provides a new and direct proof of this. Our result extends immediately to matroids and their truncations. A similar result for matroid elongations can not be hoped for, but we do obtain a weaker result for these.