Betti numbers of skeletons

TL;DR

This paper provides a formula relating the Betti numbers of a simplicial complex's skeleton to those of the original complex, offering new insights into projective dimensions and extending to matroids.

Contribution

It introduces a linear combination formula for Betti numbers of skeletons, leading to a new proof of bounds on projective dimension and extending results to matroids.

Findings

Betti numbers of skeletons can be expressed as Z-linear combinations of original Betti numbers.

The projective dimension of skeletons is at most one greater than that of the original complex.

Results extend to matroids and their truncations.

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.