Asymptotic syzygies of Stanley-Reisner rings of iterated subdivisions

TL;DR

This paper investigates the asymptotic behavior of Betti numbers in Stanley-Reisner rings derived from iterated barycentric or edgewise subdivisions, revealing bounds independent of subdivision iteration.

Contribution

It provides a detailed description of how Betti numbers behave asymptotically for iterated subdivisions of simplicial complexes, extending recent non-vanishing results.

Findings

Betti numbers of subdivided complexes are bounded independently of iteration

Asymptotic behavior of linear strands in minimal free resolutions is characterized

Results connect subdivision processes with algebraic invariants of Stanley-Reisner rings

Abstract

Inspired by recent results of Ein, Lazarsfeld, Erman and Zhou on the non-vanishing of Betti numbers of high Veronese subrings, we describe the behaviour of the Betti numbers of Stanley-Reisner rings associated with iterated barycentric or edgewise subdivisions of a given simplicial complex. Our results show that for a simplicial complex $\Delta$ of dimension $d-1$ and for $1\leq j\leq d-1$ the number of $0$'s the j-th linear strand of the minimal free resolution of the r-th barycentric or edgewise subdivision is bounded above only in terms of $d$ and $j$ (and independently of $r$).