Regularity of powers of Stanley-Reisner ideals of one-dimensional simplicial complexes

TL;DR

This paper investigates the regularity of powers and certain intermediate ideals of Stanley-Reisner ideals associated with one-dimensional simplicial complexes, establishing equalities among their regularities.

Contribution

It proves that for one-dimensional simplicial complexes, the regularity of powers and specific intermediate ideals of Stanley-Reisner ideals are equal.

Findings

Regularity of $J$, $I_ riangle^s$, and $I_ riangle^{(s)}$ are equal for all $s \\ge 1$

Intermediate ideals generated by $I_ riangle^s$ and minimal generators of $I_ riangle^{(s)}$ share the same regularity

The result applies specifically to one-dimensional simplicial complexes.

Abstract

Let $\Delta$ be a one-dimensional simplicial complex. Let $I_\Delta$ be the Stanley-Reisner ideal of $\Delta$. We prove that for all $s \ge 1$ and all intermediate ideals $J$ generated by $I_\Delta^s$ and some minimal generators of $I_\Delta^{(s)}$, we have $${\rm reg} J = {\rm reg} I_\Delta^s = {\rm reg} I_\Delta^{(s)}.$$