The $a_0$-invariants of powers of a two-dimensional squarefree monomial ideal

TL;DR

This paper provides an explicit formula for the $a_0$-invariants of powers of two-dimensional squarefree monomial ideals associated with one-dimensional simplicial complexes, depending on the girth of the complex.

Contribution

It offers a new explicit formula for $a_0$-invariants when the girth is at least 4 and characterizes complexes with specific invariants when girth is 3.

Findings

Explicit formula for $a_0(S/I_{ riangle}^n)$ when girth ≥ 4.

Characterization of complexes with $a_0$-invariants equal to $3n-1$ or $3n-2$ for girth = 3.

Provides combinatorial criteria based on girth for the invariants.

Abstract

Let $\Delta$ be an one-dimensional simplicial complex on $\{1,2,\ldots,s\}$ and $S$ the polynomial ring $K[x_1,\ldots,x_s]$ over a field $K$. The explicit formula for $a_0(S/I_{\Delta}^n)$ is presented when $\mathrm{girth}(\Delta)\geq 4$. If $\mathrm{girth}(\Delta)=3$ we characterize the simplicial complexes $\Delta$ for which $a_0(S/I_{\Delta}^n)=3n-1$ or $3n-2$.