A general formula for the index of depth stability of edge ideals

TL;DR

This paper provides the first explicit formula for the index of depth stability of squarefree monomial ideals generated by degree 2, linking it to associated graphs and introducing new techniques for analyzing edge ideals.

Contribution

It introduces a general formula for the depth stability index of degree 2 squarefree monomial ideals, connecting algebraic invariants to graph properties.

Findings

Explicit formula for stab(I) in terms of associated graphs

New techniques relating simplicial complexes and graph decompositions

Effective method for studying powers of edge ideals

Abstract

By a classical result of Brodmann, the function $\operatorname{depth} R/I^t$ is asymptotically a constant, i.e. there is a number $s$ such that $\operatorname{depth} R/I^t = \operatorname{depth} R/I^s$ for $t > s$. One calls the smallest number $s$ with this property the index of depth stability of $I$ and denotes it by $\operatorname{dstab}(I)$. This invariant remains mysterious til now. The main result of this paper gives an explicit formula for $\operatorname{dstab}(I)$ when $I$ is an arbitrary ideal generated by squarefree monomials of degree 2. That is the first general case where one can characterize $\operatorname{dstab}(I)$ explicitly. The formula expresses $\operatorname{dstab}(I)$ in terms of the associated graph. The proof involves new techniques which relate different topics such as simplicial complexes, systems of linear inequalities, graph parallelizations, and ear decompositions. It provides an effective method for the study of powers of edge ideals.