Simplicial complexes with rigid depth

TL;DR

This paper characterizes simplicial complexes with rigid depth by extending existing algebraic criteria to unmixed monomial ideals, linking algebraic properties to combinatorial structures.

Contribution

It extends Minh and Trung's result to provide criteria for when the depth of an ideal equals the depth of its radical, and characterizes all pure simplicial complexes with rigid depth.

Findings

Criteria for $ ext{depth } I = ext{depth } oot I$ for unmixed monomial ideals.

Characterization of pure simplicial complexes with rigid depth.

Connection between algebraic depth and combinatorial structure.

Abstract

We extend a result of Minh and Trung to get criteria for $\depth I=\depth\sqrt{I}$ where $I$ is an unmixed monomial ideal of the polynomial ring $S=K[x_1,..., x_n]$. As an application we characterize all the pure simplicial complexes $\Delta$ which have rigid depth, that is, which satisfy the condition that for every unmixed monomial ideal $I\subset S$ with $\sqrt{I}=I_\Delta$ one has $\depth(I)=\depth(I_\Delta).$