Square-free powers of Cohen-Macaulay simplicial forests

TL;DR

This paper proves that for Cohen-Macaulay simplicial forests, all square-free powers of their facet ideals are Cohen-Macaulay, introducing a new combinatorial concept called special leaf to analyze depth and depth functions.

Contribution

It establishes the Cohen-Macaulay property for all square-free powers of facet ideals of Cohen-Macaulay simplicial forests and introduces the special leaf concept.

Findings

All square-free powers are Cohen-Macaulay for such complexes.

Provides an explicit combinatorial formula for depth of these powers.

Shows the normalized depth function is nonincreasing.

Abstract

Let $I(\Delta)^{[k]}$ denote the $k^{\text{th}}$ square-free power of the facet ideal of a simplicial complex $\Delta$ in a polynomial ring $R$. Square-free powers are intimately related to the `Matching Theory' and `Ordinary Powers'. In this article, we show that if $\Delta$ is a Cohen-Macaulay simplicial forest, then $R/I(\Delta)^{[k]}$ is Cohen-Macaulay for all $k\ge 1$. This result is quite interesting since all ordinary powers of a graded radical ideal can never be Cohen-Macaulay unless it is a complete intersection. To prove the result, we introduce a new combinatorial notion called special leaf, and using this, we provide an explicit combinatorial formula of $\mathrm{depth}(R/I(\Delta)^{[k]})$ for all $k\ge 1$, where $\Delta$ is a Cohen-Macaulay simplicial forest. As an application, we show that the normalized depth function of a Cohen-Macaulay simplicial forest is nonincreasing.