Square-free powers of Cohen-Macaulay simplicial forests
Kanoy Kumar Das, Amit Roy, Kamalesh Saha
TL;DR
The paper investigates square-free powers $I(\Delta)^{[k]}$ of facet ideals of simplicial forests and proves that if $\Delta$ is a Cohen-Macaulay (CM) simplicial forest, then $R/I(\Delta)^{[k]}$ remains Cohen-Macaulay for all $k\ge1$. Central to the approach is the introduction of a new combinatorial tool, the special leaf, plus grafting and contraction techniques, enabling explicit depth and dimension formulas. Specifically, the authors establish the depth formula $\mathrm{depth}(R/I(\Delta)^{[k]})=|V(\Delta)|-\nu(\Delta)+k-1$ for $1\le k\le \nu(\Delta)$, where $\nu(\Delta)$ is the matching number, and deduce CM for all square-free powers. They also show the normalized depth function $g_{I(\Delta)}(k)$ is nonincreasing in this class. Together, these results extend the CM behavior of square-free powers beyond complete intersections and highlight the rich interplay between combinatorics (matchings, special leaves) and commutative algebra (depth, dimension, CM property).
Abstract
Let $I(Δ)^{[k]}$ denote the $k^{\text{th}}$ square-free power of the facet ideal of a simplicial complex $Δ$ 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 $Δ$ is a Cohen-Macaulay simplicial forest, then $R/I(Δ)^{[k]}$ 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 $\mathrm{depth}(R/I(Δ)^{[k]})$ for all $k\ge 1$, where $Δ$ is a Cohen-Macaulay simplicial forest. As an application, we show that the normalized depth function of a Cohen-Macaulay simplicial forest is nonincreasing.