Square-free powers of Cohen-Macaulay simplicial forests

Let I(Δ)[k] denote the kᵗʰ 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≥ 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 depth(R/I(Δ)[k]) for all k≥ 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.