Square-free powers of Cohen-Macaulay simplicial forests

Abstract

Let $I(\hat{I}”{)}^{[k]}$ denote the $k^{\text{th}}$ square-free power of the facet ideal of a simplicial complex $\hat{I}”$ 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 $\hat{I}”$ is a Cohen-Macaulay simplicial forest, then $R/I(\hat{I}”{)}^{[k]}$ 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 $\mathrm{depth}(R/I(\hat{I}”{)}^{[k]})$ for all $k\ge 1$, where $\hat{I}”$ is a Cohen-Macaulay simplicial forest. As an application, we show that the normalized depth function of a Cohen-Macaulay simplicial forest is nonincreasing.

Final version. To appear in Proceedings of the American Mathematical Society