Square-free powers of Cohen-Macaulay simplicial forests

Source abstract

Let I ( Δ ) [ k ] I(\Delta )^{[k]} denote the k k th square-free power of the facet ideal of a simplicial complex Δ \Delta in a polynomial ring R R . Square-free powers are intimately related to the ‘Matching Theory’ and ‘Ordinary Powers’. In this article, we show that if Δ \Delta is a Cohen-Macaulay simplicial forest, then R / I ( Δ ) [ k ] R/I(\Delta )^{[k]} is Cohen-Macaulay for all k ≥ 1 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 d e p t h ( R / I ( Δ ) [ k ] ) \mathrm {depth}(R/I(\Delta )^{[k]}) for all k ≥ 1 k\ge 1 , where Δ \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.