Square-free powers of Cohen-Macaulay simplicial forests

2025/02/25 by Das, Kanoy Kumar, Amit Roy, Roy, Amit +2 · 1 citation
Mathematics · #13C15 #13H10 #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Combinatorics (math.CO) #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2502.18396

openalex publication_date 2025/02/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let I(Δ)^[k] denote the k^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≥ 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.

Cited by

• Admissible set and squarefree-power-like function with applications to squarefree symbolic powers

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