Multiplicity of powers of squarefree monomial ideals

Abstract

Let $I$ be an arbitrary nonzero squarefree monomial ideal of dimension $d$ in a polynomial ring $S = \mathrm{k}[x_1,\ldots,x_n]$. Let $μ$ be the number of associated primes of $S/I$ of dimension $d$. We prove that the multiplicity of powers of $I$ is given by $$e_0(S/I^s) = μ\binom{n-d+s-1}{s-1},$$ for all $s \ge 1$. Consequently, we compute the multiplicity of all powers of path ideals of cycles.

Theorems & Definitions (17)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• proof
• Remark 2.2
• proof : Proof of Theorem \ref{['thm_mul']}
• Proposition 3.1
• proof
• Corollary 3.2
• proof