Multiplicity of powers of squarefree monomial ideals

Abstract

Let $I$ be an arbitrary nonzero squarefree monomial ideal of dimension $d$ in a polynomial ring $S=\mathrm{k}[x_1,\dots ,x_n]$. Let $\mu$ be the number of associated primes of $S/I$ of dimension $d$. We prove that the multiplicity of powers of $I$ is given by $$e_0(S/I^s)=\mu \left(\genfrac{}{}{0ex}{}{n-d+s-1}{s-1}\right),$$ for all $s\ge 1$. Consequently, we compute the multiplicity of all powers of path ideals of cycles.