The $k$ th symbolic power of a square free monomial ideal $\rm I$ is $\rm I^{(k)}= \cap_{p\in Min(I)}p^{k}.$

Question

Let $\rm I$ is a square free monomial ideal in $K[X_1,\ldots,X_n].$ The $k$ th symbolic power of $\rm I$, denoted by $\rm I^{(k)}$ defined to be the intersection of all primary components of $\rm I^{k}$ corresponding to the minimal primes of $\rm I^{k}$ ( Which is same as the minimal primes of $\rm I$ ) Then I need to show that $\rm I^{(k)}= \cap_{p\in Min(I)}p^{k}.$

Writing $\rm I^k=q_1\cap\cdots\cap q_s\cap\cdots \cap q_h$ where $q_i s$ are $p_i$ primary ideal and $p_1,\ldots,p_s$ are the minimal primes we want to show that $q_1\cap \cdots\cap q_s=p_1^{k}\cap\cdots\cap p_s^{k}.$ We know that $\rm I$ being a square free ideal is a radical ideal and hence $\rm I=p_1\cap\cdots p_s.$ But then I cannot proceed further. Help me, thanks.

Answer 1

Some references include https://arxiv.org/abs/1309.5082 (see section 3) and Villareal, Monomial Algebras (2nd ed) 2015, Proposition 4.3.24-25.