Strong persistence and associated prime of powers of monomial ideals

Abstract

Let $R=K[x_1,\dots ,x_n]$ be the polynomial ring in $n$ variables over a field $K$ and $I$ be a monomial ideal of degree $d\le 2$. We show that $(I^{k+1}:I)=I^k$ for all $k\ge 1$ and we disprove a motivation question that was appeared in \cite[Question 2.51]{CHHV} by providing of a counterexample. Also, by this counterexample, we give a negative answer to the question that depth function of square-free monomial ideals are non-increasing.

arXiv comment6 pages. to appear in Comm. Algebra