Strong persistence and associated primes 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 asked by of Carlini, Hà, Harbourne and Van Tuyl by providing of a counterexample. Also, by this counterexample, we give a negative answer to the question that the depth function of square-free monomial ideals are non-increasing.

Keywords:

• Associated primes
• depth
• monomial ideals
• powers of ideals

2020 Mathematics Subject Classification:

• Primary: 13A15
• Secondary: 13A30
• 13C15

Acknowledgments

We would like to thank Kamran Divaani-Aazar and Mehrdad Nasernejad for some helpful comments. Also, we would like to thank deeply grateful to the referee for the careful reading of the manuscript.