On stability properties of powers of polymatroidal ideals

Abstract

Let $R=K[x_1,\ldots ,x_n]$ be the polynomial ring in n variables over a field K with the maximal ideal $\mathfrak {m}=(x_1,\ldots ,x_n)$. Let ${\text {astab}}(I)$ and ${\text {dstab}}(I)$ be the smallest integer n for which ${\text {Ass}}(I^n)$ and ${\text {depth}}(I^n)$ stabilize, respectively. In this paper we show that ${\text {astab}}(I)={\text {dstab}}(I)$ in the following cases:

• (i) I is a matroidal ideal and $n\le 5$.

• (ii) I is a polymatroidal ideal, $n=4$ and $\mathfrak {m}\notin {\text {Ass}}^{\infty }(I)$, where ${\text {Ass}}^{\infty }(I)$ is the stable set of associated prime ideals of I.

• (iii) I is a polymatroidal ideal of degree 2.

Moreover, we give an example of a polymatroidal ideal for which ${\text {astab}}(I)\ne {\text {dstab}}(I)$. This is a counterexample to the conjecture of Herzog and Qureshi, according to which these two numbers are the same for polymatroidal ideals.