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: 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.