Resumen de On stability properties of powers of polymatroidal ideals

Shokoufe Karimi, Amir Mafi

• Let ?=?[?1,…,??] be the polynomial ring in n variables over a field K with the maximal ideal ?=(?1,…,??) . Let astab(?) and dstab(?) be the smallest integer n for which Ass(??) and depth(??) stabilize, respectively. In this paper we show that astab(?)=dstab(?) in the following cases:

  (i) I is a matroidal ideal and ?≤5 .

  (ii) I is a polymatroidal ideal, ?=4 and ?∉Ass∞(?) , where Ass∞(?) 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 astab(?)≠dstab(?) . This is a counterexample to the conjecture of Herzog and Qureshi, according to which these two numbers are the same for polymatroidal ideals.