On stability properties of powers of polymatroidal ideals

• Autores: Shokoufe Karimi, Amir Mafi
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 70, Fasc. 3, 2019, págs. 357-365
• Idioma: inglés
• DOI: 10.1007/s13348-018-0234-x
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• Resumen

  • 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.