The stable set of associated prime ideals of a polymatroidal ideal
Let \(I\) be an ideal in a Noetherian ring \(R\). The set of associated prime ideals of \(R/I\) is denoted by \(\mathrm{Ass}(I)\). In [Proc. Am. Math. Soc. 74, 16--18 (1979; Zbl 0372.13010)] \textit{M. Brodmann} showed that \(\mathrm{Ass}(I^k) = \mathrm{Ass}(I^{k+1})\) for all \(k\gg0\). The smallest number \(k_0\) for which this happens is called ``the index of stability'' and \(\mathrm{Ass}(I^{k_0})\) is called ``the stable set of associated prime ideals of \(I\)'' denoted by \(\mathrm{Ass}^{\infty}(I)\). The following natural questions arise in the contexts of Brodmann's Theorem: (1) Is there an upper bound for the index of stability depending only on \(R\)? (2) What can be said about the set \(\mathrm{Ass}^{\infty}(I)\)? Can \(\mathrm{Ass}^{\infty}(I)\) be computed in case that \(R\) is a polynomial ring and \(I\) is a graded ideal? (3) Is it true that \(\mathrm{Ass}(I)\subseteq \mathrm{Ass}(I^2)\subseteq\ldots \subseteq \mathrm{Ass}(I^k)\subseteq \ldots\) ? All these questions are widely open, even for monomial ideals. In the paper under review the authors study the associated prime ideals of powers of polymatroidal ideals, including the stable set of associated prime ideals of this class of ideals. It is shown that polymatroidal ideals have the persistence property and for transversal polymatroids and polymatroidal ideals of Veronese type the index of stability and the stable set of associated ideals is determined explicitly. This is a very clear and well-written paper.
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