Persistence and stability properties of powers of ideals (Q472996)

It is known by \textit{M. Brodmann} [Proc. Am. Math. Soc. 74, 16--18 (1979; Zbl 0372.13010)] that for an ideal \(I\) in a Notherian ring \(R\) there exists an integer \(k_0\) such that \(\mathrm{Ass}(I^k)=\mathrm{Ass}(I^{k+1})\) for \(k\geq k_0\). The smallest integer \(k_0\) with this property is called the index of stability for the associated prime ideals of \(I\) and denoted by \(\mathrm{astab}(I)\). It is natural to ask whether there exists an upper bound for \(\mathrm{astab}(I)\) in terms of \(R\). The authors due to evidence conjecture that \(\mathrm{astab}(I) <\dim(S)\) for all graded ideal \(I\) in a polynomial ring \(S\) and they prove the class of polymatroidal ideals supports this conjecture. A prime ideal \(P\in \bigcup_{k\geq 1}\mathrm{Ass}(I^k)\) is said to be persistent with respect to \(I\) if whenever \(P\in \mathrm{Ass}(I^k)\) then \(P\in \mathrm{Ass}(I^{k+1})\), and we say that \(I\) satisfies the persistence property if all prime ideals \(P\in \bigcup_{k\geq 1}\mathrm{Ass}(I^k)\) are persistent. In the paper under review, the authors introduce the concept of strong persistence. An ideal \(I\subset R\) is said to satisfy the strong persistence property with respect to \(P\in V(I)\) if for all \(k\) and all \(f\in(I^k_P : \mathfrak{m}_P )\setminus I^k_P\), where \(\mathfrak{m}_P\) is the maximal ideal of the local ring \(R_P\), there exists \(g\in I_P\) such that \(fg\notin I^{k+1}_P\). We say that the ideal \(I\) satisfies the strong persistence property if it satisfies the strong persistence property for all \(P\in V(I)\). It is obvious that the strong persistence property implies the persistence property. The authors prove that all polymatroidal ideals satisfy strong persistence. Let \(I\) be a graded ideal in a polynomial ring \(S\). We call the numerical function \(f(k)=\mathrm{depth}(S/I^k)\) the depth function of \(I\). It is known that the \(\mathrm{depth} S/I^k\) is constant for \(k\gg 0\), \textit{M. Brodmann} [Math. Proc. Camb. Philos. Soc. 86, 35--39 (1979; Zbl 0413.13011)]. The smallest integer \(k\) for which \(\mathrm{depth} S/I^k=\mathrm{depth} S/I^{k+r}\) for all \(r\geq 1\) is called the index of depth stability of \(I\) and denoted by \(\mathrm{dstab}(I)\). As for \(\mathrm{astab}(I)\), the authors expect that \(\mathrm{dstab}(I) <\dim(S)\) for all graded ideal \(I\subset S\). They prove that this inequality holds true for polymatroidal ideals.