Multiplicities associated to generalized symbolic powers (Q350807)

Let \((R, \mathfrak m)\) be a noetherian local ring of dimension \(d\). For an ideal \(\mathfrak a\) of the definition of \(R\) and an \(R\)-module \(N\), denote by \(e_{\mathfrak a}(N)\) the Samuel multiplicity of \(\mathfrak a\) with respect to \(N\). Let \(I, J\) be ideals in \(R\) and \(I_n(J) = I^n:J^\infty\). Then \( s = \dim(I_n(J)/I^n\)) is constant for \(n\gg 0\) and is called the limit dimension of family of \(R\)-module \(I_n(J)/I^n\). [Math. Proc. Camb. Philos. Soc. 148, No. 1, 55-72 (2010; Zbl 1200.13010)] gave the question: When does NEWLINE\[NEWLINE\lim_{n\rightarrow \infty}\dfrac{e_{\mathfrak m}(I_n(J)/I^n)}{n^{d - s}}NEWLINE\]NEWLINE exist? This paper reviews some results on \(e_{\mathfrak m}(I_n(J)/I^n)\) in [Zbl 1200.13010] and show that \(\lim_{n\rightarrow \infty}\dfrac{e_{\mathfrak m}(I_n(J)/I^n)}{n^{d - s}}\) exists under very general conditions by using some recent results of the author [Math. Res. Lett. 18, No. 1, 93--106 (2011; Zbl 1238.13012); ``Multiplicities Associated to Graded Families of Ideals'', \url{arXiv.1206.4077}].