Drops in joint Hilbert-Kunz multiplicities and projective equivalence
Let \((R,\mathfrak m)\) be a local or a graded ring of positive characteristic \(p\) and dimension \(d\). Let \(\mathfrak a,I\) be \(\mathfrak m\)-primary ideals of \(R\). Let \(q=p^e\). The joint Hilbert-Kunz multiplicity of \(R\) with respect to \(\mathfrak a\) is defined as \[ e_{HK}(\mathfrak a;I)=\lim_{q\to\infty}\frac{1}{q^d}\lambda\Big(\frac{R}{\mathfrak a^qI^{[q]}}\Big). \] For a real number \(t\geq 0\) write \(\mathfrak a^{tq}\) for \(\mathfrak a^{\lceil tq\rceil}\). If \(x\) is an element not in \(I\) the drop in joint Hilbert-Kunz multiplicity is defined as \[ D(\mathfrak a^t,I,x)=e_{HK}(\mathfrak a^t;I)-e_{HK}(\mathfrak a^t;(I,x)). \] The first part of the paper studies the asymptotic properties of \(D(\mathfrak a^t,I,x)\) as a function in \(t\), and it is proved that this function is bounded above by \(Ct^{d-1}\), where \(C>0\) is a constant. The author then defines the \(j^{th}\) level ideal of \(I\) with respect to \(\mathfrak a\) to be the ideal \[ I_{\mathfrak a,j}=\{x\in R\mid\exists C>0\text{ such that }D(\mathfrak a^t,I,x)\leq Ct^{d-1-j}\}, \] and asks which \(\mathfrak m\)-primary ideals \(\mathfrak a,\mathfrak b\) give rise to the same level ideals for all \(I\). The paper studies this question for monomial ideals, and in this case, relates the level ideals to the notion of projective equivalence introduced by \textit{P. Samuel} [in Ann. Math. (2) 56, 11--21 (1952; Zbl 0049.02301)].
scientific article; zbMATH DE number 5696946
scientific article