Drops in joint Hilbert-Kunz multiplicities and projective equivalence
type: Item
instance of: scholarly article
title: Drops in joint Hilbert-Kunz multiplicities and projective equivalence
zbMATH Open document ID: 1223.13004
author: Adela Vraciu
published in: Journal of Algebra
publication date: 2010-04-21T00:00:00Z
review text: 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)].
Mathematics Subject Classification ID: 13A35
Mathematics Subject Classification ID: 13H15
zbMATH DE Number: 5696946
zbMATH Keywords: tight closure
zbMATH Keywords: Hilbert-Kunz multiplicities
zbMATH Keywords: projective equivalence
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2010.02.004
OpenAlex ID: W2075120738
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DOI: 10.1016/J.JALGEBRA.2010.02.004
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description: scientific article; zbMATH DE number 5696946