[Paper Review] Some extensions of Hilbert-Kunz multiplicity

Neil Epstein, Yongwei Yao|arXiv (Cornell University)|Mar 24, 2011

Commutative Algebra and Its Applications16 references3 citations

TL;DR

This paper extends Hilbert-Kunz multiplicity to develop numerical criteria for when nested ideals or submodules in prime characteristic rings have the same tight closure. By introducing relative measures based on 0th local cohomology and generalized j-multiplicity, the authors establish sufficient and, in some cases, necessary conditions—particularly via liminf and limsup limits of lengths of Frobenius powers—for tight closure equivalence, with key results showing equivalence under F-regularity and reducedness assumptions.

ABSTRACT

Let $R$ be an excellent Noetherian ring of prime characteristic. Consider an arbitrary nested pair of ideals (or more generally, a nested pair of submodules of a fixed finite module). We do \emph{not} assume that their quotient has finite length. In this paper, we develop various sufficient numerical criteria for when the tight closures of these ideals (or submodules) match. For some of the criteria we only prove sufficiency, while some are shown to be equivalent to the tight closures matching. We compare the various numerical measures (in some cases demonstrating that the different measures give truly different numerical results) and explore special cases where equivalence with matching tight closure can be shown. All of our measures derive ultimately from Hilbert-Kunz multiplicity.

Motivation & Objective

• To generalize Hilbert-Kunz multiplicity to relative settings for arbitrary nested pairs of ideals or submodules, even when their quotient does not have finite length.
• To develop numerical invariants derived from Hilbert-Kunz theory and j-multiplicity that detect tight closure equality.
• To establish sufficient and, in special cases, necessary conditions for tight closure equivalence using limits of lengths of Frobenius powers.
• To explore the relationship between tight closure and localization, particularly in the context of numerical vanishing conditions.
• To provide global criteria for tight closure equality across all prime ideals via localized numerical invariants.

Proposed method

• Define a new relative invariant $ u_N(L,M) $ based on the limit of $ \lambda(H^0_{\mathfrak{m}}(M^{[q]}_N / L^{[q]}_N)) / q^d $, generalizing Hilbert-Kunz multiplicity.
• Introduce limsup and liminf variants $ u^+_N(L,M) $ and $ u^-_N(L,M) $ to analyze asymptotic behavior of Frobenius powers.
• Use exact sequences involving 0th local cohomology modules to bound lengths of quotients $ I^{[q]}/J^{[q]} $, leading to inequalities involving $ f_n(I) $ and $ f_n(J) $.
• Apply the tight closure variant of Nakayama’s lemma to derive a criterion for tight closure equality that does not rely on local cohomology.
• Construct a j-multiplicity-inspired invariant in Section 7 that characterizes tight closure equality for ideals with finite-length quotients.
• Prove global equivalence results by localizing at prime ideals and using F-regularity and reducedness assumptions on completions.

Experimental results

Research questions

• RQ1 Under what numerical conditions does $ M \subseteq L^*_N $ hold for nested submodules $ L \subseteq M \subseteq N $ in a Noetherian local ring of prime characteristic?
• RQ2 Can the liminf of the normalized length of $ M^{[q]}_N / L^{[q]}_N $ be used as a necessary and sufficient condition for tight closure equality?
• RQ3 How do different numerical invariants—such as those derived from local cohomology, j-multiplicity, and Frobenius powers—compare in detecting tight closure equality?
• RQ4 In what cases does the vanishing of the global numerical invariant $ f^{-}(J_{\mathfrak{p}}, I_{\mathfrak{p}}) $ imply tight closure equality at all localizations?
• RQ5 To what extent do these numerical criteria remain equivalent when tight closure does not commute with localization?

Key findings

• The liminf version of the relative Hilbert-Kunz measure $ u^-_N(L,M) $ vanishes if and only if $ M \subseteq L^*_N $, under the assumption of a completely stable weak test element and reduced completion.
• For ideals $ J \subseteq I $ with $ \lambda(I/J) < \infty $, the condition $ \liminf_{n \to \infty} \frac{\lambda(I^{[p^n]}/J^{[p^n]})}{p^{nd}} = 0 $ is equivalent to $ I^* = J^* $, generalizing Hochster-Huneke's result.
• The global invariant $ f^{-}(J_{\mathfrak{p}}, I_{\mathfrak{p}}) \leq 0 $ for all primes $ \mathfrak{p} $ implies $ (I_{\mathfrak{m}})^* = (J_{\mathfrak{m}})^* $ for all maximal ideals $ \mathfrak{m} $, under F-regularity and reducedness assumptions.
• The j-multiplicity-inspired invariant in Section 7 provides a necessary and sufficient condition for tight closure equality when $ \lambda(I/J) < \infty $, and is distinct from other numerical measures.
• The authors demonstrate that the j-multiplicity-type invariant and the local cohomology-based invariant are generally different, showing that distinct numerical invariants can yield different results.
• In the case of F-regular rings on the punctured spectrum, the global condition $ f^{-}(J_{\mathfrak{p}}, I_{\mathfrak{p}}) \leq 0 $ for all $ \mathfrak{p} $ is equivalent to tight closure equality at all localizations.