Collectanea Mathematica

, Volume 68, Issue 1, pp 69–85 | Cite as

Some extensions of Hilbert–Kunz multiplicity



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 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.


Hilbert–Kunz multiplicity Tight closure 

Mathematics Subject Classification

Primary 13A35 Secondary 13D40 


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Copyright information

© Universitat de Barcelona 2016

Authors and Affiliations

  1. 1.Department of Mathematical SciencesGeorge Mason UniversityFairfaxUSA
  2. 2.Department of Math and StatisticsGeorgia State UniversityAtlantaUSA

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