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Some extensions of Hilbert–Kunz multiplicity

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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 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.

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Notes

  1. We remark here that by the methods used in proving our Theorem 2.4, the assumption that \(\hat{R}\) is reduced is unnecessary.

  2. Note that rings with FFRT need not be F-regular (or even F-rational). Shibuta [29] proved that if R is any 1-dimensional complete local domain of prime characteristic whose residue field is either finite or algebraically closed, then R has finite F-representation type.

  3. Here too, the reducedness assumption appears to be unnecessary, in light of methods used in the proof of Theorem 2.4.

  4. Recent work of Polstra [24] indicates that this limit exists in some generality, as well as the other limits defined in this section and the next.

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Authors and Affiliations

  1. Department of Mathematical Sciences, George Mason University, Fairfax, VA, 22030, USA

    Neil Epstein

  2. Department of Math and Statistics, Georgia State University, 30 Pryor St., Atlanta, GA, 30303, USA

    Yongwei Yao

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  1. Neil Epstein
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  2. Yongwei Yao
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Correspondence to Neil Epstein.

Additional information

The second author was partially supported by the National Science Foundation DMS-0700554.

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Epstein, N., Yao, Y. Some extensions of Hilbert–Kunz multiplicity. Collect. Math. 68, 69–85 (2017). https://doi.org/10.1007/s13348-016-0174-2

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