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On the generalized Hilbert-Kunz function and multiplicity

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Abstract

Let (R, m) be a local ring of characteristic p > 0 and M a finitely generated R-module. In this note we consider the limit

$$\mathop {\lim }\limits_{n \to \infty } \frac{{l(H_m^0({F^n}(M)))}}{{{p^{n\dim R}}}}$$

where F(-) is the Peskine-Szpiro functor. A consequence of our main results shows that the limit always exists when R is excellent, equidimensional and has an isolated singularity. Furthermore, if R is a complete intersection, then the limit is 0 if and only if the projective dimension of M is less than the Krull dimension of R. We exploit this fact to give a quick proof that if R is a complete intersection of dimension 3, then the Picard group of the punctured spectrum of R is torsion-free. Our results work quite generally for other homological functors and can be used to prove that certain limits recently studied by Brenner exist over projective varieties.

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Acknowledgements

We thank Craig Huneke and University of Virginia for creating the opportunity for us to work together! We thank Holger Brenner, Srikanth Iyengar, Ryo Takahashi and Kei-ichi Watanabe for many helpful conversations. We thank the anonymous referee for valuable comments.

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Author notes

  1. Ilya Smirnov

    Present address: Department of Mathematics, Stockholm University, SE - 106 91, Stockholm, Sweden

Authors and Affiliations

  1. Department of Mathematics, University of Kansas, Lawrence, KS, 66045-7523, USA

    Hailong Dao

  2. Department of Mathematics, University of Virginia, Charlottesville, VA, 22904-4137, USA

    Ilya Smirnov

Authors
  1. Hailong Dao
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  2. Ilya Smirnov
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Correspondence to Ilya Smirnov.

Additional information

Part of this work is supported by the National Science Foundation under Grant No. 0932078 000, while the authors were in residence at the Mathematical Science Research Institute (MSRI) in Berkeley, California, during the Commutative Algebra year in 2012-2013.

The first author is partially supported by NSF grant 1104017.

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Dao, H., Smirnov, I. On the generalized Hilbert-Kunz function and multiplicity. Isr. J. Math. 237, 155–184 (2020). https://doi.org/10.1007/s11856-020-2003-2

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  • DOI: https://doi.org/10.1007/s11856-020-2003-2

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