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Transactions of the American Mathematical Society

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Hilbert-Kunz density function and Hilbert-Kunz multiplicity


Author: V. Trivedi
Journal: Trans. Amer. Math. Soc. 370 (2018), 8403-8428
MSC (2010): Primary 13D40, 14H60, 14J60, 13H15
DOI: https://doi.org/10.1090/tran/7268
Published electronically: July 20, 2018
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Abstract: For a pair $ (M, I)$, where $ M$ is a finitely generated graded module over a standard graded ring $ R$ of dimension $ d$, and $ I$ is a graded ideal with $ \ell (R/I) < \infty $, we introduce a new invariant $ HKd(M, I)$ called the Hilbert-Kunz density function. We relate this to the Hilbert-Kunz multiplicity $ e_{HK}(M, I)$ by an integral formula.

We prove that the Hilbert-Kunz density function satisfies a multiplicative formula for a Segre product of rings. This gives a formula for $ e_{HK}$ of the Segre product of rings in terms of the HKd of the rings involved. As a corollary, $ e_{HK}$ of the Segre product of any finite number of projective curves is a rational number.


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Additional Information

V. Trivedi
Affiliation: School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai-400005, India
Email: vija@math.tifr.res.in

DOI: https://doi.org/10.1090/tran/7268
Keywords: Hilbert-Kunz density, Hilbert-Kunz multiplicity
Received by editor(s): April 27, 2016
Received by editor(s) in revised form: March 24, 2017
Published electronically: July 20, 2018
Article copyright: © Copyright 2018 American Mathematical Society

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