Hilbert-Kunz density function and Hilbert-Kunz multiplicity
HTML articles powered by AMS MathViewer
- by V. Trivedi PDF
- Trans. Amer. Math. Soc. 370 (2018), 8403-8428 Request permission
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.
References
- Ian M. Aberbach, Extension of weakly and strongly F-regular rings by flat maps, J. Algebra 241 (2001), no. 2, 799–807. MR 1843326, DOI 10.1006/jabr.2001.8785
- Holger Brenner, The rationality of the Hilbert-Kunz multiplicity in graded dimension two, Math. Ann. 334 (2006), no. 1, 91–110. MR 2208950, DOI 10.1007/s00208-005-0703-x
- Kazufumi Eto and Ken-ichi Yoshida, Notes on Hilbert-Kunz multiplicity of Rees algebras, Comm. Algebra 31 (2003), no. 12, 5943–5976. MR 2014910, DOI 10.1081/AGB-120024861
- Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157
- Melvin Hochster and Craig Huneke, Tight closure, invariant theory, and the Briançon-Skoda theorem, J. Amer. Math. Soc. 3 (1990), no. 1, 31–116. MR 1017784, DOI 10.1090/S0894-0347-1990-1017784-6
- Craig Huneke, Hilbert-Kunz multiplicity and the F-signature, Commutative algebra, Springer, New York, 2013, pp. 485–525. MR 3051383, DOI 10.1007/978-1-4614-5292-8_{1}5
- Adrian Langer, Semistable sheaves in positive characteristic, Ann. of Math. (2) 159 (2004), no. 1, 251–276. MR 2051393, DOI 10.4007/annals.2004.159.251
- Mandira Mondal, In preparation.
- P. Monsky, The Hilbert-Kunz function, Math. Ann. 263 (1983), no. 1, 43–49. MR 697329, DOI 10.1007/BF01457082
- V. Trivedi, Semistability and Hilbert-Kunz multiplicities for curves, J. Algebra 284 (2005), no. 2, 627–644. MR 2114572, DOI 10.1016/j.jalgebra.2004.10.016
- Trivedi, Vijaylaxmi, Towards Hilbert-Kunz density functions in Characteristic $0$, Nagoya Math. J., (2018), 1-43 DOI 10.1017/nmj.2018.7
Additional Information
- V. Trivedi
- Affiliation: School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai-400005, India
- MR Author ID: 318259
- Email: vija@math.tifr.res.in
- Received by editor(s): April 27, 2016
- Received by editor(s) in revised form: March 24, 2017
- Published electronically: July 20, 2018
- © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 370 (2018), 8403-8428
- MSC (2010): Primary 13D40, 14H60, 14J60, 13H15
- DOI: https://doi.org/10.1090/tran/7268
- MathSciNet review: 3864381