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Some computations of the generalized Hilbert-Kunz function and multiplicity

Authors: Hailong Dao and Kei-ichi Watanabe
Journal: Proc. Amer. Math. Soc. 144 (2016), 3199-3206
MSC (2010): Primary 13A35; Secondary 13D07, 13H10
Published electronically: April 13, 2016
MathSciNet review: 3503689
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Abstract: Let $ R$ be a local ring of characteristic $ p>0$ which is $ F$-finite and has perfect residue field. We compute the generalized Hilbert-Kunz invariant for certain modules over several classes of rings: hypersurfaces of finite representation type, toric rings, and weakly $ F$-regular rings.

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  • [1] Maurice Auslander and Ragnar-Olaf Buchweitz, The homological theory of maximal Cohen-Macaulay approximations, Colloque en l'honneur de Pierre Samuel (Orsay, 1987), Mém. Soc. Math. France (N.S.) 38 (1989), 5-37 (English, with French summary). MR 1044344 (91h:13010)
  • [2] Holger Brenner, The Hilbert-Kunz function in graded dimension two, Comm. Algebra 35 (2007), no. 10, 3199-3213. MR 2356151 (2008h:13028),
  • [3] Lindsay Burch, Codimension and analytic spread, Proc. Cambridge Philos. Soc. 72 (1972), 369-373. MR 0304377 (46 #3512)
  • [4] S. D. Cutkosky, Multiplicities of graded families of linear series and ideals, arXiv:1301.5613, preprint.
  • [5] Hailong Dao, Decent intersection and Tor-rigidity for modules over local hypersurfaces, Trans. Amer. Math. Soc. 365 (2013), no. 6, 2803-2821. MR 3034448,
  • [6] H. Dao, T. Se, Finite F-type and F-abundant modules,
  • [7] H. Dao, I. Smirnov, On generalized HIlbert-Kunz function and multiplicity, arxiv:1305.1833, preprint.
  • [8] N. Epstein, Y. Yao, Some extensions of Hilbert-Kunz multiplicity, arXiv:1103.4730, preprint.
  • [9] Craig Huneke, Tight closure and its applications, With an appendix by Melvin Hochster, CBMS Regional Conference Series in Mathematics, vol. 88, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1996. MR 1377268 (96m:13001)
  • [10] Daniel Katz and Javid Validashti, Multiplicities and Rees valuations, Collect. Math. 61 (2010), no. 1, 1-24. MR 2604855 (2011c:13004),
  • [11] Jack Jeffries and Jonathan Montaño, The $ j$-multiplicity of monomial ideals, Math. Res. Lett. 20 (2013), no. 4, 729-744. MR 3188029,
  • [12] Zsolt Patakfalvi and Karl Schwede, Depth of $ F$-singularities and base change of relative canonical sheaves, J. Inst. Math. Jussieu 13 (2014), no. 1, 43-63. MR 3134015,
  • [13] Gerhard Seibert, The Hilbert-Kunz function of rings of finite Cohen-Macaulay type, Arch. Math. (Basel) 69 (1997), no. 4, 286-296. MR 1466822 (98h:13022),
  • [14] Yongwei Yao, Modules with finite $ F$-representation type, J. London Math. Soc. (2) 72 (2005), no. 1, 53-72. MR 2145728 (2006b:13012),
  • [15] Yuji Yoshino, Cohen-Macaulay modules over Cohen-Macaulay rings, London Mathematical Society Lecture Note Series, vol. 146, Cambridge University Press, Cambridge, 1990. MR 1079937 (92b:13016)
  • [16] Keiichi Watanabe, $ F$-regular and $ F$-pure normal graded rings, J. Pure Appl. Algebra 71 (1991), no. 2-3, 341-350. MR 1117644 (92g:13003),

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Additional Information

Hailong Dao
Affiliation: Department of Mathematics, University of Kansas, Lawrence, Kansas 66045-7523

Kei-ichi Watanabe
Affiliation: Department of Mathematics, College of Human and Science, Nihon University, Setagaya, Tokyo, 156-0045, Japan

Keywords: Frobenius endomorphism, generalized Hilbert-Kunz multiplicity, toric rings, isolated singularity
Received by editor(s): March 3, 2015
Received by editor(s) in revised form: July 28, 2015
Published electronically: April 13, 2016
Additional Notes: The first author was partially supported by NSF grant 1104017
The second author was partially supported by JSPS Grant-in-Aid for Scientific Research (C) Grant Number 26400053
Communicated by: Irena Peeva
Article copyright: © Copyright 2016 American Mathematical Society

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