Hostname: page-component-6d856f89d9-8l2sj Total loading time: 0 Render date: 2024-07-16T05:45:25.319Z Has data issue: false hasContentIssue false
  • English
  • Français

CONTINUITY OF HILBERT–KUNZ MULTIPLICITY AND F-SIGNATURE

Part of: Local theory Commutative algebra: Homological methods General commutative ring theory Local rings and semilocal rings

Published online by Cambridge University Press:  27 December 2018

THOMAS POLSTRA  and
ILYA SMIRNOV Open the ORCID record for ILYA SMIRNOV [Opens in a new window]
THOMAS POLSTRA
Affiliation:
Department of Mathematics, University of Utah, Salt Lake City, USA email polstra@math.utah.edu
ILYA SMIRNOV
Affiliation:
Department of Mathematics, Stockholm University, S-106 91, Stockholm, Sweden email smirnov@math.su.se
Get access Rights & Permissions [Opens in a new window]

Abstract

We establish the continuity of Hilbert–Kunz multiplicity and F-signature as functions from a Cohen–Macaulay local ring $(R,\mathfrak{m},k)$ of prime characteristic to the real numbers at reduced parameter elements with respect to the $\mathfrak{m}$-adic topology.

MSC classification

Primary: 13D40: Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series 13A35: Characteristic $p$ methods (Frobenius endomorphism) and reduction to characteristic $p$; tight closure 13H15: Multiplicity theory and related topics 14B05: Singularities
Type
Article
Information
Nagoya Mathematical Journal , Volume 239 , September 2020 , pp. 322 - 345
Check for updates
Copyright
© 2018 Foundation Nagoya Mathematical Journal

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Polstra was supported in part by NSF Postdoctoral Research Fellowship DMS #1703856.

References

Aberbach, I. M. and Enescu, F., Lower bounds for Hilbert–Kunz multiplicities in local rings of fixed dimension, Michigan Math. J. 57 (2008), 1–16. Special volume in honor of Melvin Hochster.10.1307/mmj/1220879393Google Scholar
Aberbach, I. M. and Leuschke, G. J., The F-signature and strong F-regularity, Math. Res. Lett. 10(1) (2003), 51–56.10.4310/MRL.2003.v10.n1.a6Google Scholar
Adamus, J. and Patel, A., On finite determinacy of complete intersection singularities, preprint, 2017, arXiv:1705.08985.Google Scholar
Blickle, M. and Enescu, F., On rings with small Hilbert–Kunz multiplicity, Proc. Amer. Math. Soc. 132(9) (2004), 2505–2509 (electronic).10.1090/S0002-9939-04-07469-6Google Scholar
Bruns, W. and Herzog, J., Cohen–Macaulay Rings, Cambridge Studies in Advanced Mathematics 39, Cambridge University Press, Cambridge, 1993.Google Scholar
Cutkosky, S. D. and Srinivasan, H., An intrinsic criterion for isomorphism of singularities, Amer. J. Math. 115(4) (1993), 789–821.10.2307/2375013Google Scholar
De Stefani, A., Polstra, T. and Yao, Y., Globalizing F-invariants, preprint, 2016, arXiv:1608.08580.Google Scholar
Enescu, F. and Yao, Y., The lower semicontinuity of the Frobenius splitting numbers, Math. Proc. Cambridge Philos. Soc. 150(1) (2011), 35–46.10.1017/S0305004110000484Google Scholar
de Fernex, T., Ein, L. and Mustaţă, M., Shokurov’s ACC conjecture for log canonical thresholds on smooth varieties, Duke Math. J. 152(1) (2010), 93–114.10.1215/00127094-2010-008Google Scholar
Gabber, O. and Orgogozo, F., Sur la p-dimension des corps, Invent. Math. 174(1) (2008), 47–80.10.1007/s00222-008-0133-yGoogle Scholar
Hacon, C. D., McKernan, J. and Xu, C., ACC for log canonical thresholds, Ann. of Math. (2) 180(2) (2014), 523–571.10.4007/annals.2014.180.2.3Google Scholar
Hernández, D. J., Núñez-Betancourt, L. and Witt, E. E., Local m-adic constancy of F-pure thresholds and test ideals, Math. Proc. Cambridge Philos. Soc. 164(2) (2018), 285–295.10.1017/S0305004117000196Google Scholar
Hironaka, H., “On the equivalence of singularities. I”, in Arithmetical Algebraic Geometry (Proc. Conf. Purdue Univ., 1963), Harper & Row, New York, 1965, 153–200.Google Scholar
Hochster, M., Cyclic purity versus purity in excellent Noetherian rings, Trans. Amer. Math. Soc. 231(2) (1977), 463–488.10.1090/S0002-9947-1977-0463152-5Google Scholar
Hochster, M., The structure theory of complete local rings. Supplementary notes to Math 615 Winter 2017, available at http://www.math.lsa.umich.edu/∼hochster/615W17/615.html.Google Scholar
Hochster, M., Foundations of tight closure. Lecture notes available at http://www.math.lsa.umich.edu/∼hochster/mse.html.Google Scholar
Hochster, M. and Huneke, C., Tight closure, invariant theory, and the Briançon–Skoda theorem, J. Amer. Math. Soc. 3(1) (1990), 31–116.Google Scholar
Huneke, C. and Leuschke, G. J., Two theorems about maximal Cohen–Macaulay modules, Math. Ann. 324(2) (2002), 391–404.10.1007/s00208-002-0343-3Google Scholar
Kunz, E., Characterizations of regular local rings of characteristic p, Amer. J. Math. 91 (1969), 772–784.10.2307/2373351Google Scholar
Kunz, E., On Noetherian rings of characteristic p, Amer. J. Math. 98(4) (1976), 999–1013.Google Scholar
Kurano, K. and Shimomoto, K., An elementary proof of Cohen–Gabber theorem in the equal characteristic p > 0 case, Tohoku Math. J. (2) 70(3) (2018), 377–389.10.2748/tmj/1537495352+0+case,+Tohoku+Math.+J.+(2)+70(3)+(2018),+377–389.10.2748/tmj/1537495352>Google Scholar
Matsumura, H., Commutative Ring Theory, Cambridge Studies in Advanced Mathematics 8, Cambridge University Press, Cambridge, 1986. Translated from the Japanese by M. Reid.Google Scholar
Monsky, P., The Hilbert–Kunz function, Math. Ann. 263(1) (1983), 43–49.10.1007/BF01457082Google Scholar
Monsky, P., Hilbert–Kunz functions in a family: point-S 4 quartics, J. Algebra 208(1) (1998), 343–358.Google Scholar
Polstra, T. and Tucker, K., F-signature and Hilbert–Kunz multiplicity: a combined approach and comparison, Algebra Number Theory 12(1) (2018), 61–97.10.2140/ant.2018.12.61Google Scholar
Samuel, P., Algébricité de certains points singuliers algébroïdes, J. Math. Pures Appl. (9) 35 (1956), 1–6.Google Scholar
Sato, K., Ascending chain condition for $F$-pure thresholds on a fixed strongly $F$-regular germ, preprint, 2017, arXiv:1710.05331.Google Scholar
Sato, K., Ascending chain condition for $F$-pure thresholds with fixed embedding dimension, preprint, 2018, arXiv:1805.07066.10.1093/imrn/rnz031Google Scholar
Shokurov, V. V., Three-dimensional log perestroikas, Izv. Ross. Akad. Nauk Ser. Mat. 56(1) (1992), 105–203.Google Scholar
Smith, K. E. and Van den Bergh, M., Simplicity of rings of differential operators in prime characteristic, Proc. Lond. Math. Soc. (3) 75(1) (1997), 32–62.10.1112/S0024611597000257Google Scholar
Srinivas, V. and Trivedi, V., The invariance of Hilbert functions of quotients under small perturbations, J. Algebra 186(1) (1996), 1–19.10.1006/S0021-8693(96)90000-9Google Scholar
Takagi, S. and Watanabe, K.-i., On F-pure thresholds, J. Algebra 282(1) (2004), 278–297.10.1016/j.jalgebra.2004.07.011Google Scholar
Tucker, K., F-signature exists, Invent. Math. 190(3) (2012), 743–765.Google Scholar
Watanabe, K.-i. and Yoshida, K.-i., Hilbert–Kunz multiplicity and an inequality between multiplicity and colength, J. Algebra 230(1) (2000), 295–317.10.1006/jabr.1999.7956Google Scholar