Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-17T14:42:39.115Z Has data issue: false hasContentIssue false
  • English
  • Français

Local 𝔪-adic constancy of F-pure thresholds and test ideals

Published online by Cambridge University Press:  02 May 2017

DANIEL J. HERNÁNDEZ ,
LUIS NÚÑEZ-BETANCOURT  and
EMILY E. WITT
DANIEL J. HERNÁNDEZ
Affiliation:
University of Kansas, Department of Mathematics, 405 Snow Hall, 1460 Jayhawk Blvd, Lawrence, KS 66045–7594, USA. e-mail: hernandez@ku.edu
LUIS NÚÑEZ-BETANCOURT
Affiliation:
Centro de Investigación en Matemáticas, A.C., Jalisco S/N, Col. Valenciana CP: 36023 Guanajuato, Gto, Mexico, Apartado Postal 402, CP 36000. e-mail: luisnub@cimat.mx
EMILY E. WITT
Affiliation:
University of Kansas, Department of Mathematics, 405 Snow Hall, 1460 Jayhawk Blvd, Lawrence, KS 66045–7594, USA. e-mail: witt@ku.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we consider a corollary of the ACC conjecture for F-pure thresholds. Specifically, we show that the F-pure threshold (and more generally, the test ideals) associated to a polynomial with an isolated singularity are locally constant in the 𝔪-adic topology of the corresponding local ring. As a by-product of our methods, we also describe a simple algorithm for computing all of the F-jumping numbers and test ideals associated to an arbitrary polynomial over an F-finite field.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 164 , Issue 2 , March 2018 , pp. 285 - 295
Copyright
Copyright © Cambridge Philosophical Society 2017 

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.)

References

REFERENCES

[1] Montaner, J. À., Blickle, M. and Lyubeznik, G. Generators of D-modules in positive characteristic. Math. Res. Lett. 12 (2005), no. 4, 459–473.CrossRefGoogle Scholar
[2] Benito, A., Faber, E. and Smith, K. E. Measuring singularities with Frobenius: the basics. Commutative Algebra. (Springer, New York, 2013), pp. 57–97.CrossRefGoogle Scholar
[3] Blickle, M., Mustaţǎ, M. and Smith, K. E. Discreteness and rationality of F-thresholds. Michigan Math. J. 57 (2008), 43–61. Special volume in honor of Melvin Hochster.CrossRefGoogle Scholar
[4] Blickle, M., Mustaţǎ, M. and Smith, K. E. F-thresholds of hypersurfaces. Trans. Amer. Math. Soc. 361 (2009), no. 12, 6549–6565.CrossRefGoogle Scholar
[5] Boix, A. F. and Katzman, M. An algorithm for producing F-pure ideals. Arch. Math. (Basel) 103 (2014), no. 5, 421–433.CrossRefGoogle Scholar
[6] de Fernex, T., Ein, L. and Mustaţǎ, M. Shokurov's ACC conjecture for log canonical thresholds on smooth varieties. Duke Math. J. 152 (2010), no. 1, 93–114.CrossRefGoogle Scholar
[7] de Fernex, T., Ein, L. and Mustaţǎ, M. Log canonical thresholds on varieties with bounded singularities. Classification of algebraic varieties. EMS Ser. Congr. Rep., Eur. Math. Soc. (Zürich, 2011), pp. 221–257.Google Scholar
[8] Hacon, C. D., McKernan, J. and Xu, C. Acc for log canonical thresholds, arXiv:1208:4150 (2012).Google Scholar
[9] Hara, N. and Yoshida, K.-I. A generalization of tight closure and multiplier ideals. Trans. Amer. Math. Soc. 355 (2003), no. 8, 3143–3174 (electronic).CrossRefGoogle Scholar
[10] Hernández, D. J. F-invariants of diagonal hypersurfaces. To appear in Proc. Amer. Math. Soc. Google Scholar
[11] Hernández, D. J. F-pure thresholds of binomial hypersurfaces. Proc. Amer. Math. Soc. 142 no. 7, (2014), 2227–2242.CrossRefGoogle Scholar
[12] Hernández, D. J., Núñez-Betancourt, L., Witt, E. E. and Zhang, W. F-pure thresholds of quasi-homogeneous polynomials, Preprint (2013).Google Scholar
[13] Hochster, M. and Huneke, C. Tight closure, invariant theory, and the Briançon-Skoda theorem. J. Amer. Math. Soc. 3 (1990), no. 1, 31–116.Google Scholar
[14] Hochster, M. and Huneke, C. F-regularity, test elements and smooth base change. Trans. Amer. Math. Soc. 346 (1994), no. 1, 1–62.Google Scholar
[15] Katzman, M. Parameter-test-ideals of Cohen-Macaulay rings. Composition Math. 144, no. 4 (2008), 933–948.CrossRefGoogle Scholar
[16] Katzman, M., Lyubeznik, G. and Zhang, W. An upper bound on the number of F-jumping coefficients of a principal ideal. Proc. Amer. Math. Soc. 139, no. 12 (2011), 4193–4197.CrossRefGoogle Scholar
[17] Katzman, M. and Schwede, K. An algorithm for computing compatibly Frobenius split subvarieties. J. Symbolic Comput. 47 no. 8, (2012), 996–1008.CrossRefGoogle Scholar
[18] Mustaţǎ, M., Takagi, S. and Watanabe, K.-I. F-thresholds and Bernstein-Sato polynomials. European Congress of Mathematics, Eur. Math. Soc. (Zürich 2005), pp. 341–364.Google Scholar
[19] Schwede, K. and Tucker, K. Test ideals of non-principal ideals: computations, jumping numbers, alterations and division theorems. J. Math. Pures Appl. (9) 102 (2014), no. 5, 891–929.CrossRefGoogle Scholar
[20] Shokurov, V. V. Three-dimensional log perestroikas. Izv. Ross. Akad. Nauk Ser. Mat. 56 (1992), no. 1, 105–203.Google Scholar
[21] Takagi, S. and Watanabe, K.-I. On F-pure thresholds. J. Algebra 282 (2004), no. 1, 278–297.CrossRefGoogle Scholar