Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-13T07:48:01.705Z Has data issue: false hasContentIssue false
  • English
  • Français

Test Ideals Vs. Multiplier Ideals

Published online by Cambridge University Press:  11 January 2016

Mircea Mustaţă  and
Ken-Ichi Yoshida
Mircea Mustaţă
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA, mmustata@umich.edu
Ken-Ichi Yoshida
Affiliation:
Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya, 464-8602, Japan, yoshida@math.nagoya-u.ac.jp
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The generalized test ideals introduced in [HY] are related to multiplier ideals via reduction to characteristic p. In addition, they satisfy many of the subtle properties of the multiplier ideals, which in characteristic zero follow via vanishing theorems. In this note we give several examples to emphasize the different behavior of test ideals and multiplier ideals. Our main result is that every ideal in an F-finite regular local ring can be written as a generalized test ideal. We also prove the semicontinuity of F-pure thresholds (though the analogue of the Generic Restriction Theorem for multiplier ideals does not hold).

Keywords

13A3514B05
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 193 , 2009 , pp. 111 - 128
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2009

References

[BMS1] Blickle, M., Mustaţǎ, M. and Smith, K. E., Discreteness and rationality of F-thresholds, Michigan Math. J., 57 (2008), 43–61.Google Scholar
[BMS2] Blickle, M., Mustaţǎ, M. and Smith, K. E., F-thresholds of hypersurfaces, Trans. Amer. Math. Soc., to appear.Google Scholar
[ELSV] Ein, L., Lazarsfeld, R., Smith, K. E., and Varolin, D., Jumping coefficients of multiplier ideals, Duke Math. J., 123 (2004), 469–506.Google Scholar
[FJ] Favre, C. and Jonsson, M., Valuations and multiplier ideals, J. Amer. Math. Soc., 18 (2005), 655–684.Google Scholar
[Ha] Hara, N., with an appendix by Monsky, P., F-pure thresholds and F-jumping coefficients in dimension two, Math. Res. Lett., 13 (2006), 747–760.Google Scholar
[HT] Hara, N. and Takagi, S., On a generalization of test ideals, Nagoya Math. J., 175 (2004), 59–74.CrossRefGoogle Scholar
[HY] Hara, N. and Yoshida, K., A generalization of tight closure and multiplier ideals, Trans. Amer. Math. Soc., 355 (2003), 3143–3174.Google Scholar
[HWY] Hara, N., Watanabe, K.-i. and Yoshida, K., F-rationality of Rees algebras, J. Algebra, 247 (2002), 153–190.Google Scholar
[Ku] Kunz, E., Characterization of regular local rings in characteristic p, Amer. J. Math., 91 (1969), 772–784.Google Scholar
[Laz] Lazarsfeld, R., Positivity in Algebraic Geometry II, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, A Series of Modern Surveys in Mathematics, Vol. 49, Springer-Verlag, Berlin, 2004.Google Scholar
[LL] Lazarsfeld, R. and Lee, K., Local syzygies of multiplier ideals, Invent. Math., 167 (2007), 409–418.Google Scholar
[LW] Lipman, J. and Watanabe, K.-i., Integrally closed ideals in two-dimensional regular local rings are multiplier ideals, Math. Res. Letters, 10 (2003), 423–434.Google Scholar
[MTW] 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
[Sch] Schwede, K., Generalized test ideals, sharp F-purity, and sharp test elements, Math. Res. Lett., 15 (2008), 1251–1262.Google Scholar
[Ta] Takagi, S., Formulas for multiplier ideals on singular varieties, Amer. J. Math., 128 (2006), 1345–1362.Google Scholar
[TW] Takagi, S. and Watanabe, K.-i., On F-pure thresholds, J. Algebra, 282 (2004), 278–297.Google Scholar