Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

An interpretation of multiplier ideals via tight closure


Author: Shunsuke Takagi
Translated by:
Journal: J. Algebraic Geom. 13 (2004), 393-415
DOI: https://doi.org/10.1090/S1056-3911-03-00366-7
Published electronically: December 4, 2003
MathSciNet review: 2047704
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Abstract | References | Additional Information

Abstract: Hara [Trans. Amer. Math. Soc. 353 (2001), 1885-1906] and Smith [Comm. Algebra 28 (2000), 5915-5929] independently proved that in a normal ${\mathbb Q}$-Gorenstein ring of characteristic $p \gg 0$, the test ideal coincides with the multiplier ideal associated to the trivial divisor. We extend this result for a pair $(R, \Delta)$ of a normal ring $R$ and an effective ${\mathbb Q}$-Weil divisor $\Delta$ on $\operatorname{Spec}R$. As a corollary, we obtain the equivalence of strongly $\text{F}$-regular pairs and $\text{klt}$ pairs.


References [Enhancements On Off] (What's this?)

  • [AM] I. Aberbach and B. MacCrimmon, Some results on test ideals, Proc. Edinburgh Math. Soc. (2) 42 (1999), 541-549.
  • [BS] J. Briançon and H. Skoda, Sur la clôture intégrale d'un idéal de germes de fonctions holomorphes en un point de $C^n$, C. R. Acad. Sci. Paris. Sér. A 278 (1974), 949-951.
  • [DEL] J.-P. Demailly, L. Ein and R. Lazarsfeld, A subadditivity property of multiplier ideals, Michigan. Math. J. 48 (2000), 137-156.
  • [ELS] L. Ein, R. Lazarsfeld, and K. Smith, Uniform bounds and symbolic powers on smooth varieties, Invent. Math. 144 (2001), 241-252.
  • [Fe] R. Fedder, F-purity and rational singularity, Trans. Amer. Math. Soc. 278 (1983), 461-480.
  • [Ha1] N. Hara, F-regularity and F-purity of graded rings, J. Algebra, 172 (1995), 804-818.
  • [Ha2] -, A characterization of rational singularities in terms of injectivity of Frobenius maps, Amer. J. Math. 120 (1998), 981-996.
  • [Ha3] -, Geometric interpretation of tight closure and test ideals, Trans. Amer. Math. Soc. 353 (2001), 1885-1906.
  • [HW] N. Hara and K.-i. Watanabe, F-regular and F-pure rings vs. log terminal and log canonical singularities, J. Alg. Geom. 11 (2002), 363-392.
  • [HY] N. Hara and K. Yoshida, A generalization of tight closure and multiplier ideals, Trans. Amer. Math. Soc. 355 (2003), 3143-3174.
  • [HH1] M. Hochster and C. Huneke, Tight closure, invariant theory and the Briançon-Skoda theorem, J. Amer. Math. Soc. 3 (1990), 31-116.
  • [HH2] -, Tight closure and strong F-regularity, Mem. Soc. Math. France 38 (1989), 119-133.
  • [HH3] -, Comparison of symbolic and ordinary powers of ideals, Invent. Math. 147 (2002), 349-369.
  • [HR] M. Hochster and J. Roberts, The purity of the Frobenius and local cohomology, Adv. Math. 21 (1976), 117-172.
  • [Hu] C. Huneke, ``Tight closure and its applications,'' CBMS Regional Conf. Ser. Math. 88, Amer. Math. Soc., Providence (1996).
  • [Ko] J. Kollár, Singularities of pairs: in ``Algebraic Geometry-Santa Cruz 1995", Proc. Symp. Pure Math. 62 (1997), 221-287.
  • [KM] J. Kollár and S. Mori, ``Birational Geometry of Algebraic Varieties,'' Cambridge Tracts in Math. 134, Cambridge University Press, 1998.
  • [Ku] E. Kunz, On Noetherian rings of characteristic $p$, Amer. J. Math. 98 (1976), 999-1013.
  • [La] R. Lazarsfeld, Multiplier ideals for algebraic geometers, preprint.
  • [Mc] B. MacCrimmon, Weak F-regularity is strong F-regularity for rings with isolated non- ${\mathbb Q}$-Gorenstein points, Trans. Amer. Math. Soc. (to appear).
  • [MS] V. B. Mehta and V. Srinivas, A characterization of rational singularities, Asian. J. Math. 1 (1997), 249-278.
  • [Sm1] K. Smith, F-rational rings have rational singularities, Amer. J. Math. 119 (1997), 159-180.
  • [Sm2] -, The multiplier ideal is a universal test ideal, Comm. Algebra 28 (2000), 5915-5929.
  • [Wa] K.-i. Watanabe, A characterization of ``bad'' singularities via the Frobenius map, Proceedings of the 18-th symposium on commutative algebra (Toyama, 1996), 122-126, 1996. (in Japanese).
  • [Wi] L. J. Williams, Uniform stability of kernels of Koszul cohomology indexed by the Frobenius endomorphism, J. Algebra 172 (1995), 721-743.


Additional Information

Shunsuke Takagi
Affiliation: Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1, Komaba, Meguro, Tokyo 153-8914, Japan
Email: stakagi@ms.u-tokyo.ac.jp

DOI: https://doi.org/10.1090/S1056-3911-03-00366-7
Received by editor(s): December 17, 2001
Published electronically: December 4, 2003

American Mathematical Society