Journal of Algebraic Geometry

Author(s): Shunsuke Takagi
Journal: J. Algebraic Geom. 13 (2004), 393-415.
Posted: December 4, 2003
Retrieve article in: PDF DVI PostScript

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

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.

[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. 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 , 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.

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

PII: S 1056-3911(03)00366-7
Received by editor(s): December 17, 2001
Posted: December 4, 2003

Previous issue \| Table of contents \| Next issue Recently posted articles \| Most recent issue \| All issues