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
Journal: J. Algebraic Geom. 13 (2004), 393-415
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.

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    [AM]AM I. Aberbach and B. MacCrimmon, Some results on test ideals, Proc. Edinburgh Math. Soc. (2) 42 (1999), 541–549. [BS]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]DEL J.-P. Demailly, L. Ein and R. Lazarsfeld, A subadditivity property of multiplier ideals, Michigan. Math. J. 48 (2000), 137–156. [ELS]ELS L. Ein, R. Lazarsfeld, and K. Smith, Uniform bounds and symbolic powers on smooth varieties, Invent. Math. 144 (2001), 241–252. [Fe]Fe R. Fedder, F-purity and rational singularity, Trans. Amer. Math. Soc. 278 (1983), 461–480. [Ha1]Ha1 N. Hara, F-regularity and F-purity of graded rings, J. Algebra, 172 (1995), 804–818. [Ha2]Ha2 ---, A characterization of rational singularities in terms of injectivity of Frobenius maps, Amer. J. Math. 120 (1998), 981–996. [Ha3]Ha3 ---, Geometric interpretation of tight closure and test ideals, Trans. Amer. Math. Soc. 353 (2001), 1885–1906. [HW]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]HY N. Hara and K. Yoshida, A generalization of tight closure and multiplier ideals, Trans. Amer. Math. Soc. 355 (2003), 3143–3174. [HH1]HH1 M. Hochster and C. Huneke, Tight closure, invariant theory and the Briançon-Skoda theorem, J. Amer. Math. Soc. 3 (1990), 31–116. [HH2]HH2 ---, Tight closure and strong F-regularity, Mem. Soc. Math. France 38 (1989), 119–133. [HH3]HH3 ---, Comparison of symbolic and ordinary powers of ideals, Invent. Math. 147 (2002), 349–369. [HR]HR M. Hochster and J. Roberts, The purity of the Frobenius and local cohomology, Adv. Math. 21 (1976), 117–172. [Hu]Hu C. Huneke, “Tight closure and its applications,” CBMS Regional Conf. Ser. Math. 88, Amer. Math. Soc., Providence (1996). [Ko]Ko J. Kollár, Singularities of pairs: in “Algebraic Geometry-Santa Cruz 1995", Proc. Symp. Pure Math. 62 (1997), 221–287. [KM]KM J. Kollár and S. Mori, “Birational Geometry of Algebraic Varieties,” Cambridge Tracts in Math. 134, Cambridge University Press, 1998. [Ku]Ku E. Kunz, On Noetherian rings of characteristic $p$, Amer. J. Math. 98 (1976), 999–1013. [La]La R. Lazarsfeld, Multiplier ideals for algebraic geometers, preprint. [Mc]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]MS V. B. Mehta and V. Srinivas, A characterization of rational singularities, Asian. J. Math. 1 (1997), 249–278. [Sm1]Sm1 K. Smith, F-rational rings have rational singularities, Amer. J. Math. 119 (1997), 159–180. [Sm2]Sm2 ---, The multiplier ideal is a universal test ideal, Comm. Algebra 28 (2000), 5915–5929. [Wa]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]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

Received by editor(s): December 17, 2001
Published electronically: December 4, 2003