An interpretation of multiplier ideals via tight closure
Author:
Shunsuke Takagi
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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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.
[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.
[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
Email:
stakagi@ms.u-tokyo.ac.jp
Received by editor(s):
December 17, 2001
Published electronically:
December 4, 2003
