An interpretation of multiplier ideals via tight closure
Author:
Shunsuke Takagi
Translated by:
Journal:
J. Algebraic Geom. 13 (2004), 393415
Published electronically:
December 4, 2003
MathSciNet review:
2047704
Fulltext PDF
Abstract 
References 
Additional Information
Abstract: Hara [Trans. Amer. Math. Soc. 353 (2001), 18851906] and Smith [Comm. Algebra 28 (2000), 59155929] independently proved that in a normal Gorenstein ring of characteristic , the test ideal coincides with the multiplier ideal associated to the trivial divisor. We extend this result for a pair of a normal ring and an effective Weil divisor on . As a corollary, we obtain the equivalence of strongly regular pairs and pairs.
 [AM]
I. Aberbach and B. MacCrimmon, Some results on test ideals, Proc. Edinburgh Math. Soc. (2) 42 (1999), 541549.
 [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), 949951.
 [DEL]
J.P. Demailly, L. Ein and R. Lazarsfeld, A subadditivity property of multiplier ideals, Michigan. Math. J. 48 (2000), 137156.
 [ELS]
L. Ein, R. Lazarsfeld, and K. Smith, Uniform bounds and symbolic powers on smooth varieties, Invent. Math. 144 (2001), 241252.
 [Fe]
R. Fedder, Fpurity and rational singularity, Trans. Amer. Math. Soc. 278 (1983), 461480.
 [Ha1]
N. Hara, Fregularity and Fpurity of graded rings, J. Algebra, 172 (1995), 804818.
 [Ha2]
, A characterization of rational singularities in terms of injectivity of Frobenius maps, Amer. J. Math. 120 (1998), 981996.
 [Ha3]
, Geometric interpretation of tight closure and test ideals, Trans. Amer. Math. Soc. 353 (2001), 18851906.
 [HW]
N. Hara and K.i. Watanabe, Fregular and Fpure rings vs. log terminal and log canonical singularities, J. Alg. Geom. 11 (2002), 363392.
 [HY]
N. Hara and K. Yoshida, A generalization of tight closure and multiplier ideals, Trans. Amer. Math. Soc. 355 (2003), 31433174.
 [HH1]
M. Hochster and C. Huneke, Tight closure, invariant theory and the BriançonSkoda theorem, J. Amer. Math. Soc. 3 (1990), 31116.
 [HH2]
, Tight closure and strong Fregularity, Mem. Soc. Math. France 38 (1989), 119133.
 [HH3]
, Comparison of symbolic and ordinary powers of ideals, Invent. Math. 147 (2002), 349369.
 [HR]
M. Hochster and J. Roberts, The purity of the Frobenius and local cohomology, Adv. Math. 21 (1976), 117172.
 [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 GeometrySanta Cruz 1995", Proc. Symp. Pure Math. 62 (1997), 221287.
 [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), 9991013.
 [La]
R. Lazarsfeld, Multiplier ideals for algebraic geometers, preprint.
 [Mc]
B. MacCrimmon, Weak Fregularity is strong Fregularity for rings with isolated non 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), 249278.
 [Sm1]
K. Smith, Frational rings have rational singularities, Amer. J. Math. 119 (1997), 159180.
 [Sm2]
, The multiplier ideal is a universal test ideal, Comm. Algebra 28 (2000), 59155929.
 [Wa]
K.i. Watanabe, A characterization of ``bad'' singularities via the Frobenius map, Proceedings of the 18th symposium on commutative algebra (Toyama, 1996), 122126, 1996. (in Japanese).
 [Wi]
L. J. Williams, Uniform stability of kernels of Koszul cohomology indexed by the Frobenius endomorphism, J. Algebra 172 (1995), 721743.
 [AM]
 I. Aberbach and B. MacCrimmon, Some results on test ideals, Proc. Edinburgh Math. Soc. (2) 42 (1999), 541549.
 [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), 949951.
 [DEL]
 J.P. Demailly, L. Ein and R. Lazarsfeld, A subadditivity property of multiplier ideals, Michigan. Math. J. 48 (2000), 137156.
 [ELS]
 L. Ein, R. Lazarsfeld, and K. Smith, Uniform bounds and symbolic powers on smooth varieties, Invent. Math. 144 (2001), 241252.
 [Fe]
 R. Fedder, Fpurity and rational singularity, Trans. Amer. Math. Soc. 278 (1983), 461480.
 [Ha1]
 N. Hara, Fregularity and Fpurity of graded rings, J. Algebra, 172 (1995), 804818.
 [Ha2]
 , A characterization of rational singularities in terms of injectivity of Frobenius maps, Amer. J. Math. 120 (1998), 981996.
 [Ha3]
 , Geometric interpretation of tight closure and test ideals, Trans. Amer. Math. Soc. 353 (2001), 18851906.
 [HW]
 N. Hara and K.i. Watanabe, Fregular and Fpure rings vs. log terminal and log canonical singularities, J. Alg. Geom. 11 (2002), 363392.
 [HY]
 N. Hara and K. Yoshida, A generalization of tight closure and multiplier ideals, Trans. Amer. Math. Soc. 355 (2003), 31433174.
 [HH1]
 M. Hochster and C. Huneke, Tight closure, invariant theory and the BriançonSkoda theorem, J. Amer. Math. Soc. 3 (1990), 31116.
 [HH2]
 , Tight closure and strong Fregularity, Mem. Soc. Math. France 38 (1989), 119133.
 [HH3]
 , Comparison of symbolic and ordinary powers of ideals, Invent. Math. 147 (2002), 349369.
 [HR]
 M. Hochster and J. Roberts, The purity of the Frobenius and local cohomology, Adv. Math. 21 (1976), 117172.
 [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 GeometrySanta Cruz 1995", Proc. Symp. Pure Math. 62 (1997), 221287.
 [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), 9991013.
 [La]
 R. Lazarsfeld, Multiplier ideals for algebraic geometers, preprint.
 [Mc]
 B. MacCrimmon, Weak Fregularity is strong Fregularity for rings with isolated non 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), 249278.
 [Sm1]
 K. Smith, Frational rings have rational singularities, Amer. J. Math. 119 (1997), 159180.
 [Sm2]
 , The multiplier ideal is a universal test ideal, Comm. Algebra 28 (2000), 59155929.
 [Wa]
 K.i. Watanabe, A characterization of ``bad'' singularities via the Frobenius map, Proceedings of the 18th symposium on commutative algebra (Toyama, 1996), 122126, 1996. (in Japanese).
 [Wi]
 L. J. Williams, Uniform stability of kernels of Koszul cohomology indexed by the Frobenius endomorphism, J. Algebra 172 (1995), 721743.
Additional Information
Shunsuke Takagi
Affiliation:
Graduate School of Mathematical Sciences, University of Tokyo, 381, Komaba, Meguro, Tokyo 1538914, Japan
Email:
stakagi@ms.utokyo.ac.jp
DOI:
http://dx.doi.org/10.1090/S1056391103003667
PII:
S 10563911(03)003667
Received by editor(s):
December 17, 2001
Published electronically:
December 4, 2003
