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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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.

[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