F-regular and F-pure rings vs. log terminal and log canonical singularities

Authors:
Nobuo Hara and Kei-ichi Watanabe

Journal:
J. Algebraic Geom. **11** (2002), 363-392

DOI:
https://doi.org/10.1090/S1056-3911-01-00306-X

Published electronically:
December 17, 2001

MathSciNet review:
1874118

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Abstract |
References |
Additional Information

Abstract: We investigate the relationship of F-regular (resp. F-pure) rings and log terminal (resp. log canonical) singularities. Also, we extend the notions of F-regularity and F-purity to “F-singularities of pairs." The notions of F-regular and F-pure rings in characteristic $p > 0$ are characterized by a splitting of the Frobenius map, and define some classes of rings having “mild" singularities. On the other hand, there are notions of log terminal and log canonical singularities defined via resolution of singularities in characteristic zero. These are defined also for pairs of a normal variety and a $\mathbb Q$-divisor on it, and play important roles in birational algebraic geometry. As an analog of these singularities of pairs, we introduce the concept of “F-singularities of pairs," namely strong F-regularity, divisorial F-regularity and F-purity for a pair $(A,\Delta )$ of a normal ring $A$ of characteristic $p > 0$ and an effective $\mathbb Q$-divisor $\Delta$ on $\operatorname {Spec} A$. The main theorem of this paper asserts that, if $K_{A}+\Delta$ is $\mathbb Q$-Cartier, then the above three variants of F-singularities of pairs imply KLT, PLT and LC properties, respectively. We also prove some results for F-singularities of pairs which are analogous to singularities of pairs in characteristic zero.

[AKM]AKM Aberbach, I., Katzman, M. and MacCrimmon, B., *Weak F-regularity deforms in $\mathbb Q$-Gorenstein rings*, J. Algebra **204** (1998), 281–285.
[A]A Alexeev, V., *Classification of log-canonical surface singularities*, in “Flips and Abundance for Algebraic Threefolds—Salt Lake City, Utah, August 1991," Asterisque No. 211, Soc. Math. France, 1992, pp. 47–58.
[E]E Ein, L., *Multiplier ideals, vanishing theorems and applications: in “Algebraic Geometry—Santa Cruz 1995,"* Proc. Symp. Pure Math. **62** (1997), 203–219.
[EGA]EGA Grothendieck, A. and Dieudonné, J., *Éléments de Géométrie Algébrique*, Chap. IV, Publ. Math. I.H.E.S. Vol. 28, 1966.
[F]F Fedder, R., *F-purity and rational singularity*, Trans. Amer. Math. Soc. **278** (1983), 461–480.
[FW]FW Fedder, R. and Watanabe, K.-i., *A characterization of F-regularity in terms of F-purity*, in “Commutative Algebra," Math. Sci. Res. Inst. Publ. Vol. 15, Springer-Verlag, New York, 1989, pp. 227–245.
[Gl]Gl Glassbrenner, D., *Strong F-regularity in images of regular local rings*, Proc. Amer. Math. Soc. **124** (1996), 345–353.
[Ha1]Ha1 Hara, N., *F-regularity and F-purity of graded rings*, J. Algebra **172** (1995), 804–818.
[Ha2]Ha2 ---, *Classification of two-dimensional F-regular and F-pure singularities*, Adv. Math. **133** (1998), 33–53.
[Ha3]Ha3 ---, *A characterization of rational singularities in terms of injectivity of Frobenius maps*, Amer. J. Math. **120** (1998), 981–996.
[Ha4]Ha4 ---, *Geometric interpretation of tight closure and test ideals*, Trans. Amer. Math. Soc. **353** (2001), 1885–1906.
[HH1]HH1 Hochster, M. and Huneke, C., *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 ---, *F-regularity, test elements, and smooth base change*, Trans. Amer. Math. Soc. **346** (1994), 1–62.
[HR]HR Hochster, M. and Roberts, J., *The purity of the Frobenius and local cohomology*, Adv. Math. **21** (1976), 117–172.
[Ka]Ka Kawamata, Y., *Crepant blowing-up of 3-dimensional canonical singularities and its applications to degeneration of surfaces*, Ann. Math. **127** (1988), 93–163.
[KMM]KMM Kawamata, Y., Matsuda, K. and Matsuki, K., *Introduction to the minimal model problem: in “Algebraic Geometry, Sendai 1985,"* Adv. Stud. Pure Math. **10** (1987), 283–360.
[Ko]Ko Kollár, J., *Singularities of pairs: in “Algebraic Geometry—Santa Cruz 1995,"* Proc. Symp. Pure Math. **62** (1997), 221–287.
[Mc]Mc MacCrimmon, B., *Weak F-regularity is strong F-regularity for rings with isolated non-$\mathbb Q$-Gorenstein points*, Trans. Amer. Math. Soc. (to appear).
[MR]MR Mehta, V. B. and Ramanathan, A., *Frobenius splitting and cohomology vanishing for Schubert varieties*, Ann. Math. **122** (1985), 27–40.
[MS1]MS1 Mehta, V. B. and Srinivas, V., *Normal F-pure surface singularities*, J. Algebra **143** (1991), 130–143.
[MS2]MS2 ---, *A characterization of rational singularities*, Asian J. Math. **1** (1997), 249–278.
[N]N Nakayama, N., *Zariski-decomposition and abundance*, RIMS preprint series 1142 (1997).
[Si]Si Singh, A., *F-regularity does not deform*, Amer. J. Math. **121** (1999), 919–929.
[Sh]Sh Shokurov V. V., *$3$-fold log flips*, Izv. Russ. A. N. Ser. Mat. **56** (1992), 105–203.
[S1]S1 Smith, K. E., *F-rational rings have rational singularities*, Amer. J. Math. **119** (1997), 159–180.
[S2]S2 ---, *The multiplier ideal is a universal test ideal*, special volume in honor of R. Hartshorne, Comm. Algebra **28** (2000), no. 12, 5915–5929.
[S3]S3 ---, *Vanishing, singularities and effective bound via prime characteristic local algebra: in “Algebraic Geometry—Santa Cruz 1995,"* Proc. Symp. Pure Math. **62** (1997).
[W1]W1 Watanabe, K.-i., *Some remarks concerning Demazure’s construction of normal graded rings*, Nagoya Math. J. **83** (1981), 203–211.
[W2]W2 ---, *F-regular and F-pure normal graded rings*, J. Pure Appl. Algebra **71** (1991), 341-350.
[W3]W3 ---, *F-regular and F-pure rings vs. log-terminal and log-canonical singularities*, (an earlier version of the present paper).
[Wi]Wi Williams, L. J., *Uniform stability of kernels of Koszul cohomology indexed by the Frobenius endomorphism*, J. Algebra **172** (1995), 721–743.

[AKM]AKM Aberbach, I., Katzman, M. and MacCrimmon, B., *Weak F-regularity deforms in $\mathbb Q$-Gorenstein rings*, J. Algebra **204** (1998), 281–285.
[A]A Alexeev, V., *Classification of log-canonical surface singularities*, in “Flips and Abundance for Algebraic Threefolds—Salt Lake City, Utah, August 1991," Asterisque No. 211, Soc. Math. France, 1992, pp. 47–58.
[E]E Ein, L., *Multiplier ideals, vanishing theorems and applications: in “Algebraic Geometry—Santa Cruz 1995,"* Proc. Symp. Pure Math. **62** (1997), 203–219.
[EGA]EGA Grothendieck, A. and Dieudonné, J., *Éléments de Géométrie Algébrique*, Chap. IV, Publ. Math. I.H.E.S. Vol. 28, 1966.
[F]F Fedder, R., *F-purity and rational singularity*, Trans. Amer. Math. Soc. **278** (1983), 461–480.
[FW]FW Fedder, R. and Watanabe, K.-i., *A characterization of F-regularity in terms of F-purity*, in “Commutative Algebra," Math. Sci. Res. Inst. Publ. Vol. 15, Springer-Verlag, New York, 1989, pp. 227–245.
[Gl]Gl Glassbrenner, D., *Strong F-regularity in images of regular local rings*, Proc. Amer. Math. Soc. **124** (1996), 345–353.
[Ha1]Ha1 Hara, N., *F-regularity and F-purity of graded rings*, J. Algebra **172** (1995), 804–818.
[Ha2]Ha2 ---, *Classification of two-dimensional F-regular and F-pure singularities*, Adv. Math. **133** (1998), 33–53.
[Ha3]Ha3 ---, *A characterization of rational singularities in terms of injectivity of Frobenius maps*, Amer. J. Math. **120** (1998), 981–996.
[Ha4]Ha4 ---, *Geometric interpretation of tight closure and test ideals*, Trans. Amer. Math. Soc. **353** (2001), 1885–1906.
[HH1]HH1 Hochster, M. and Huneke, C., *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 ---, *F-regularity, test elements, and smooth base change*, Trans. Amer. Math. Soc. **346** (1994), 1–62.
[HR]HR Hochster, M. and Roberts, J., *The purity of the Frobenius and local cohomology*, Adv. Math. **21** (1976), 117–172.
[Ka]Ka Kawamata, Y., *Crepant blowing-up of 3-dimensional canonical singularities and its applications to degeneration of surfaces*, Ann. Math. **127** (1988), 93–163.
[KMM]KMM Kawamata, Y., Matsuda, K. and Matsuki, K., *Introduction to the minimal model problem: in “Algebraic Geometry, Sendai 1985,"* Adv. Stud. Pure Math. **10** (1987), 283–360.
[Ko]Ko Kollár, J., *Singularities of pairs: in “Algebraic Geometry—Santa Cruz 1995,"* Proc. Symp. Pure Math. **62** (1997), 221–287.
[Mc]Mc MacCrimmon, B., *Weak F-regularity is strong F-regularity for rings with isolated non-$\mathbb Q$-Gorenstein points*, Trans. Amer. Math. Soc. (to appear).
[MR]MR Mehta, V. B. and Ramanathan, A., *Frobenius splitting and cohomology vanishing for Schubert varieties*, Ann. Math. **122** (1985), 27–40.
[MS1]MS1 Mehta, V. B. and Srinivas, V., *Normal F-pure surface singularities*, J. Algebra **143** (1991), 130–143.
[MS2]MS2 ---, *A characterization of rational singularities*, Asian J. Math. **1** (1997), 249–278.
[N]N Nakayama, N., *Zariski-decomposition and abundance*, RIMS preprint series 1142 (1997).
[Si]Si Singh, A., *F-regularity does not deform*, Amer. J. Math. **121** (1999), 919–929.
[Sh]Sh Shokurov V. V., *$3$-fold log flips*, Izv. Russ. A. N. Ser. Mat. **56** (1992), 105–203.
[S1]S1 Smith, K. E., *F-rational rings have rational singularities*, Amer. J. Math. **119** (1997), 159–180.
[S2]S2 ---, *The multiplier ideal is a universal test ideal*, special volume in honor of R. Hartshorne, Comm. Algebra **28** (2000), no. 12, 5915–5929.
[S3]S3 ---, *Vanishing, singularities and effective bound via prime characteristic local algebra: in “Algebraic Geometry—Santa Cruz 1995,"* Proc. Symp. Pure Math. **62** (1997).
[W1]W1 Watanabe, K.-i., *Some remarks concerning Demazure’s construction of normal graded rings*, Nagoya Math. J. **83** (1981), 203–211.
[W2]W2 ---, *F-regular and F-pure normal graded rings*, J. Pure Appl. Algebra **71** (1991), 341-350.
[W3]W3 ---, *F-regular and F-pure rings vs. log-terminal and log-canonical singularities*, (an earlier version of the present paper).
[Wi]Wi Williams, L. J., *Uniform stability of kernels of Koszul cohomology indexed by the Frobenius endomorphism*, J. Algebra **172** (1995), 721–743.

Additional Information

**Nobuo Hara**

Affiliation:
Department of Mathematical Sciences, Waseda University, Okubo, Shinjuku-ku, Tokyo 169-8555, Japan

Address at time of publication:
Mathematical Institute, Tohoku University, Sendai 980-8578, Japan

Email:
hara@math.tohoku.ac.jp

**Kei-ichi Watanabe**

Affiliation:
Department of Mathematics, College of Humanities and Sciences, Nihon University, Sakura-josui, Setagaya-ku, Tokyo 156-0045, Japan

MR Author ID:
216208

Email:
watanabe@math.chs.nihon-u.ac.jp

Received by editor(s):
January 17, 2000

Received by editor(s) in revised form:
August 21, 2000

Published electronically:
December 17, 2001

Additional Notes:
Both authors are partially supported by Grant-in-Aid for Scientific Research, Japan