Fregular and Fpure rings vs. log terminal and log canonical singularities
Authors:
Nobuo Hara and Keiichi Watanabe
Journal:
J. Algebraic Geom. 11 (2002), 363392
Published electronically:
December 17, 2001
MathSciNet review:
1874118
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Abstract 
References 
Additional Information
Abstract: We investigate the relationship of Fregular (resp. Fpure) rings and log terminal (resp. log canonical) singularities. Also, we extend the notions of Fregularity and Fpurity to ``Fsingularities of pairs." The notions of Fregular and Fpure rings in characteristic 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 divisor on it, and play important roles in birational algebraic geometry. As an analog of these singularities of pairs, we introduce the concept of ``Fsingularities of pairs," namely strong Fregularity, divisorial Fregularity and Fpurity for a pair of a normal ring of characteristic and an effective divisor on . The main theorem of this paper asserts that, if is Cartier, then the above three variants of Fsingularities of pairs imply KLT, PLT and LC properties, respectively. We also prove some results for Fsingularities of pairs which are analogous to singularities of pairs in characteristic zero.
 [AKM]
Aberbach, I., Katzman, M. and MacCrimmon, B., Weak Fregularity deforms in Gorenstein rings, J. Algebra 204 (1998), 281285.
 [A]
Alexeev, V., Classification of logcanonical surface singularities, in ``Flips and Abundance for Algebraic ThreefoldsSalt Lake City, Utah, August 1991," Asterisque No. 211, Soc. Math. France, 1992, pp. 4758.
 [E]
Ein, L., Multiplier ideals, vanishing theorems and applications: in ``Algebraic GeometrySanta Cruz 1995," Proc. Symp. Pure Math. 62 (1997), 203219.
 [EGA]
Grothendieck, A. and Dieudonné, J., Éléments de Géométrie Algébrique, Chap. IV, Publ. Math. I.H.E.S. Vol. 28, 1966.
 [F]
Fedder, R., Fpurity and rational singularity, Trans. Amer. Math. Soc. 278 (1983), 461480.
 [FW]
Fedder, R. and Watanabe, K.i., A characterization of Fregularity in terms of Fpurity, in ``Commutative Algebra," Math. Sci. Res. Inst. Publ. Vol. 15, SpringerVerlag, New York, 1989, pp. 227245.
 [Gl]
Glassbrenner, D., Strong Fregularity in images of regular local rings, Proc. Amer. Math. Soc. 124 (1996), 345353.
 [Ha1]
Hara, N., Fregularity and Fpurity of graded rings, J. Algebra 172 (1995), 804818.
 [Ha2]
, Classification of twodimensional Fregular and Fpure singularities, Adv. Math. 133 (1998), 3353.
 [Ha3]
, A characterization of rational singularities in terms of injectivity of Frobenius maps, Amer. J. Math. 120 (1998), 981996.
 [Ha4]
, Geometric interpretation of tight closure and test ideals, Trans. Amer. Math. Soc. 353 (2001), 18851906.
 [HH1]
Hochster, M. and Huneke, C., 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]
, Fregularity, test elements, and smooth base change, Trans. Amer. Math. Soc. 346 (1994), 162.
 [HR]
Hochster, M. and Roberts, J., The purity of the Frobenius and local cohomology, Adv. Math. 21 (1976), 117172.
 [Ka]
Kawamata, Y., Crepant blowingup of 3dimensional canonical singularities and its applications to degeneration of surfaces, Ann. Math. 127 (1988), 93163.
 [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), 283360.
 [Ko]
Kollár, J., Singularities of pairs: in ``Algebraic GeometrySanta Cruz 1995," Proc. Symp. Pure Math. 62 (1997), 221287.
 [Mc]
MacCrimmon, B., Weak Fregularity is strong Fregularity for rings with isolated nonGorenstein points, Trans. Amer. Math. Soc. (to appear).
 [MR]
Mehta, V. B. and Ramanathan, A., Frobenius splitting and cohomology vanishing for Schubert varieties, Ann. Math. 122 (1985), 2740.
 [MS1]
Mehta, V. B. and Srinivas, V., Normal Fpure surface singularities, J. Algebra 143 (1991), 130143.
 [MS2]
, A characterization of rational singularities, Asian J. Math. 1 (1997), 249278.
 [N]
Nakayama, N., Zariskidecomposition and abundance, RIMS preprint series 1142 (1997).
 [Si]
Singh, A., Fregularity does not deform, Amer. J. Math. 121 (1999), 919929.
 [Sh]
Shokurov V. V., fold log flips, Izv. Russ. A. N. Ser. Mat. 56 (1992), 105203.
 [S1]
Smith, K. E., Frational rings have rational singularities, Amer. J. Math. 119 (1997), 159180.
 [S2]
, The multiplier ideal is a universal test ideal, special volume in honor of R. Hartshorne, Comm. Algebra 28 (2000), no. 12, 59155929.
 [S3]
, Vanishing, singularities and effective bound via prime characteristic local algebra: in ``Algebraic GeometrySanta Cruz 1995," Proc. Symp. Pure Math. 62 (1997).
 [W1]
Watanabe, K.i., Some remarks concerning Demazure's construction of normal graded rings, Nagoya Math. J. 83 (1981), 203211.
 [W2]
, Fregular and Fpure normal graded rings, J. Pure Appl. Algebra 71 (1991), 341350.
 [W3]
, Fregular and Fpure rings vs. logterminal and logcanonical singularities, (an earlier version of the present paper).
 [Wi]
Williams, L. J., Uniform stability of kernels of Koszul cohomology indexed by the Frobenius endomorphism, J. Algebra 172 (1995), 721743.
 [AKM]
 Aberbach, I., Katzman, M. and MacCrimmon, B., Weak Fregularity deforms in Gorenstein rings, J. Algebra 204 (1998), 281285.
 [A]
 Alexeev, V., Classification of logcanonical surface singularities, in ``Flips and Abundance for Algebraic ThreefoldsSalt Lake City, Utah, August 1991," Asterisque No. 211, Soc. Math. France, 1992, pp. 4758.
 [E]
 Ein, L., Multiplier ideals, vanishing theorems and applications: in ``Algebraic GeometrySanta Cruz 1995," Proc. Symp. Pure Math. 62 (1997), 203219.
 [EGA]
 Grothendieck, A. and Dieudonné, J., Éléments de Géométrie Algébrique, Chap. IV, Publ. Math. I.H.E.S. Vol. 28, 1966.
 [F]
 Fedder, R., Fpurity and rational singularity, Trans. Amer. Math. Soc. 278 (1983), 461480.
 [FW]
 Fedder, R. and Watanabe, K.i., A characterization of Fregularity in terms of Fpurity, in ``Commutative Algebra," Math. Sci. Res. Inst. Publ. Vol. 15, SpringerVerlag, New York, 1989, pp. 227245.
 [Gl]
 Glassbrenner, D., Strong Fregularity in images of regular local rings, Proc. Amer. Math. Soc. 124 (1996), 345353.
 [Ha1]
 Hara, N., Fregularity and Fpurity of graded rings, J. Algebra 172 (1995), 804818.
 [Ha2]
 , Classification of twodimensional Fregular and Fpure singularities, Adv. Math. 133 (1998), 3353.
 [Ha3]
 , A characterization of rational singularities in terms of injectivity of Frobenius maps, Amer. J. Math. 120 (1998), 981996.
 [Ha4]
 , Geometric interpretation of tight closure and test ideals, Trans. Amer. Math. Soc. 353 (2001), 18851906.
 [HH1]
 Hochster, M. and Huneke, C., 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]
 , Fregularity, test elements, and smooth base change, Trans. Amer. Math. Soc. 346 (1994), 162.
 [HR]
 Hochster, M. and Roberts, J., The purity of the Frobenius and local cohomology, Adv. Math. 21 (1976), 117172.
 [Ka]
 Kawamata, Y., Crepant blowingup of 3dimensional canonical singularities and its applications to degeneration of surfaces, Ann. Math. 127 (1988), 93163.
 [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), 283360.
 [Ko]
 Kollár, J., Singularities of pairs: in ``Algebraic GeometrySanta Cruz 1995," Proc. Symp. Pure Math. 62 (1997), 221287.
 [Mc]
 MacCrimmon, B., Weak Fregularity is strong Fregularity for rings with isolated nonGorenstein points, Trans. Amer. Math. Soc. (to appear).
 [MR]
 Mehta, V. B. and Ramanathan, A., Frobenius splitting and cohomology vanishing for Schubert varieties, Ann. Math. 122 (1985), 2740.
 [MS1]
 Mehta, V. B. and Srinivas, V., Normal Fpure surface singularities, J. Algebra 143 (1991), 130143.
 [MS2]
 , A characterization of rational singularities, Asian J. Math. 1 (1997), 249278.
 [N]
 Nakayama, N., Zariskidecomposition and abundance, RIMS preprint series 1142 (1997).
 [Si]
 Singh, A., Fregularity does not deform, Amer. J. Math. 121 (1999), 919929.
 [Sh]
 Shokurov V. V., fold log flips, Izv. Russ. A. N. Ser. Mat. 56 (1992), 105203.
 [S1]
 Smith, K. E., Frational rings have rational singularities, Amer. J. Math. 119 (1997), 159180.
 [S2]
 , The multiplier ideal is a universal test ideal, special volume in honor of R. Hartshorne, Comm. Algebra 28 (2000), no. 12, 59155929.
 [S3]
 , Vanishing, singularities and effective bound via prime characteristic local algebra: in ``Algebraic GeometrySanta Cruz 1995," Proc. Symp. Pure Math. 62 (1997).
 [W1]
 Watanabe, K.i., Some remarks concerning Demazure's construction of normal graded rings, Nagoya Math. J. 83 (1981), 203211.
 [W2]
 , Fregular and Fpure normal graded rings, J. Pure Appl. Algebra 71 (1991), 341350.
 [W3]
 , Fregular and Fpure rings vs. logterminal and logcanonical singularities, (an earlier version of the present paper).
 [Wi]
 Williams, L. J., Uniform stability of kernels of Koszul cohomology indexed by the Frobenius endomorphism, J. Algebra 172 (1995), 721743.
Additional Information
Nobuo Hara
Affiliation:
Department of Mathematical Sciences, Waseda University, Okubo, Shinjukuku, Tokyo 1698555, Japan
Address at time of publication:
Mathematical Institute, Tohoku University, Sendai 9808578, Japan
Email:
hara@math.tohoku.ac.jp
Keiichi Watanabe
Affiliation:
Department of Mathematics, College of Humanities and Sciences, Nihon University, Sakurajosui, Setagayaku, Tokyo 1560045, Japan
Email:
watanabe@math.chs.nihonu.ac.jp
DOI:
http://dx.doi.org/10.1090/S105639110100306X
PII:
S 10563911(01)00306X
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 GrantinAid for Scientific Research, Japan
