Cyclic covers of rings with rational singularities
Author:
Anurag K. Singh
Journal:
Trans. Amer. Math. Soc. 355 (2003), 10091024
MSC (2000):
Primary 13A35, 13A02; Secondary 13H10, 14B05
Published electronically:
November 1, 2002
MathSciNet review:
1938743
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: We examine some recent work of Phillip Griffith on étale covers and fibered products from the point of view of tight closure theory. While it is known that cyclic covers of Gorenstein rings with rational singularities are CohenMacaulay, we show this is not true in general in the absence of the Gorenstein hypothesis. Specifically, we show that the canonical cover of a Gorenstein ring with rational singularities need not be CohenMacaulay.
 [Bo]
J.F. Boutot, Singularités rationnelles et quotients par les groupes réductifs, Invent. Math. 88 (1987), 6568. MR 88a:14005
 [Ch]
W.L. Chow, On unmixedness theorem, Amer. J. Math. 86 (1964), 799822. MR 30:2031
 [De]
M. Demazure, Anneaux gradués normaux, in: Le Dûng Tráng (ed.) Introduction à la théorie des singularités II, Hermann, Paris, (1988), 3568. MR 91k:14004
 [GoW]
S. Goto and K.i. Watanabe, On graded rings I, J. Math. Soc. Japan 30 (1978), 179213. MR 81m:13021
 [GrW]
P. Griffith and D. Weston, Restrictions of torsion divisor classes to hypersurfaces, J. Algebra 167 (1994), 473487. MR 95c:13008
 [Gr]
P. Griffith, Induced formal deformations and the CohenMacaulay property, Trans. Amer. Math. Soc. 353 (2001), 7793. MR 2001b:13020
 [Ha]
N. Hara, A characterisation of rational singularities in terms of injectivity of Frobenius maps, Amer. J. Math. 120 (1998), 981996. MR 99h:13005
 [HW]
N. Hara and K.i. Watanabe, Fregular and Fpure rings vs. log terminal and log canonical singularities, J. Algebraic Geom. 11 (2002), 363392.
 [HH1]
M. Hochster and C. Huneke, Tight closure and strong Fregularity, Mem. Soc. Math. France 38 (1989), 119133. MR 91i:13025
 [HH2]
M. Hochster and C. Huneke, Tight closure, invariant theory, and the BriançonSkoda theorem, J. Amer. Math. Soc. 3 (1990), 31116. MR 91g:13010
 [HH3]
M. Hochster and C. Huneke, Fregularity, test elements, and smooth base change, Trans. Amer. Math. Soc. 346 (1994), 162. MR 95d:13007
 [HH4]
M. Hochster and C. Huneke, Tight closure of parameter ideals and splitting in modulefinite extensions, J. Algebraic Geom. 3 (1994), 599670. MR 95k:13002
 [HH5]
M. Hochster and C. Huneke, Tight closure in equal characteristic zero, in preparation.
 [HR]
M. Hochster and J. Roberts, Rings of invariants of reductive groups acting on regular rings are CohenMacaulay, Adv. in Math. 13 (1974), 115175. MR 50:311
 [Ka]
Y. Kawamata, The cone of curves of algebraic varieties, Ann. of Math. (2) 119 (1984), 603633. MR 86c:14013b
 [Li]
J. Lipman, Rational singularities, with applications to algebraic surfaces and unique factorization, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 195279. MR 43:1986
 [LS]
G. Lyubeznik and K. E. Smith, Strong and weak Fregularity are equivalent for graded rings, Amer. J. Math. 121 (1999), 12791290. MR 2000m:13006
 [Ma]
H. Matsumura, Commutative ring theory, Cambridge Studies in Advanced Mathematics 8, Cambridge University Press, CambridgeNew York, 1986. MR 88h:13001
 [Si1]
A. K. Singh, Gorenstein splinter rings of characteristic are Fregular, Mathematical Proceedings of the Cambridge Philosophical Society 127 (1999), 201205. MR 2000j:13006
 [Si2]
A. K. Singh, Multisymbolic Rees algebras and strong Fregularity, Math. Z. 235 (2000), 335344. MR 2001j:13006
 [Sm1]
K. E. Smith, Frational rings have rational singularities, Amer. J. Math. 119 (1997), 159180. MR 97k:13004
 [Sm2]
K. E. Smith, Vanishing theorems, singularities, and effective bounds in algebraic geometry via prime characteristic local algebra, in: J. Kollár, R. Lazarsfeld and David R. Morrison (eds.) Proc. Sympos. Pure Math. 62 (1997), 289325. MR 99a:14026
 [TW]
M. Tomari and K.i. Watanabe, Normal graded rings and normal cyclic covers, Manuscripta Math. 76 (1992), 325340. MR 93j:13002
 [Wa1]
K.i. Watanabe, Some remarks concerning Demazure's construction of normal graded rings, Nagoya Math. J. 83 (1981), 203211. MR 83g:13016
 [Wa2]
K.i. Watanabe, Rational singularities with action, in: Commutative Algebra (Trento, 1981), Lecture Notes in Pure and Appl. Math. 84, Marcel Dekker, New York, (1983), 339351. MR 84e:14005
 [Wa3]
K.i. Watanabe, Fregular and Fpure normal graded rings, J. Pure Appl. Algebra 71 (1991), 341350. MR 92g:13003
 [Wa4]
K.i. Watanabe, Infinite cyclic covers of strongly Fregular rings, Contemp. Math. 159 (1994), 423432. MR 95c:13030
 [Bo]
 J.F. Boutot, Singularités rationnelles et quotients par les groupes réductifs, Invent. Math. 88 (1987), 6568. MR 88a:14005
 [Ch]
 W.L. Chow, On unmixedness theorem, Amer. J. Math. 86 (1964), 799822. MR 30:2031
 [De]
 M. Demazure, Anneaux gradués normaux, in: Le Dûng Tráng (ed.) Introduction à la théorie des singularités II, Hermann, Paris, (1988), 3568. MR 91k:14004
 [GoW]
 S. Goto and K.i. Watanabe, On graded rings I, J. Math. Soc. Japan 30 (1978), 179213. MR 81m:13021
 [GrW]
 P. Griffith and D. Weston, Restrictions of torsion divisor classes to hypersurfaces, J. Algebra 167 (1994), 473487. MR 95c:13008
 [Gr]
 P. Griffith, Induced formal deformations and the CohenMacaulay property, Trans. Amer. Math. Soc. 353 (2001), 7793. MR 2001b:13020
 [Ha]
 N. Hara, A characterisation of rational singularities in terms of injectivity of Frobenius maps, Amer. J. Math. 120 (1998), 981996. MR 99h:13005
 [HW]
 N. Hara and K.i. Watanabe, Fregular and Fpure rings vs. log terminal and log canonical singularities, J. Algebraic Geom. 11 (2002), 363392.
 [HH1]
 M. Hochster and C. Huneke, Tight closure and strong Fregularity, Mem. Soc. Math. France 38 (1989), 119133. MR 91i:13025
 [HH2]
 M. Hochster and C. Huneke, Tight closure, invariant theory, and the BriançonSkoda theorem, J. Amer. Math. Soc. 3 (1990), 31116. MR 91g:13010
 [HH3]
 M. Hochster and C. Huneke, Fregularity, test elements, and smooth base change, Trans. Amer. Math. Soc. 346 (1994), 162. MR 95d:13007
 [HH4]
 M. Hochster and C. Huneke, Tight closure of parameter ideals and splitting in modulefinite extensions, J. Algebraic Geom. 3 (1994), 599670. MR 95k:13002
 [HH5]
 M. Hochster and C. Huneke, Tight closure in equal characteristic zero, in preparation.
 [HR]
 M. Hochster and J. Roberts, Rings of invariants of reductive groups acting on regular rings are CohenMacaulay, Adv. in Math. 13 (1974), 115175. MR 50:311
 [Ka]
 Y. Kawamata, The cone of curves of algebraic varieties, Ann. of Math. (2) 119 (1984), 603633. MR 86c:14013b
 [Li]
 J. Lipman, Rational singularities, with applications to algebraic surfaces and unique factorization, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 195279. MR 43:1986
 [LS]
 G. Lyubeznik and K. E. Smith, Strong and weak Fregularity are equivalent for graded rings, Amer. J. Math. 121 (1999), 12791290. MR 2000m:13006
 [Ma]
 H. Matsumura, Commutative ring theory, Cambridge Studies in Advanced Mathematics 8, Cambridge University Press, CambridgeNew York, 1986. MR 88h:13001
 [Si1]
 A. K. Singh, Gorenstein splinter rings of characteristic are Fregular, Mathematical Proceedings of the Cambridge Philosophical Society 127 (1999), 201205. MR 2000j:13006
 [Si2]
 A. K. Singh, Multisymbolic Rees algebras and strong Fregularity, Math. Z. 235 (2000), 335344. MR 2001j:13006
 [Sm1]
 K. E. Smith, Frational rings have rational singularities, Amer. J. Math. 119 (1997), 159180. MR 97k:13004
 [Sm2]
 K. E. Smith, Vanishing theorems, singularities, and effective bounds in algebraic geometry via prime characteristic local algebra, in: J. Kollár, R. Lazarsfeld and David R. Morrison (eds.) Proc. Sympos. Pure Math. 62 (1997), 289325. MR 99a:14026
 [TW]
 M. Tomari and K.i. Watanabe, Normal graded rings and normal cyclic covers, Manuscripta Math. 76 (1992), 325340. MR 93j:13002
 [Wa1]
 K.i. Watanabe, Some remarks concerning Demazure's construction of normal graded rings, Nagoya Math. J. 83 (1981), 203211. MR 83g:13016
 [Wa2]
 K.i. Watanabe, Rational singularities with action, in: Commutative Algebra (Trento, 1981), Lecture Notes in Pure and Appl. Math. 84, Marcel Dekker, New York, (1983), 339351. MR 84e:14005
 [Wa3]
 K.i. Watanabe, Fregular and Fpure normal graded rings, J. Pure Appl. Algebra 71 (1991), 341350. MR 92g:13003
 [Wa4]
 K.i. Watanabe, Infinite cyclic covers of strongly Fregular rings, Contemp. Math. 159 (1994), 423432. MR 95c:13030
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Additional Information
Anurag K. Singh
Affiliation:
Department of Mathematics, University of Utah, 155 S. 1400 E., Salt Lake City, Utah 841120090
Address at time of publication:
Mathematical Sciences Research Institute, 1000 Centennial Drive, #5070, Berkeley, California 947205070
Email:
asingh@msri.org
DOI:
http://dx.doi.org/10.1090/S0002994702031860
PII:
S 00029947(02)031860
Received by editor(s):
August 21, 2002
Published electronically:
November 1, 2002
Additional Notes:
This manuscript is based on work supported in part by the National Science Foundation under Grant No. DMS 0070268. I would like to thank the referee for a careful reading of the manuscript and for helpful suggestions.
Dedicated:
Dedicated to Professor Phillip Griffith on the occasion of his sixtieth birthday
Article copyright:
© Copyright 2002
American Mathematical Society
