A Frobenius characterization of rational singularity in $2$-dimensional graded rings
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- by Richard Fedder PDF
- Trans. Amer. Math. Soc. 340 (1993), 655-668 Request permission
Abstract:
A ring $R$ is said to be $F$-rational if, for every prime $P$ in $R$, the local ring ${R_P}$ has the property that every system of parameters ideal is tightly closed (as defined by Hochster-Huneke). It is proved that if $R$ is a $2$-dimensional graded ring with an isolated singularity at the irrelevant maximal ideal $m$, then the following are equivalent: (1) $R$ has a rational singularity at $m$. (2) $R$ is $F$-rational. (3) $a(R) < 0$. Here $a(R)$ (as defined by Goto-Watanabe) denotes the least nonvanishing graded piece of the local cohomology module ${H_m}(R)$. The proof of this result relies heavily on the properties of derivations of $R$, and suggests further questions in that direction; paradigmatically, if one knows that $D(a)$ satisfies a certain property for every derivation $D$, what can one conclude about the original ring element $a$?References
- David A. Buchsbaum and David Eisenbud, What makes a complex exact?, J. Algebra 25 (1973), 259–268. MR 314819, DOI 10.1016/0021-8693(73)90044-6
- Richard Fedder, $F$-purity and rational singularity, Trans. Amer. Math. Soc. 278 (1983), no. 2, 461–480. MR 701505, DOI 10.1090/S0002-9947-1983-0701505-0
- Richard Fedder, $F$-purity and rational singularity in graded complete intersection rings, Trans. Amer. Math. Soc. 301 (1987), no. 1, 47–62. MR 879562, DOI 10.1090/S0002-9947-1987-0879562-4 R. Fedder, R. Hübl, and C. Huneke, Zeros of differential forms along one-fibered ideals, preprint.
- Hubert Flenner, Rationale quasihomogene Singularitäten, Arch. Math. (Basel) 36 (1981), no. 1, 35–44 (German). MR 612234, DOI 10.1007/BF01223666 R. Fedder and K.-i. Watanabe, A characterization of $F$-regularity in terms of $F$-purity, Proceedings of the Program in Commutative Algebra at MSRI (held in June and July 1987), Springer-Verlag, 1989.
- Melvin Hochster and Craig Huneke, Tightly closed ideals, Bull. Amer. Math. Soc. (N.S.) 18 (1988), no. 1, 45–48. MR 919658, DOI 10.1090/S0273-0979-1988-15592-9
- Melvin Hochster and Craig Huneke, Tight closure, Commutative algebra (Berkeley, CA, 1987) Math. Sci. Res. Inst. Publ., vol. 15, Springer, New York, 1989, pp. 305–324. MR 1015524, DOI 10.1007/978-1-4612-3660-3_{1}5 —, Tight closure and strong $F$-regularity, Mém. Soc. Math. France (numero Consacré au colloque en l’honneur de P. Samuel). —, Tight closure, invariant theory and the Briancon-Skoda Theorem. I, preprint.
- Melvin Hochster and Joel L. Roberts, The purity of the Frobenius and local cohomology, Advances in Math. 21 (1976), no. 2, 117–172. MR 417172, DOI 10.1016/0001-8708(76)90073-6
- Shiro Goto and Keiichi Watanabe, On graded rings. I, J. Math. Soc. Japan 30 (1978), no. 2, 179–213. MR 494707, DOI 10.2969/jmsj/03020179
- Ernst Kunz, Kähler differentials, Advanced Lectures in Mathematics, Friedr. Vieweg & Sohn, Braunschweig, 1986. MR 864975, DOI 10.1007/978-3-663-14074-0
- Joseph Lipman and Bernard Teissier, Pseudorational local rings and a theorem of Briançon-Skoda about integral closures of ideals, Michigan Math. J. 28 (1981), no. 1, 97–116. MR 600418
- Eben Matlis, Injective modules over Noetherian rings, Pacific J. Math. 8 (1958), 511–528. MR 99360
- Hideyuki Matsumura, Commutative algebra, W. A. Benjamin, Inc., New York, 1970. MR 0266911 V. B. Mehta and V. Srinivas, Normal $F$-pure surface singularities, preprint, Tata Institute, Bombay.
- C. Peskine and L. Szpiro, Dimension projective finie et cohomologie locale. Applications à la démonstration de conjectures de M. Auslander, H. Bass et A. Grothendieck, Inst. Hautes Études Sci. Publ. Math. 42 (1973), 47–119 (French). MR 374130 J.-P. Serre, Algèbre locale. Multiplicités, Lecture Notes in Math., vol. 11, Springer-Verlag, Berlin, Heidelberg, and New York, 1965. K.-i. Watanabe, Rational singularities with ${K^\ast }$-action, Lecture Notes in Pure and Appl. Math., 84, Dekker, 1983, pp. 339-351. —, Study of $F$-purity in dimension $2$, preprint, Tokai Univ., Hiratsuka, 259-12, Japan. O. Zariski and P. Samuel, Commutative algebra, vols. I and II, Van Nostrand, Princeton, N. J., 1958 and 1960.
Additional Information
- © Copyright 1993 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 340 (1993), 655-668
- MSC: Primary 13A35; Secondary 13D45, 13N05, 14H20
- DOI: https://doi.org/10.1090/S0002-9947-1993-1116312-3
- MathSciNet review: 1116312