A Frobenius characterization of rational singularity in -dimensional graded rings

Author:
Richard Fedder

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

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A ring is said to be -rational if, for every prime in , the local ring has the property that every system of parameters ideal is tightly closed (as defined by Hochster-Huneke). It is proved that if is a -dimensional graded ring with an isolated singularity at the irrelevant maximal ideal , then the following are equivalent:

(1) has a rational singularity at .

(2) is -rational.

(3) .

Here (as defined by Goto-Watanabe) denotes the least nonvanishing graded piece of the local cohomology module .

The proof of this result relies heavily on the properties of derivations of , and suggests further questions in that direction; paradigmatically, if one knows that satisfies a certain property for every derivation , what can one conclude about the original ring element ?

**[BE]**D. Buchsbaum and D. Eisenbud,*What makes a complex exact*?, J. Algebra**25**(1973), 259-268. MR**0314819 (47:3369)****[F1]**R. Fedder, -*purity and rational singularity*, Trans. Amer. Math. Soc.**278**(1983), 461-480. MR**701505 (84h:13031)****[F2]**-, -*purity and rational singularity in graded complete intersection rings*, Trans. Amer. Math. Soc.**301**(1987), 47-62. MR**879562 (88h:14002)****[FHH]**R. Fedder, R. Hübl, and C. Huneke,*Zeros of differential forms along one-fibered ideals*, preprint.**[Fl]**H. Flenner,*Quasihomogene Rationale Singularitaten*, Arch. Math.**36**(1981), 35-44. MR**612234 (82g:14036)****[FW]**R. Fedder and K.-i. Watanabe,*A characterization of*-*regularity in terms of*-*purity*, Proceedings of the Program in Commutative Algebra at MSRI (held in June and July 1987), Springer-Verlag, 1989.**[HH1]**M. Hochster and C. Huneke,*Tightly closed ideals*, Bull. Amer. Math. Soc.**18**(1988), 45-48. MR**919658 (89b:13003)****[HH2]**-,*Tight closure*, Proceedings of the Program in Commutative Algebra at MSRI (held in June and July 1987), Springer-Verlag, 1989. MR**1015524 (91f:13022)****[HH3]**-,*Tight closure and strong*-*regularity*, Mém. Soc. Math. France (numero Consacré au colloque en l'honneur de P. Samuel).**[HH4]**-,*Tight closure, invariant theory and the Briancon-Skoda Theorem*. I, preprint.**[HR]**M. Hochster and J. L. Roberts,*The purity of the Frobenius and local cohomology*, Adv. in Math.**21**(1976), 117-172. MR**0417172 (54:5230)****[GW]**S. Goto and K.-i. Watanabe,*On graded rings*. I, J. Math. Soc. Japan**30**(1978), 179-213. MR**494707 (81m:13021)****[K]**E. Kunz,*Kähler differentials*, Vieweg, Braunschweig, Weisbaden, 1986. MR**864975 (88e:14025)****[LT]**J. Lipman and B. Teissier,*Pseudo-rational local rings and a theorem of Briancon-Skoda about integral closure of ideals*, Michigan Math. J.**28**(1981), 97-116. MR**600418 (82f:14004)****[Matl]**E. Matlis,*Injective modules over Noetherian rings*, Pacific J. Math.**8**(1958), 511-528. MR**0099360 (20:5800)****[Mats]**H. Matsumura,*Commutative algebra*, Benjamin, 1970. MR**0266911 (42:1813)****[MeSr]**V. B. Mehta and V. Srinivas,*Normal*-*pure surface singularities*, preprint, Tata Institute, Bombay.**[PS]**C. Peskine and L. Szpiro,*Dimension projective finie et cohomologie locale*, Inst. Hautes Études Sci. Publ. Math.**42**(1973), 323-395. MR**0374130 (51:10330)****[Se]**J.-P. Serre,*Algèbre locale. Multiplicités*, Lecture Notes in Math., vol. 11, Springer-Verlag, Berlin, Heidelberg, and New York, 1965.**[W1]**K.-i. Watanabe,*Rational singularities with*-*action*, Lecture Notes in Pure and Appl. Math., 84, Dekker, 1983, pp. 339-351.**[W2]**-,*Study of*-*purity in dimension*, preprint, Tokai Univ., Hiratsuka, 259-12, Japan.**[ZS]**O. Zariski and P. Samuel,*Commutative algebra*, vols. I and II, Van Nostrand, Princeton, N. J., 1958 and 1960.

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
13A35,
13D45,
13N05,
14H20

Retrieve articles in all journals with MSC: 13A35, 13D45, 13N05, 14H20

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1993-1116312-3

Article copyright:
© Copyright 1993
American Mathematical Society