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A Frobenius characterization of rational singularity in $ 2$-dimensional graded rings

Author: Richard Fedder
Journal: Trans. Amer. Math. Soc. 340 (1993), 655-668
MSC: Primary 13A35; Secondary 13D45, 13N05, 14H20
MathSciNet review: 1116312
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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$?

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