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

A ring R R is said to be F F -rational if, for every prime P P in R R, the local ring R P RP has the property that every system of parameters ideal is tightly closed (as defined by Hochster-Huneke). It is proved that if R R is a 2 2 -dimensional graded ring with an isolated singularity at the irrelevant maximal ideal m m, then the following are equivalent: (1) R R has a rational singularity at m m. (2) R R is F F -rational. (3) a (R) > 0 a (R) > 0. Here a (R) a (R) (as defined by Goto-Watanabe) denotes the least nonvanishing graded piece of the local cohomology module H m (R) Hₘ (R). The proof of this result relies heavily on the properties of derivations of R R, and suggests further questions in that direction; paradigmatically, if one knows that