Abstract
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 {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 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 ) {H_m}(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 D ( a ) D(a) satisfies a certain property for every derivation D D , what can one conclude about the original ring element a a ?
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