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

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 $a$?