F-rationality of certain Rees algebras and counterexamples to “Boutot's Theorem” for F-rational rings

Abstract

F-rational rings are defined for rings of characteristic p > 0 using the Frobenius endomorphism and corresponds to rational singularities in characteristic 0. We study F-rationality of certain Rees algebras and prove that every Cohen-Macaulay local ring with isolated singularity and negative a-invariant has a Rees algebra which is F-rational. As a consequence, we find that “Boutot's Theorem” asserting that a pure subring of a rational singularity is a rational singularity is not true for a F-rational ring.