F-Rationality of Rees Algebras

Abstract

In this paper, using the notion of the tight integral closure, we will give a criterion for F-rationality of Rees algebras of m-primary ideals in a Cohen–Macaulay local ring. As its application, we prove the following results: (1) In dimension two, if A is F-rational and I is integrally closed, then the Rees algebra R(I) is F-rational. On the other hand, in higher dimensions, we construct many examples of Cohen–Macaulay, normal Rees algebras which are not F rational. (2) If both A and R(I) are F-rational, then so is the extended Rees algebra R′(I). (3) If R(I) is F-rational and a(G(I))≠−1, then A is F-rational.On the other hand, using resolution of singularities, we will prove that a two-dimensional rational singularity always admits F-rational Rees algebras. In particular, this theorem gives another way than that devised by Watanabe (1997, J. Pure Appl. Algebra122, 323–328) to construct counterexamples to the Boutot-type theorem for F-rational rings.