A characterization of rational singularities in terms of injectivity of Frobenius maps

Abstract

The notions of F -rational and F -regular rings are defined via tight closure, which is a closure operation for ideals in a commutative ring of positive characteristic. The geometric significance of these notions has persisted, and K. E. Smith proved that F -rational rings have rational singularities. We now ask about the converse implication. The answer to this question is yes and no. For a fixed positive characteristic, there is a rational singularity which is not F -rational, so the answer is no. In this paper, however, we aim to show that the answer is yes in the following sense: If a ring of characteristic zero has rational singularity, then its modulo p reduction is F -rational for almost all characteristic p . This result leads us to the correspondence of F -regular rings and log terminal singularities.