N. Hara, A CHARACTERIZATION OF RATIONAL-SINGULARITIES IN TERMS OF INJECTIVITY OF FROBENIUS MAPS, American journal of mathematics, 120(5), 1998, pp. 981-996
The notions of F-rational and F-regular rings are defined via tight cl
osure, which is a closure operation for ideals in a commutative ring o
f positive characteristic. The geometric significance of these notions
has persisted, and K. E. Smith proved that F-rational rings have rati
onal singularities. We now ask about the converse implication. The ans
wer to this question is yes and no. For a fixed positive characteristi
c, there is a rational singularity which is not F-rational, so the ans
wer is no. In this paper, however, we aim to show that the answer is y
es in the following sense: If a ring of characteristic zero has ration
al singularity, then its module p reduction is F-rational for almost a
ll characteristic p. This result leads us to the correspondence of F-r
egular rings and log terminal singularities.