TY - JOUR

T1 - A characterization of rational singularities in terms of injectivity of Frobenius maps

AU - Hara, Nobuo

N1 - Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

PY - 1998/10

Y1 - 1998/10

N2 - 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.

AB - 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.

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U2 - 10.1353/ajm.1998.0037

DO - 10.1353/ajm.1998.0037

M3 - Article

AN - SCOPUS:0001515067

VL - 120

SP - 981

EP - 996

JO - American Journal of Mathematics

JF - American Journal of Mathematics

SN - 0002-9327

IS - 5

ER -