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

Nobuo Hara

Research output: Contribution to journalArticlepeer-review

86 Citations (Scopus)

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.

Original languageEnglish
Pages (from-to)981-996
Number of pages16
JournalAmerican Journal of Mathematics
Volume120
Issue number5
DOIs
Publication statusPublished - 1998 Oct

ASJC Scopus subject areas

  • Mathematics(all)

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