## Abstract

Let R be a two-dimensional normal graded ring over a field of characteristic p > 0. We want to describe the tight closure of (0) in the local cohomology module H^{2}_{R+}(R) using the graded module structure of H^{2}_{R+}(R). For this purpose we explore the condition that the Frobenius map F : [H^{2}_{R+}(R)]_{n} → [H^{2}_{R+}(R)]_{pn} induced on graded pieces of H^{2}_{R+}(R) is injective. This problem is treated geometrically as follows: There exists an ample fractional divisor D on X = Proj(R) such that R = R(X, D) = ⊕_{n≥0}H^{0}(X, script O sign_{X}(nD)). Then the above map is identified with the induced Frobenius on the cohomology groups F_{n} : H^{1}(X, script O sign_{X}(nD)) → H^{1}(X, script O sign_{X}(pnD)). Our interest is the case n < 0, and in this case, a generalization of Tango's method for integral divisors enables us to show that F_{n} is injective if p is greater than a certain bound given explicitly by X and nD. This result is useful to study F-rationality of R. The notion of F-rational rings in characteristic p > 0 is defined via tight closure and is expected to characterize rational singularities. We ask if a modulo p reduction of a rational singularity in characteristic 0 is F-rational for p ≫ 0. Our result answers to this question affirmatively and also sheds light to behavior of F-rationality in small p.

Original language | English |
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Pages (from-to) | 301-315 |

Number of pages | 15 |

Journal | manuscripta mathematica |

Volume | 90 |

Issue number | 3 |

Publication status | Published - 1996 Jul 1 |

## ASJC Scopus subject areas

- Mathematics(all)