論文

査読有り
1996年7月

The injectivity of Frobenius acting on cohomology and local cohomology modules

MANUSCRIPTA MATHEMATICA
  • N Hara
  • ,
  • K Watanabe

90
3
開始ページ
301
終了ページ
315
記述言語
英語
掲載種別
研究論文(学術雑誌)
出版者・発行元
SPRINGER VERLAG

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-R+(2)(R) using the graded module structure of H-R+(2) (R). For this purpose we explore the condition that the Frobenius map F : [H-R+(2)(R)](n) --> [H-R+(2)(R)](pn) induced on graded pieces of H-R+(2) (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) = +H-n greater than or equal to 0(0)(X,O-X(nD)) Then the above map is identified with the induced Frobenius on the cohomology groups
F-n : H-1(X, O-X(nD)) --> H-1(X, O-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 no. 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 module p reduction of a rational singularity in characteristic 0 is F-rational for p much greater than 0. Our result answers to this question affirmatively and also sheds light to behavior of F-rationality in small p.

Web of Science ® 被引用回数 : 12

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https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcAuth=JSTA_CEL&SrcApp=J_Gate_JST&DestLinkType=FullRecord&KeyUT=WOS:A1996UZ02300003&DestApp=WOS_CPL

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