The injectivity of Frobenius acting on cohomology and local cohomology modules |
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Authors: | Nobuo Hara Kei-ichi Watanabe |
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Institution: | (1) Department of Mathematics, Waseda University, 3-4-1 Okubo Shinjuku, 169 Tokyo, Japan;(2) Department of Mathematical Science, Tokai University, Hiratsuka, 259-12 Kanagawa, Japan |
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Abstract: | LetR be a two-dimensional normal graded ring over a field of characteristicp>0. We want to describe the tight closure of (O) in the local cohomology moduleH
R+
2
(R) using the graded module structure ofH
R+
2
(R). For this purpose we explore the condition that the Frobenius mapF: H
R+
2
(R)]n→H
R+
2
(R)]pninduced on graded pieces ofH
R+
2
(R) is injective. This problem is treated geometrically as follows: There exists an ample fractional divisorD onX=Proj (R) such thatR=R (X, D)= ⊕
n≥0H0(X
O
X
(n D)). Then the above map is identified with the induced Frobenius on the cohomology groups
Our interest is the casen<0, and in this case, a generalization of Tango's method for integral divisors enables us to show thatF
n is injective ifp is greater than a certain bound given explicitly byX andnD. This result is useful to studyF-rationality ofR. The notion ofF-rational rings in characteristicp>0 is defined via tight closure and is expected to characterize rational singularities. We ask if a modulop reduction of a rational signularity in characteristic 0 isF-rational forp≫0. Our result answers to this question affirmatively and also sheds light to behavior ofF-rationality in smallp. |
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Keywords: | |
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