The injectivity of Frobenius acting on cohomology and local cohomology modules
The injectivity of Frobenius acting on cohomology and local cohomology modules
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Frobenius 作用于上同调和局部上同调模的单射性
DOI：
10.1007/bf02568308
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发表时间：
1996
影响因子：
0.6
通讯作者：
Kei
中科院分区：
文献类型：
--
作者：
Nobuo Hara;Kei
在特征为>的场上画一个二维标准梯度环。利用hr +2(R)的梯度模结构描述局部上同模hr +2(R)中(O)的紧闭包。为此，我们探讨了在HR+2(R)的梯度片段上诱导的Frobenius映射f: [HR+2(R)]n→[HR+2(R)]是内射的条件。这个问题被几何地处理如下：存在一个足够的分数除数donx =Proj (R)，使得R=R (X， D)=⊕n≥0H0（XOX(n D)）。然后在上同调群上用诱导的Frobenius识别上述映射。我们感兴趣的是case <0，在这种情况下，对Tango的积分除数方法的推广使我们能够证明fnis的内射ifpi大于xandnd明确给出的某个界。这一结果对研究r的f -合理性有一定的指导意义。特征>0中的off -有理环概念是通过紧闭定义的，并期望表征有理奇点。我们问特征0上的一个有理奇点的模生成是否对p < 0是f -有理的。我们的研究结果肯定地回答了这一问题，并对小型动物的非理性行为提供了启示。
LetRbe 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 moduleHR+2(R) using the graded module structure ofHR+2(R). For this purpose we explore the condition that the Frobenius mapF: [HR+2(R)]n→[HR+2(R)]pninduced on graded pieces ofHR+2(R) is injective. This problem is treated geometrically as follows: There exists an ample fractional divisorDonX=Proj (R) such thatR=R (X, D)= ⊕n≥0H0(XOX(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 thatFnis injective ifpis greater than a certain bound given explicitly byXandnD. 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 modulopreduction 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.