Openness of the F-rational locus and smooth base change
Openness of the F-rational locus and smooth base change
复制标题
F-有理数轨迹的开放性和平稳的碱基变化
DOI：
10.1016/s0021-8693(05)80010-9
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发表时间：
1995
影响因子：
0.9
通讯作者：
J. D. Velez
中科院分区：
文献类型：
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作者：
J. D. Velez
在整个本文中，环被假定为交换的，结合的，与身份，和诺特的主要特征p。在大多数情况下，我们还将假设，环减少。在[HH 0] Hochster和Huneke介绍了概念紧封闭的理想Noether环的积极的总理特征，并为restaurant生成代数领域的特征0。紧闭包技术产生了新的证明，线性约化群的不变量环是Cohen-Macaulay（参见。[HR，Bou，HOE]），Brian § on-Skoda定理关于正则环中理想的积分闭包在等特征情形下的加强形式（cf. [BrS，HH 1，LS，LT]）和各种局部同调结构（cf. [Rol-Ro 4，Hol，Ho 2，EvG，Du]）.在[HH 5]中Hochster和Huneke引入了强F-正则环的概念，在[HH 2]中他们证明了对于优秀局部环上的R-G生成代数的环，R的素理想P的集合，其中RP是Gorenstein和弱F-正则的（这等价于强F-正则的性质）在Zerkki拓扑中是开的。本文的主要目的是证明F-有理环的一个类似结果。我们将证明（定理3.5）：对于优局部环上的约化生成代数的约化环，F-有理轨迹在Zeromki拓扑中是开的。为了证明这一点，我们将发展一个强F-有理环的理论，它应该被看作是强F-正则性理论的类似物。在第1节中，我们将给出某些类型的环（关于容许环的定义，见1.5）的这个性质的一个特征，用它们的标准模来表示。这
Throughout this paper rings are assumed to be commutative, associative, with identity, and Noetherian of prime characteristic p. For the most part we will also assume that rings are reduced. In [HH0] Hochster and Huneke introduced the notion of tight closure for ideals of a Noetherian ring of positive prime characteristic and for finitely generated algebras over a field of characteristic 0. Tight closure techniques have produced new proofs that rings of invariants of linearly reductive groups are Cohen-Macaulay (cf.[HR, Bou, HOE]), of the Brian § on-Skoda theorem on integral closures of ideals in regular rings in a strengthened form in the equicharacteristic case (cf.[BrS, HH1, LS, LT]) and various local homological conjectures (cf.[Rol-Ro4, Hol, Ho2, EvG, Du]).In [HH5] Hochster and Huneke introduced the notion of strongly F-regular rings and in [HH2] they proved that for rings which are finitely generated algebras over excellent local rings, the set of prime ideals P of R for which Rp is Gorenstein and weakly F-regular (this is equivalent to the property of being strongly F-regular) is open in the Zariski topology. The main objective of this paper is to prove an analogous result for F-rational rings. We will prove (Theorem 3.5) that the F-rational locus is open in the Zariski topology for reduced rings which are finitely generated algebras over excellent local rings. In order to prove this, we will develop a theory of strongly F-rational rings, which should be regarded as an analogue of the theory of strong F-regularity. In Section 1 we will give a characterization of this property for certain types of rings (see 1.5 for a definition of an admissible ring) in terms of their canonical modules. This