F-regularity does not deform
F-regularity does not deform
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F-规整不变形
DOI：
10.1353/ajm.1999.0029
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发表时间：
1999
影响因子：
1.7
通讯作者：
Anurag Singh
中科院分区：
文献类型：
--
作者：
Anurag Singh
我们证明了f正则性不变形的性质，从而解决了紧闭理论中一个长期悬而未决的问题。具体地说，我们构造了一个三维梯度域R，它不是f正则的（甚至不是f纯的），但它有一个商R tR，它是f正则的。在特征为p 0的域上构造示例，以及在特征为0的域上构造示例。1. 介绍。在本文中，所有环都是可交换的，是诺伊式的，并且有一个单位元。紧闭理论是由Melvin Hochster和Craig Huneke在（h2）中提出的，它把人们的注意力吸引到那些所有理想都是紧闭的环上，这些环被称为弱f正则环。“f正则”一词保留给所有定位都是弱f正则的环。随着理论的发展，一个自然的问题是F-正则性的性质是否变形，即，如果（R, m， K）是一个局部环，使得R tR对于某些非零因子m是F-正则的，m ustR是F-正则的？（参见（Ho）的后记。）Hochster和Huneke表明，如果环R是Gorenstein， （HH3），这确实是正确的，并且他们的工作之后已经进行了各种尝试来扩展这个结果，参见（AKM）， （Si）， （Sm3）。我们这里的主要目标是通过构造一组例子来证明f正则性不会变形来解决这个问题。我们将始终考虑-分级环，但在所有情况下，局部例子都可以通过在齐次极大理想处的局部化得到。我们的主要结果是：定理1.1。存在一个三维的-分级环R（在特征p 2的域R 0 = K上有限生成），它不是f纯的，但有一个f正则商R tR，其中t m是齐次非零因子。具体来说，对于正整数m和n满足m m n 2，考虑环R = K(A， B， C， D， T) I，其中I是由
We show that the property of F-regularity does not deform, and thereby settle a long- standing open question in the theory of tight closure. Specifically, we construct a three dimensional -graded domain R which is not F-regular (or even F-pure), but has a quotient R tR which is F-regular. Examples are constructed over fields of characteristic p 0, as well as over fields of characteristic zero. 1. Introduction. Throughout this paper, all rings are commutative, Noethe- rian, and have an identity element. The theory of tight closure was developed by Melvin Hochster and Craig Huneke in (HH2) and draws attention to rings which have the property that all their ideals are tightly closed, called weakly F-regular rings. The term F-regular is reserved for rings all of whose localizations are weakly F-regular. A natural question that arose with the development of the the- ory was whether the property of F-regularity deforms, i.e., if (R, m, K) is a local ring such that R tR is F-regular for some nonzerodivisor t m ,m ustR be F- regular? (See the Epilogue of (Ho).) Hochster and Huneke showed that this is indeed true if the ring R is Gorenstein, (HH3), and their work has been followed by various attempts at extending this result, see (AKM), (Si), (Sm3). Our pri- mary goal here is to settle this question by constructing a family of examples to show that F-regularity does not deform. We shall throughout be considering -graded rings, but local examples can be obtained, in all cases, by localizing at the homogeneous maximal ideals. Our main result is: THEOREM 1.1. There exists an -graded ring R of dimension three (finitely generated over a field R 0 = K of characteristic p 2) which is not F-pure, but has an F-regular quotient R tR where t m is a homogeneous nonzerodivisor. Specifically, for positive integers m and n satisfying m m n 2, consider the ring R = K(A, B, C, D, T) I where I is generated by the size two minors of the