On the commutation of the test ideal with localization and completion
On the commutation of the test ideal with localization and completion
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论测试理想的转换与定位与完成
DOI：
10.1090/s0002-9947-01-02643-5
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发表时间：
2001
影响因子：
1.3
通讯作者：
Karen E. Smith
中科院分区：
文献类型：
--
作者：
G. Lyubeznik;Karen E. Smith
设R是素特征的优良正则局部环上的本质有限型约化环。证明了R的检验理想与局部化交换，如果R是局部的，则R的检验理想与完备化交换，条件是R的每个局部环的剩余域的内射船体E中零的紧闭包等于E中零的有限紧闭包.这是证明后一个条件适用于所有的局部环的素特征;这是证明在这里的所有科恩-麦考利奇与最孤立的非Gorenstein奇，并在一般情况下为所有孤立的奇。为了证明具有局部化和完备化的测试理想的可换性，引入并研究了与每个R-模相关联的Frobenius算子环。这个理论产生了一个R的理想，它定义了非强F-正则轨迹，并且与局部化和完备化互换。证明了这个理想是R的一般检验理想，并且在R的每个局部环中，在0 E = 0 fg <$E的假设下，证明了这个理想与检验理想是等价的.
Let R be a reduced ring that is essentially of finite type over an excellent regular local ring of prime characteristic. Then it is shown that the test ideal of R commutes with localization and, if R is local, with completion, under the additional hypothesis that the tight closure of zero in the injective hull E of the residue field of every local ring of R is equal to the finitistic tight closure of zero in E. It is conjectured that this latter condition holds for all local rings of prime characteristic; it is proved here for all Cohen-Macaulay singularities with at most isolated non-Gorenstein singularities, and in general for all isolated singularities. In order to prove the result on the commutation of the test ideal with localization and completion, a ring of Frobenius operators associated to each R-module is introduced and studied. This theory gives rise to an ideal of R which defines the non-strongly F-regular locus, and which commutes with localization and completion. This ideal is conjectured to be the test ideal of R in general, and shown to equal the test ideal under the hypothesis that 0E = 0 fg∗ E in every local ring of R.