Tight closure test exponents for certain parameter ideals
Tight closure test exponents for certain parameter ideals
复制标题
某些理想参数的紧闭测试指数
DOI：
10.1307/mmj/1156345596
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发表时间：
2005
影响因子：
0.9
通讯作者：
R. Y. Sharp
中科院分区：
文献类型：
--
作者：
R. Y. Sharp
研究了素数特征为p的交换诺瑟环R上理想I$的紧闭包。从表面上看，形式定义要求进行无数次检查，以确定$R$中的元素是否属于$I$的紧闭包。当$R$是特征$p$上的有限型约简代数时，存在的Hochster和Huneke紧闭的检验元大大改善了这方面的情况。
This paper is concerned with the tight closure of an ideal $I$ in a commutative Noetherian ring $R$ of prime characteristic $p$. The formal definition requires, on the face of things, an infinite number of checks to determine whether or not an element of $R$ belongs to the tight closure of $I$. The situation in this respect is much improved by Hochster's and Huneke's test elements for tight closure, which exist when $R$ is a reduced algebra of finite type over an excellent local ring of characteristic $p$.
More recently, Hochster and Huneke have introduced the concept of test exponent for tight closure: existence of these test exponents would mean that one would have to perform just one single check to determine whether or not an element of $R$ belongs to the tight closure of $I$. However, to quote Hochster and Huneke, 'it is not at all clear whether to expect test exponents to exist; roughly speaking, test exponents exist if and only if tight closure commutes with localization'.
The main purpose of this paper is to provide a short direct proof that test exponents exist for parameter ideals in a reduced excellent equidimensional local ring of characteristic $p$.