THE NOTION OF TIGHT CLOSURE IN EQUAL CHARACTERISTIC ZERO
THE NOTION OF TIGHT CLOSURE IN EQUAL CHARACTERISTIC ZERO
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等特性零的紧闭概念
DOI：
10.1016/j.jalgebra.2004.07.011
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发表时间：
2014
影响因子：
0.9
通讯作者：
M. Hochster
中科院分区：
文献类型：
--
作者：
M. Hochster
我们在这里所描述的理论的详细处理在[HH 6]中给出。事实上，在一个包含有理数Q的任意诺特环上，我们得到了一个定义在nitely生成的模的子模上的闭包运算。该操作具有与特征p情况相同的持久性属性。对于正则环，每个理想（以及每个nitely生成模的每个子模）都是紧闭的。理想的紧闭包包含在积分闭包中，并且通常要小得多。一个具有与特征p中相同的结肠捕获特性，并且更一般地，一个具有类似的幻影同源理论。结果，我们得到了一个理论，它产生了特征p中所做的相同的特征0版本：我们得到了一个非常简短的证明，证明了正则环的直和式是科恩-麦考利（以及更多：它们是F-正则的），改进了所谓的“局部同调代数”（这些代数现在大部分都是定理），以及Brian con-Skoda定理的紧闭版本。
The detailed treatment of the theory we sketch here is given in [HH6]. One does in fact get, over an arbitrary Noetherian ring containing the rational numbers, Q, a closure operation de ned on submodules of nitely generated modules. The operation has the same kind of persistence properties as in the characteristic p case. For regular rings, every ideal (and every submodule of every nitely generated module) is tightly closed. The tight closure of an ideal is contained in the integral closure and is usually much smaller. One has the same kind of colon-capturing properties as in characteristic p, and, more generally, one has an analogous phantom homology theory. In consequence, one has a theory that yields equal characteristic 0 versions of what has been done in characteristic p: one gets a very short proof that direct summands of regular rings are Cohen-Macaulay (and more: they are F-regular), improved versions of the so-called \local homological conjectures" (these conjectures are now theorems, for the most part), and a tight closure version of the Brian con-Skoda theorem.