Localization and test exponents for tight closure.
Localization and test exponents for tight closure.
复制标题
紧密闭合的本地化和测试指数。
DOI：
10.1307/mmj/1030132721
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发表时间：
2002
影响因子：
0.9
通讯作者：
C. Huneke
中科院分区：
文献类型：
--
作者：
M. Hochster;C. Huneke
我们介绍了一个测试指数紧封闭的概念，并探讨其关系的问题表明，紧封闭与本地化，一个长期悬而未决的问题。粗略地说，测试指数存在当且仅当紧闭包与局部化交换：需要环上的温和条件来证明这一点。我们给其他的，独立的，条件是必要的和充分的紧封闭，在一般情况下，与本地化交换，在某些相关的素数和行为的指数需要湮灭当地的上同调。虽然某些相关的条件（这里给出的条件较弱）以前已知是充分的，但这些是实际上等价的这种类型的第一个条件。§4的困难计算使用了多重性的结合性和许多其他工具来表明，可以给出局部化与紧闭包交换的充分条件，其中关于使用Frobenius自同态迭代定义的模的长度的渐近陈述取代了§3中引入的素数集的有限性条件。结果是局部的，并且需要环上的特殊条件：一个是可数素数回避成立。然而，这不是一个非常严格的条件：例如，环包含不可数域就足够了。可数素数回避在任何完备局部环中也成立。但我们也需要存在一个强测试理想（见
We introduce the notion of a test exponent for tight closure, and explore its relationship with the problem of showing that tight closure commutes with localization, a longstanding open question. Roughly speaking, test exponents exist if and only if tight closure commutes with localization: mild conditions on the ring are needed to prove this. We give other, independent, conditions that are necessary and sufficient for tight closure to commute with localization in the general case, in terms of behavior of certain associated primes and behavior of exponents needed to annihilate local cohomology. While certain related conditions (the ones given here are weaker) were previously known to be sufficient, these are the first conditions of this type that are actually equivalent. The difficult calculation of§4 uses associativity of multiplicities and many other tools to show that sufficient conditions for localization to commute with tight closure can be given in which asymptotic statements about lengths of modules defined using the iterates of the Frobenius endomorphism replace the finiteness conditions on sets of primes introduced in §3. The result is local and requires special conditions on the rings: one is that countable prime avoidance holds. This is not a very restrictive condition however: it suffices, for example, for the ring to contain an uncountable field. Countable prime avoidance also holds in any complete local ring. But we also need the existence of a strong test ideal (see