The Frobenius structure of local cohomology
The Frobenius structure of local cohomology
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局部上同调的 Frobenius 结构
DOI：
10.2140/ant.2008.2.721
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发表时间：
2007
影响因子：
1.3
通讯作者：
M. Hochster
中科院分区：
文献类型：
--
作者：
Florian Enescu;M. Hochster
本文所给的环都是交换的、结合单位元的、Noether环。在全文中，p表示正素数。在大多数情况下，我们将研究本地环，即，具有唯一极大理想的Noether环。同样，我们的主要兴趣是在环的正素特征p。如果（R，m）是当地的特征p，有一个自然的行动的Frobenius自同态R的每一个局部上同调模H m（R）。如果F的作用将N映射到其自身，则我们称这些局部上同调模之一的R-子模N为F-稳定的。我们的目标之一是了解当局部环（R，m），特别是约化Cohen-Macaulay局部环，具有局部上同调模的R-子模的F-稳定的性质。当这种情况发生时，我们说R是FH-有限的。当然，当环是Cohen-Macaulay环时，只有一个非零局部上同调模Hm（R），其中d = dim（R）。研究H m（R）的F-稳定子模的问题自然地出现在紧闭包理论中，采用K.史密斯[Sm 1，2，3]。例如，在一个示例中，如果R是完备的，约化的，并且Gorenstein，Hm（R）的最大真F-稳定子模对应于0的紧闭包（在有限意义下：参见[HH 2]，§8），并且它的零化子是R的测试理想。也可参见此处的讨论2.10。
All given rings in this paper are commutative, associative with identity, and Noetherian. Throughout, p denotes a positive prime integer. For the most part, we shall be studying local rings, i.e., Noetherian rings with a unique maximal ideal. Likewise our main interest is in rings of positive prime characteristic p. If (R, m) is local of characteristic p, there is a natural action of the Frobenius endomorphism of R on each of its local cohomology modules H m(R). We call an R-submodule N of one of these local cohomology modules F-stable if the action of F maps N into itself. One of our objectives is to understand when a local ring, (R, m), especially a reduced Cohen-Macaulay local ring, has the property that only finitely many R-submodules of its local cohomology modules F-stable. When this occurs we say that R is FH-finite. Of course, when the ring is Cohen-Macaulay there is only one non-vanishing local cohomology module, H m(R), where d = dim (R). The problem of studying the F-stable submodules of H m(R) arises naturally in tight closure theory, taking a point of view pioneered by K. Smith [Sm1,2,3]. E.g., if R is complete, reduced, and Gorenstein, the largest proper F-stable submodule of H m(R) corresponds to the tight closure of 0 (in the finitistic sense: see [HH2], §8), and its annihilator is the test ideal of R. Also see Discussion 2.10 here.