Absolute integral closures are big Cohen-Macaulay algebras in characteristic $P$
Absolute integral closures are big Cohen-Macaulay algebras in characteristic $P$
复制标题
绝对积分闭包是具有 $P$ 特征的大 Cohen-Macaulay 代数
DOI：
10.1090/s0273-0979-1991-15970-7
复制
发表时间：
1991
影响因子：
1.3
通讯作者：
C. Huneke
中科院分区：
文献类型：
--
作者：
M. Hochster;C. Huneke
在本文中，“环”是指具有单位元且模是唯一的交换环。我们主要研究局部环，即具有唯一极大理想m的noether环（R, m）。在这样一个环中，xx，…，如果m在理想（x{，…）中有幂，则xn em是一个参数系统。， xn)R， n是R的Krull维数。当R完备且包含域时，这意味着R在形式幂级数子K[[xx，…]上是模有限的。， xn]] = A。我吗?如果某些（等价地，每个）参数系统是i中的正则序列，则称为Cohen-Macaulay。，这意味着每个xi+x都是非零因子R/{xx，…， xt)R，对于0 < i < n 1。在形式幂级数子A上R是模有限的情况下，这意味着R是一个自由的^4模。对于交换代数和代数几何的许多定理，Cohen-Macaulay条件（可能是在各种局部环上）正是使理论成立所需要的。我们的主要结果,下面给出一个代数形式在以几何形式定理1和定理2,断言,在温和条件下局部环R上的积极' p特点,可以“正确”的失败Cohen-Macaulay条件R本身通过一个非常大的积分的延伸R .值得强调的是,这是非常错误的0的特点,是一个惊喜,我们相信,我们可以从下面的观察得到这些定理离特征为0的域上的真值有多远：在定理2的情况下，当X是射影正规时，上同的映射总是内射的，这是一个简单的迹论证。尽管定理1和定理2
Throughout this paper "ring" means commutative ring with identity and modules are unital. Our main interest is in local rings, i.e., Noetherian rings (R, m) with a unique maximal ideal m . In such a ring, xx, . . . , xn e m is a system of parameters if m has a power in the ideal (x{, . . . , xn)R and n is the Krull dimension of R. When R is complete and contains a field, this means that R is module-finite over a formal power series subring K[[xx, . . . , xn]] = A . i? is called Cohen-Macaulay if some (equivalently, every) system of parameters is a regular sequence in i?, which means that every xi+x is a nonzerodivisor on R/{xx, . . . , xt)R, for 0 < i < n 1. In the case where R is module-finite over the formal power series subring A , this means that R is a free ^4-module. For many theorems of commutative algebra and of algebraic geometry, the Cohen-Macaulay condition (possibly on the local rings of a variety) is just what is needed to make the theory work. Our main result, which is given below in an algebraic form in Theorem 1 and in a geometric form in Theorem 2, asserts that, under mild conditions on a local ring R of positive prime characteristic p, one can "correct" the failure of the Cohen-Macaulay condition in R itself by passing to a very large integral extension of R. It is worth emphasizing that this is quite false in characteristic 0, and comes as a surprise, we believe, in characteristic p. One can get an idea of how far these theorems are from the truth over a field of characteristic 0 from the following observation: In the situation of Theorem 2, when X is projectively normal, the maps of cohomology are always injective, by an easy trace argument. Despite the fact that both Theorem 1 and Theorem 2