Commutative algebras and cohomology
Commutative algebras and cohomology
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DOI：
10.1090/s0002-9947-1962-0142607-6
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发表时间：
1962-02
影响因子：
1.3
通讯作者：
D. Harrison
中科院分区：
文献类型：
--
作者：
D. Harrison
本文有三个目的：(1)发展交换代数的交换上同调理论的机制；(2)应用该上同调理论给出交换代数的环扩展的工作理论；(3)通过将某些自然上同调条件与代数几何中的相应条件联系起来，应用并检验该理论。第一个目的在第一节和第三节中讨论。我们的上同调有几个Hochschild理论所没有的性质（张量积的显式关系，从代数到因子代数变化的可行序列，以及局部化的显式公式）；然而，与Hochschild理论不同的是，我们没有维度转换技术。因此，我们只考虑第一、第二和第三上同调模块。第二部分将考虑第二个目的。对于B和A交换代数，我们令M(A)表示HomA(A， A)，取A中元素相乘的理想模。根据Eilenberg和Mac Lane关于群的原型理论，我们将每个代数的同态从B关联到M(A)， M(A)是第三个上同态模中的一个元素。如果这个元素为零，则同态称为无阻碍。第四节也是最后一节的目的最好的表达是考虑仿射变量V上的一个点p在一个完全域k上。设a是V的坐标环，，是与p相关的素理想，Q是a /p的商域。证明p是一个简单点当且仅当a在Q中有系数的第二个上同调模为零。用这种方法给出正则局部环的一个刻划。我们还证明了当且仅当第一和第二上同调模（a的系数都在Q中）的维数之差等于V的维数时，p将是一个完全交集。我们的第三个主要结论是当且仅当a对所有有限生成的模具有平凡的第二上同调时，V将是非奇异的。
This paper has three purposes: (1) to develop the machinery of a commutative cohomology theory for commutative algebras, (2) to apply this cohomology theory to give a working theory of ring extensions for commutative algebras, and (3) to employ and test the theory by relating certain natural cohomological conditions to corresponding conditions in algebraic geometry. The first purpose is approached in the first and third sections. Our cohomology has several properties not enjoyed by the Hochschild theory (an explicit relation for the tensor product, a workable sequence for changes from an algebra to a factor algebra, and an explicit formula for going local); however, unlike the Hochschild theory, we have no dimension shifting techniques. For this reason, we restrict consideration to the first, second, and third cohomology modules. The second purpose is considered in the second section. For B and A commutative algebras, we let M(A) denote HomA(A, A), taken modulo the ideal of multiplications by elements in A. Following the prototype theory which Eilenberg and Mac Lane developed for groups, we associate to each algebra homomorphism from B into M(A), an element in a third cohomology module. The homomorphism is called unobstructed if this element is zero. The unobstructed homomorphisms, together with the elements of a certain second cohomology module, are associated in a one-one fashion with the commutative algebra extensions of A by B. The purpose of the fourth and last section is best expressed by considering a point p on an affine variety V over a perfect field k. Let A be the coordinate ring of V, , be the prime ideal associated with p, and Q be the quotient field of A/p. We prove that p is a simple point if and only if the second cohomology module of A with coefficients in Q is zero. In this manner we give a characterization of regular local rings. We also prove that p will be a complete intersection if and only if the difference between the dimensions of the first and second cohomology modules (both of A with coefficients in Q) is the dimension of V. Our third main result is that V will be nonsingular if and only if A has trivial second cohomology with respect to all finitely generated Amodules.