On the complete relative homology and cohomology of Frobenius extensions
On the complete relative homology and cohomology of Frobenius extensions
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论Frobenius扩张的完全相对同调和上同调
DOI：
10.21099/tkbjm/1496162799
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发表时间：
1995
期刊：
影响因子：
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通讯作者：
T. Nozawa
中科院分区：
文献类型：
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作者：
T. Nozawa
设G是有限群，K是G的子群，M是左G-模.然后在[6]中定义了reZ的完全相对同调群Hr（G，K，M）和上同调群Hr（G，K，M）。设1是G的单位元。对于K={1\ Hr（G，K，M）^H~ r '\G，K，M）的情形成立。但对任意G，K，M和r，不存在从Hr（G，K，M）到H~r-（G，K，M）的同构。事实上，在[6，p.262]中，存在G，K和M，使得Hr（G，K，M）=Z/2 Z，并且Hr（G，K，Af）=0，对所有reZ。如果我们在[6，p.262]中设置M=Q/Z，则Hr（G，K，M）=0且Hr（G，K，M）^Z/2 Z对所有reZ成立。设A是交换环K上的代数，F是子代数，使得环扩张A/F是Frobenius扩张.在第一节中，我们将引入r ∈ Z的完全相对上同调群Hr（A，F，-）和同调群Hr（A，F，-）。当环扩张F/K也是Frobenius扩张时，我们可以定义一个if-同态W% r '。Hr{A，F，（―）A）―> H~r~\A，F，―），其中A是Nakayama自同构.本文的主要目的是给出图A/r是同构的必要条件和等价条件。定理6.3，7.1和7.2提供了必要和必然的条件。在第8节中，我们将所得结果应用于由有限群G和子群K定义的扩张。推广了Tate上同调的对偶性，证明了Hr（G，K，-）= Hr\G，K，-）当且仅当K是G的Hall子群.
Let G be a finite group, K a subgroup of G and M a left G-module. Then for reZ the complete relative homology group Hr(G, K, M) and cohomology group Hr(G, K, M) are defined in [6]. Let 1 be the unit element of G. For the case of K={1\ Hr(G, K, M)^H~r'\G, K, M) holds. But itis not true that for any G, K, M and r there exists an isomorphism from Hr(G, K, M) into H~r-\G, K, M). In fact, in [6, p. 262] there are G, K and M such that Hr(G, K, M)=Z/2Z and Hr{G, K, Af)=0 for all reZ. And if we set M=Q/Z in [6, p. 262], Hr(G, K, M)=0 and Hr(G, K, M)^Z/2Z hold for all reZ. Let A be an algebra over a commutative ring K and F a subalgebra such that the ring extension A/F is a Frobenius extension. In section 1 we shall introduce the complete relative cohomology group Hr(A, F, ―) and homology group Hr(A, F, ―) for r£Z. When the ring extension F/K is also a Frobenius extension, We can define a if-homomorphism W%r'. Hr{A, F, (―)A)―> H~r~\A,F, ―) for reZ, where A is the Nakayama automorphism. The main purpose of this paper is to show necessary and sufficientconditions on which WrA/r is an isomorphism. Theorems 6.3, 7.1 and 7.2 provide the necessary and sufficientconditions. In section 8 we apply our results to extensions defined by a finitegroup G and a subgroup K. In generalization of the well-known duality for Tate cohomology we show that Hr{G, K, ―)=H~r~\G,K, ―) if and only if K is a Hall subgroup of G.