On the cohomology of Frobenius algebras
On the cohomology of Frobenius algebras
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关于 Frobenius 代数的上同调
DOI：
10.1016/0022-4049(92)90070-v
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发表时间：
1992
影响因子：
0.8
通讯作者：
K. Sanada
中科院分区：
文献类型：
--
作者：
K. Sanada
设R为交换环，a为R上有限生成的自由Frobenius代数。在1986年，Nakayama利用他的自同构构造了a的完全（Hochschild）上同调理论。本文的目的是研究A的完全上同群H ' (A,-)（r EZ）的一些结构，特别是关于r的周期性；如果H ' (A， M) 2: Hn+ d （A， M）对所有的双侧da模M和所有的整数n都成立，则A具有周期为d# 0的周期上同调。有限群上同调理论中最重要的部分之一可能是上同调的周期性。在这种情况下，一个基本的方法是在Tate上同调上有一个杯积（参见[2,3,5]）。现在[3]第十一章，习题1和21给出了一个原型，杯子积的“第二类乘积”之一，它似乎适合于处理其周期性的完全（Hochschild）上同调。对它进行修正，与有限群的上同调理论并行，我们将得到这里需要的一个一般理论我们将证明上同调周期性的一个基本定理。在第1节中，我们解释了Frobenius代数的完全上同调理论和在Nakayama[4]之后上同调的维移。第二节给出了完全上同上的杯积的定义，并给出了杯积的显式构造。在后半部分，我们证明了两个基本性质：反交换性和结合性。在第3节中，沿着与Cartan和Eilenberg[3]，第十二章，第61节相同的路线，我们证明了
Let R be a commutative ring and let A be a finitely generated free Frobenius algebra over R. In [4], Nakayama constructed a complete (Hochschild) cohomology theory of A by means of his automorphism. The purpose of the paper is to investigate some structures of the complete cohomology groups H’(A,-)(r EZ) of A, in particular a periodicity with respect to r; A is said to have periodic cohomology of period d# 0 if H”(A, M) 2: Hn+ d (A, M) holds for all two-sidedA-modules M and all integers n. One of the most important parts of the cohomology theory of finite groups may be the periodicity of the cohomology. In such a case a fundamental approach is one with a cup product on the Tate cohomology (cf.[2, 3, 5]). Now [3, Chapter XI, Exercises 1 and 21 gives a prototype, one of the ‘products of the second kind’, of the cup product which seems to be appropriate to the complete (Hochschild) cohomology in dealing with its periodicity. Modifying this, in parallel with the cohomology theory of finite groups, we proceed to a general theory which is needed here and we will prove a basic theorem for the periodicity of the cohomology. In Section 1 we explain the complete cohomology theory of Frobenius algebras and dimension-shifting of the cohomology in a bit more detail after Nakayama [4]. In Section 2 we introduce the definition of the cup product on the complete cohomology and construct it explicitly. In the latter half we prove two basic properties: anti-commutativity and associativity. In Section 3, along the same lines as Cartan and Eilenberg[3, Chapter XII, Section 61, we prove the