Let Q be a finite group acting on a group G as a group automorphisms, C∗(G) the bar complex, H∗Q(G,A) the homology of invariant group chains and HQ∗(G,A) the cohomology invariant, both defined in Knudson’s paper “The homology of invariant group chains”. In this paper, we define the Tate homology of invariants Ĥ∗Q(G,A) and the Tate cohomology of invariants ĤQ∗(G,A). When the coefficient A is the abelian group of the integers, we proved that these groups are isomorphics, ĤQi(G, ℤ)≅Ĥ i−1Q(G, ℤ). Further, we prove that the homology and cohomology of invariant group chains are duals, HQi(G, ℤ)≅H i−1Q(G, ℤ), i ≥ 2.

中文翻译：

---

不变群链的 Tate (Co) 同源性

让问是作用于群的有限群G作为群自同构，C*(G)酒吧综合体，H*问(G,一种)不变群链的同源性和H问*(G,一种)上同调不变量，均在 Knudson 的论文“不变群链的同调”中定义。在本文中，我们定义了不变量的 Tate 同调H*问(G,一种)和不变量的 Tate 上同调H问*(G,一种). 当系数一种是整数的阿贝尔群，我们证明了这些群是同构的，H问一世(G, ℤ)≅H 一世-1问(G, ℤ). 此外，我们证明不变群链的同调和上同调是对偶的，H问一世(G, ℤ)≅H 一世-1问(G, ℤ),一世 ≥ 2.