Tate cohomology (as well as Borel homology and cohomology) of connective K-theory for \(G=({\mathbb {Z}}/2)^n\) was completely calculated by Bruner and Greenlees (The connective K-theory of finite groups, 2003). In this note, we essentially redo the calculation by a different, more elementary method, and we extend it to \(p>2\) prime. We also identify the resulting spectra, which are products of Eilenberg–Mac Lane spectra, and finitely many finite Postnikov towers. For \(p=2\), we also reconcile our answer completely with the result of [2], which is in a different form, and hence the comparison involves some non-trivial combinatorics.

中文翻译：

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再次探讨基本阿贝尔群的关联k理论的Tate同调

\（G =（{\\ mathbb {Z}} / 2）^ n \）的结缔K理论的泰特（Tate）同调（以及Borel同源性和同调）由Bruner和Greenlees（有限组，2003年）。在本注释中，我们实质上通过另一种更基本的方法来重做计算，并将其扩展为\（p> 2 \）素数。我们还确定了产生的光谱，这是Eilenberg–Mac Lane光谱和有限个有限的Postnikov塔的乘积。对于\（p = 2 \），我们也将答案与[2]的结果完全调和，后者的形式不同，因此比较涉及一些非平凡的组合。