A generalization of Tate-Nakayama theorem by using hypercohomology
A generalization of the theorem of Tate and Nakayama on class field theory is obtained, which is stated as follows: Theorem. (Generalized Tate-Nakayama theorem) Let \(G\) be a finite group, and let \(A^ \bullet\) denote a bounded complex of \(G\)-modules. Let \(a\) be an element of \(\widehat{H}^ 2 (G,A^ \bullet)\). Assume that for each \(p\)-Sylow subgroup \(G_ p\) of \(G\), the following conditions are satisfied: (1) \(\widehat{H}^ 1 (G_ p, A^ \bullet)=0\). (2) \(\widehat{H}^ 2 (G_ p,A^ \bullet)\) is generated by the element \(\text{Res}_{G/G_ p}(a)\) whose order is equal to \(| G_ p|\). Then for every integer \(q\) and every subgroup \(H\) of \(G\), one has \[ \widehat{H}^{q-2} (H,\mathbb{Z}) \simeq \widehat{H}^ q (H,A^ \bullet). \] The idea of proof is to use a modified hypercohomology group of \(G\) with coefficients \(A^ \bullet\). This generalizes the previous result by the same author [Invent. Math. 101, 705-715 (1990; Zbl 0751.11055)] proved for two-term complexes to arbitrary bounded complexes.
scientific article; zbMATH DE number 488097