class formation

《Local Fields》 · 第 166–167 页 · Chapter XI，Section 2「Class Formations」，定义与 Axioms I–II

A class formation consists of a formation $(G,\{G_E\}_{E\in X},X,A)$, together with a homomorphism $\operatorname{inv}_E:H^2(F/E)\to\mathbb Q/\mathbb Z$ for each Galois extension $F/E$, satisfying axioms I and II below: Axiom I. For every Galois extension $F/E$, $H^1(F/E)=0$. Axiom II. (a) The homomorphism $\operatorname{inv}_E$ is injective, and maps $H^2(F/E)$ onto the unique subgroup of $\mathbb Q/\mathbb Z$ of order equal to $[F:E]$. (b) For any extension $E'/E$, $\operatorname{inv}_{E'}\circ\operatorname{Res}_{E/E'}=[E':E]\operatorname{inv}_E$.