Class formation

In mathematics, a class formation is a topological group acting on a module satisfying certain conditions. Class formations were introduced by Emil Artin and John Tate to organize the various Galois groups and modules that appear in class field theory. A formation is a topological group G together with a topological G-module A on which G acts continuously. A layer E/F of a formation is a pair of open subgroups E, F of G such that F is a finite index subgroup of E. It is called a normal layer if F is a normal subgroup of E, and a cyclic layer if in addition the quotient group is cyclic. If E is a subgroup of G, then AE is defined to be the elements of A fixed by E. We write Hn(E/F) for the Tate cohomology group Hn(E/F, AF) whenever E/F is a normal layer. (Some authors think of E and F as fixed fields rather than subgroup of G, so write F/E instead of E/F.) In applications, G is often the absolute Galois group of a field, and in particular is profinite, and the open subgroups therefore correspond to the finite extensions of the field contained in some fixed separable closure. A class formation is a formation such that for every normal layer E/F H1(E/F) is trivial, and H2(E/F) is cyclic of order |E/F|. In practice, these cyclic groups come provided with canonical generators uE/F ∈ H2(E/F), called fundamental classes, that are compatible with each other in the sense that the restriction (of cohomology classes) of a fundamental class is another fundamental class. Often the fundamental classes are considered to be part of the structure of a class formation. A formation that satisfies just the condition H1(E/F)=1 is sometimes called a field formation. For example, if G is any finite group acting on a field L and A=L×, then this is a field formation by Hilbert's theorem 90. The most important examples of class formations (arranged roughly in order of difficulty) are as follows: Archimedean local class field theory: The module A is the group of non-zero complex numbers, and G is either trivial or is the cyclic group of order 2 generated by complex conjugation.