On the Galois Cohomology Group of the Ring of Integers in a Global Field and its Adele Ring

Abstract

By a global field we mean a field which is either an algebraic number field, or an algebraic function field in one variable over a finite constant field.The purpose of the present note is to show that the Galois cohomology group of the ring of integers of a global field is isomorphic to that of the ring of integers of its adele ring and is reduced to asking for that of the ring of local integers.

Galois invariants of K_1-groups of Iwasawa algebras

“…From [10] we know, that H 1 (Gal(M 1 /M 2 ), O M 1 ) = 1 for every finite unramified extension M 1 of Q p . Since ∆ can be written as the inverse limit of finite groups corresponding to the finite unramified subextensions of S, and S[C G ] as a ∆module is isomorphic to the direct sum of copies of S, we get the statement above by using standard properties of the cohomology of groups.…”