Cohomology groups of the class groups over a Z_pextension

On the Galois cohomology of ideal class groups

“…This isomorphism recovers the main result (Corollary 1.4) of the second named author in [8]. We remark that the latter result was proved in loc.…”

“…In several interesting cases, explicit class-field theoretic computations have been used to describe the cohomology of the ideal class group of the ring of algebraic integers of K as a Gal(K/k)-module. For example, such explicit descriptions play a key role in important recent work of Schoof [6,7], of Cornacchia and Greither [3] and of Greither [4] and also in work of the second named author [8]. In this note we wish to point out thatétale cohomology provides an alternative approach to the derivation of all such explicit descriptions and also allows one to make some new observations that are in much the same spirit.…”