Compact Galois module

A compact Galois module is a compact profinite abelian group equipped with a continuous Galois group action. An abelian pro-p group is naturally a compact ${\mathbb{Z}}_p$-module. A continuous action of the Galois group of a Zp-extension extends to its Iwasawa algebra of a Zp-extension. This compact topology differs from the discrete topology usually used in Galois cohomology.