I am trying to understand the cohomology groups $H^1(\Gamma, \mathbb{Z}^r)$, where $\Gamma$ is a finite (or profinite) group with a continuous action on $\mathbb{Z}^r$ (in my setting, $\Gamma$ will be the Galois group of some extension of nonarchimedean fields).
Does anyone have a resource I can start with?
Most of what I have found (e.g. here) has the free abelian group as the domain, and not codomain of cocycles.
Edit (in response to Eric Wofsey's comment): This is a fair point, so I will try to explain my motivation.
It is known that the category of algebraic tori over a given field $F$ is anti-equivalent to the category of finitely-generated free abelian groups over $F$, equipped with a continuous action of $\operatorname{Gal}(F_s / F)$, where $F_s$ is the separable closure of $F$; passing to finite subextensions $F \subset E \subset F_s$ allows us to consider tori defined over $F$ which split over $E$.
Modulo $F$-isomorphism, the $E$-isomorphism classes of these tori can be shown to correspond with finite disjoint unions of sets in bijection with the set $H^1(\operatorname{Gal}(E/F), \mathbb{Z}^r)$ (where $r$ is the dimension of the torus).
Because there are only finitely many such isomorphism classes (at least, in the examples I have computed), this implies that the groups $H^1(\operatorname{Gal}(E/F), \mathbb{Z}^r)$ are themselves finite; I am trying to understand if there is an intrinsic reason for this.
