Definition 59.57.2. Let $G$ be a topological group. Let $M$ be a discrete $G$-module with continuous $G$-action. In other words, $M$ is an object of the category $\text{Mod}_ G$ introduced in Definition 59.57.1.

The right derived functors $H^ i(G, M)$ of $H^0(G, M)$ on the category $\text{Mod}_ G$ are called the

*continuous group cohomology groups*of $M$.If $G$ is an abstract group endowed with the discrete topology then the $H^ i(G, M)$ are called the

*group cohomology groups*of $M$.If $G$ is a Galois group, then the groups $H^ i(G, M)$ are called the

*Galois cohomology groups*of $M$.If $G$ is the absolute Galois group of a field $K$, then the groups $H^ i(G, M)$ are sometimes called the

*Galois cohomology groups of $K$ with coefficients in $M$*. In this case we sometimes write $H^ i(K, M)$ instead of $H^ i(G, M)$.

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