The Stacks project

Lemma 59.57.6. Let $G$ be a profinite topological group. Then

  1. $H^ i(G, M)$ is torsion for $i > 0$ and any $G$-module $M$, and

  2. $H^ i(G, M) = 0$ if $M$ is a $\mathbf{Q}$-vector space.

Proof. Proof of (1). By dimension shifting we see that it suffices to show that $H^1(G, M)$ is torsion for every $G$-module $M$. Choose an exact sequence $0 \to M \to I \to N \to 0$ with $I$ an injective object of the category of $G$-modules. Then any element of $H^1(G, M)$ is the image of an element $y \in N^ G$. Choose $x \in I$ mapping to $y$. The stabilizer $U \subset G$ of $x$ is open, hence has finite index $r$. Let $g_1, \ldots , g_ r \in G$ be a system of representatives for $G/U$. Then $\sum g_ i(x)$ is an invariant element of $I$ which maps to $ry$. Thus $r$ kills the element of $H^1(G, M)$ we started with. Part (2) follows as then $H^ i(G, M)$ is both a $\mathbf{Q}$-vector space and torsion. $\square$

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