Lemma 59.57.5. Let $G$ be a topological group. Let $k$ be a field. Let $V$ be a $k\text{-}G$-module. If $G$ is topologically finitely generated and $\dim _ k(V) < \infty $, then $\dim _ k H^1(G, V) < \infty $.
Proof. Let $g_1, \ldots , g_ r \in G$ be elements which topologically generate $G$, i.e., this means that the subgroup generated by $g_1, \ldots , g_ r$ is dense. By Lemma 59.57.4 we see that $H^1(G, V)$ is the $k$-vector space of extensions
\[ 0 \to V \to E \to k \to 0 \]
of $k\text{-}G$-modules. Choose $e \in E$ mapping to $1 \in k$. Write
\[ g_ i \cdot e = v_ i + e \]
for some $v_ i \in V$. This is possible because $g_ i \cdot 1 = 1$. We claim that the list of elements $v_1, \ldots , v_ r \in V$ determine the isomorphism class of the extension $E$. Once we prove this the lemma follows as this means that our Ext vector space is isomorphic to a subquotient of the $k$-vector space $V^{\oplus r}$; some details omitted. Since $E$ is an object of the category defined in Definition 59.57.1 we know there is an open subgroup $U$ such that $u \cdot e = e$ for all $u \in U$. Now pick any $g \in G$. Then $gU$ contains a word $w$ in the elements $g_1, \ldots , g_ r$. Say $gu = w$. Since the element $w \cdot e$ is determined by $v_1, \ldots , v_ r$, we see that $g \cdot e = (gu) \cdot e = w \cdot e$ is too. $\square$
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