The Stacks project

Lemma 93.6.4. In Example 93.6.1 assume $\Gamma $ is topologically finitely generated. Let $\rho _0 : \Gamma \to \text{GL}_ k(V)$ be a finite dimensional representation. Assume $\Lambda $ is a complete local ring with residue field $k$ (the classical case). Then the functor

\[ F : \mathcal{C}_\Lambda \longrightarrow \textit{Sets},\quad A \longmapsto \mathop{\mathrm{Ob}}\nolimits (\mathcal{D}\! \mathit{ef}_{V, \rho _0}(A))/\cong \]

of isomorphism classes of objects has a hull. If $H^0(\Gamma , \text{End}_ k(V)) = k$, then $F$ is prorepresentable.

Proof. The proof is exactly the same as the proof of Lemma 93.5.4. $\square$

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