Lemma 93.5.4. In Example 93.5.1 assume $\Gamma $ 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 existence of a hull follows from Lemmas 93.5.2 and 93.5.3 and Formal Deformation Theory, Lemma 90.16.6 and Remark 90.15.7.
Assume $H^0(\Gamma , \text{End}_ k(V)) = k$. To see that $F$ is prorepresentable it suffices to show that $F$ is a deformation functor, see Formal Deformation Theory, Theorem 90.18.2. In other words, we have to show $F$ satisfies (RS). For this we can use the criterion of Formal Deformation Theory, Lemma 90.16.7. The required surjectivity of automorphism groups will follow if we show that
\[ A \cdot \text{id}_ M = \text{End}_{A[\Gamma ]}(M) \]
for any object $(A, M, \rho )$ of $\mathcal{F}$ such that $M \otimes _ A k$ is isomorphic to $V$ as a representation of $\Gamma $. Since the left hand side is contained in the right hand side, it suffices to show $\text{length}_ A \text{End}_{A[\Gamma ]}(M) \leq \text{length}_ A A$. Choose pairwise distinct ideals $(0) = I_ n \subset \ldots \subset I_1 \subset A$ with $n = \text{length}(A)$. By correspondingly filtering $M$, we see that it suffices to prove $\mathop{\mathrm{Hom}}\nolimits _{A[\Gamma ]}(M, I_ tM/I_{t + 1}M)$ has length $1$. Since $I_ tM/I_{t + 1}M \cong M \otimes _ A k$ and since any $A[\Gamma ]$-module map $M \to M \otimes _ A k$ factors uniquely through the quotient map $M \to M \otimes _ A k$ to give an element of
\[ \text{End}_{A[\Gamma ]}(M \otimes _ A k) = \text{End}_{k[\Gamma ]}(V) = k \]
we conclude.
$\square$
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