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
Comments (0)