The Stacks project

Lemma 93.5.2. Example 93.5.1 satisfies the Rim-Schlessinger condition (RS). In particular, $\mathcal{D}\! \mathit{ef}_{V, \rho _0}$ is a deformation category for any finite dimensional representation $\rho _0 : \Gamma \to \text{GL}_ k(V)$.

Proof. Let $A_1 \to A$ and $A_2 \to A$ be morphisms of $\mathcal{C}_\Lambda $. Assume $A_2 \to A$ is surjective. According to Formal Deformation Theory, Lemma 90.16.4 it suffices to show that the functor $\mathcal{F}(A_1 \times _ A A_2) \to \mathcal{F}(A_1) \times _{\mathcal{F}(A)} \mathcal{F}(A_2)$ is an equivalence of categories.

Consider an object

\[ ((A_1, M_1, \rho _1), (A_2, M_2, \rho _2), (\text{id}_ A, \varphi )) \]

of the category $\mathcal{F}(A_1) \times _{\mathcal{F}(A)} \mathcal{F}(A_2)$. Then, as seen in the proof of Lemma 93.4.2, we can consider the finite projective $A_1 \times _ A A_2$-module $M_1 \times _\varphi M_2$. Since $\varphi $ is compatible with the given actions we obtain

\[ \rho _1 \times \rho _2 : \Gamma \longrightarrow \text{GL}_{A_1 \times _ A A_2}(M_1 \times _\varphi M_2) \]

Then $(M_1 \times _\varphi M_2, \rho _1 \times \rho _2)$ is an object of $\mathcal{F}(A_1 \times _ A A_2)$. This construction determines a quasi-inverse to our functor. $\square$

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