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. $\square$
Post a comment
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)