93.6 Continuous representations
A very interesting thing one can do is to take an infinite Galois group and study the deformation theory of its representations, see [Mazur-deforming].
Example 93.6.1 (Representations of a topological group). Let \Gamma be a topological group. Let \mathcal{F} be the category defined as follows
an object is a triple (A, M, \rho ) consisting of an object A of \mathcal{C}_\Lambda , a finite projective A-module M, and a continuous homomorphism \rho : \Gamma \to \text{GL}_ A(M) where \text{GL}_ A(M) is given the discrete topology1, and
a morphism (f, g) : (B, N, \tau ) \to (A, M, \rho ) consists of a morphism f : B \to A in \mathcal{C}_\Lambda together with a map g : N \to M which is f-linear and \Gamma -equivariant and induces an isomorpism N \otimes _{B, f} A \cong M.
The functor p : \mathcal{F} \to \mathcal{C}_\Lambda sends (A, M, \rho ) to A and (f, g) to f. It is clear that p is cofibred in groupoids. Given a finite dimensional k-vector space V and a continuous representation \rho _0 : \Gamma \to \text{GL}_ k(V), let x_0 = (k, V, \rho _0) be the corresponding object of \mathcal{F}(k). We set
\mathcal{D}\! \mathit{ef}_{V, \rho _0} = \mathcal{F}_{x_0}
Since every finite projective module over a local ring is finite free (Algebra, Lemma 10.78.2) we see that
\begin{matrix} \text{isomorphism classes}
\\ \text{of objects of }\mathcal{F}(A)
\end{matrix} = \coprod \nolimits _{n \geq 0}\quad \begin{matrix} \text{GL}_ n(A)\text{-conjugacy classes of}
\\ \text{continuous homomorphisms }\rho : \Gamma \to \text{GL}_ n(A)
\end{matrix}
Lemma 93.6.2. Example 93.6.1 satisfies the Rim-Schlessinger condition (RS). In particular, \mathcal{D}\! \mathit{ef}_{V, \rho _0} is a deformation category for any finite dimensional continuous representation \rho _0 : \Gamma \to \text{GL}_ k(V).
Proof.
The proof is exactly the same as the proof of Lemma 93.5.2.
\square
Lemma 93.6.3. In Example 93.6.1 let \rho _0 : \Gamma \to \text{GL}_ k(V) be a finite dimensional continuous representation. Then
T\mathcal{D}\! \mathit{ef}_{V, \rho _0} = H^1(\Gamma , \text{End}_ k(V)) \quad \text{and}\quad \text{Inf}(\mathcal{D}\! \mathit{ef}_{V, \rho _0}) = H^0(\Gamma , \text{End}_ k(V))
Thus \text{Inf}(\mathcal{D}\! \mathit{ef}_{V, \rho _0}) is always finite dimensional and T\mathcal{D}\! \mathit{ef}_{V, \rho _0} is finite dimensional if \Gamma is topologically finitely generated.
Proof.
The proof is exactly the same as the proof of Lemma 93.5.3.
\square
In Example 93.6.1 if \Gamma is topologically finitely generated and (V, \rho _0) is a finite dimensional continuous representation of \Gamma over k, then \mathcal{D}\! \mathit{ef}_{V, \rho _0} admits a presentation by a smooth prorepresentable groupoid in functors over \mathcal{C}_\Lambda and a fortiori has a (minimal) versal formal object. This follows from Lemmas 93.6.2 and 93.6.3 and the general discussion in Section 93.3.
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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