The Stacks project

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

  1. 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

  2. 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. $\square$

[1] An alternative would be to require the $A$-module $M$ with $G$-action given by $\rho $ is an $A\text{-}G$-module as defined in √Čtale Cohomology, Definition 59.57.1. However, since $M$ is a finite $A$-module, this is equivalent.

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