Lemma 93.6.3 (0DVW)—The Stacks project

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. $\square$

The Stacks project

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