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The Stacks project

Lemma 99.15.8. Let $k$ be a field and let $x = (X \to \mathop{\mathrm{Spec}}(k))$ be an object of $\mathcal{X} = \mathcal{C}\! \mathit{urves}$ over $\mathop{\mathrm{Spec}}(k)$.

  1. If $k$ is of finite type over $\mathbf{Z}$, then the vector spaces $T\mathcal{F}_{\mathcal{X}, k, x}$ and $\text{Inf}(\mathcal{F}_{\mathcal{X}, k, x})$ (see Artin's Axioms, Section 98.8) are finite dimensional, and

  2. in general the vector spaces $T_ x(k)$ and $\text{Inf}_ x(k)$ (see Artin's Axioms, Section 98.21) are finite dimensional.

Proof. This is immediate from the fully faithful embedding (99.15.1.1) and the corresponding fact for $\mathcal{S}\! \mathit{paces}'_{fp, flat, proper}$ (Lemma 99.13.8). $\square$

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