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