The Stacks project

Lemma 106.2.3. In Situation 106.2.1 let $x_0 : \mathop{\mathrm{Spec}}(k) \to \mathcal{X}$ be a morphism, where $k$ is a finite type field over $S$. Then $\mathcal{F}_{\mathcal{X}, k, x_0}$ is a deformation category and $T\mathcal{F}_{\mathcal{X}, k, x_0}$ and $\text{Inf}(\mathcal{F}_{\mathcal{X}, k, x_0})$ are finite dimensional $k$-vector spaces.

Proof. Choose an affine open $\mathop{\mathrm{Spec}}(\Lambda ) \subset S$ such that $\mathop{\mathrm{Spec}}(k) \to S$ factors through it. By Artin's Axioms, Section 97.3 we obtain a predeformation category $\mathcal{F}_{\mathcal{X}, k, x_0}$ over the category $\mathcal{C}_\Lambda $. (As pointed out in locus citatus this category only depends on the morphism $\mathop{\mathrm{Spec}}(k) \to S$ and not on the choice of $\Lambda $.) By Artin's Axioms, Lemmas 97.6.1 and 97.5.2 $\mathcal{F}_{\mathcal{X}, k, x_0}$ is actually a deformation category. By Artin's Axioms, Lemma 97.8.1 we find that $T\mathcal{F}_{\mathcal{X}, k, x_0}$ and $\text{Inf}(\mathcal{F}_{\mathcal{X}, k, x_0})$ are finite dimensional $k$-vector spaces. $\square$


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