Lemma 99.5.9. In Situation 99.5.1 assume that $S$ is a locally Noetherian scheme and $B \to S$ is locally of finite presentation. Let $k$ be a finite type field over $S$ and let $x_0 = (\mathop{\mathrm{Spec}}(k), g_0, \mathcal{G}_0)$ be an object of $\mathcal{X} = \mathcal{C}\! \mathit{oh}_{X/B}$ over $k$. Then the spaces $T\mathcal{F}_{\mathcal{X}, k, x_0}$ and $\text{Inf}(\mathcal{F}_{\mathcal{X}, k, x_0})$ (Artin's Axioms, Section 98.8) are finite dimensional.
Proof. Observe that by Lemma 99.5.7 our stack in groupoids $\mathcal{X}$ satisfies property (RS*) defined in Artin's Axioms, Section 98.21. In particular $\mathcal{X}$ satisfies (RS). Hence all associated predeformation categories are deformation categories (Artin's Axioms, Lemma 98.6.1) and the statement makes sense.
In this paragraph we show that we can reduce to the case $B = \mathop{\mathrm{Spec}}(k)$. Set $X_0 = \mathop{\mathrm{Spec}}(k) \times _{g_0, B} X$ and denote $\mathcal{X}_0 = \mathcal{C}\! \mathit{oh}_{X_0/k}$. In Remark 99.5.5 we have seen that $\mathcal{X}_0$ is the $2$-fibre product of $\mathcal{X}$ with $\mathop{\mathrm{Spec}}(k)$ over $B$ as categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. Thus by Artin's Axioms, Lemma 98.8.2 we reduce to proving that $B$, $\mathop{\mathrm{Spec}}(k)$, and $\mathcal{X}_0$ have finite dimensional tangent spaces and infinitesimal automorphism spaces. The tangent space of $B$ and $\mathop{\mathrm{Spec}}(k)$ are finite dimensional by Artin's Axioms, Lemma 98.8.1 and of course these have vanishing $\text{Inf}$. Thus it suffices to deal with $\mathcal{X}_0$.
Let $k[\epsilon ]$ be the dual numbers over $k$. Let $\mathop{\mathrm{Spec}}(k[\epsilon ]) \to B$ be the composition of $g_0 : \mathop{\mathrm{Spec}}(k) \to B$ and the morphism $\mathop{\mathrm{Spec}}(k[\epsilon ]) \to \mathop{\mathrm{Spec}}(k)$ coming from the inclusion $k \to k[\epsilon ]$. Set $X_0 = \mathop{\mathrm{Spec}}(k) \times _ B X$ and $X_\epsilon = \mathop{\mathrm{Spec}}(k[\epsilon ]) \times _ B X$. Observe that $X_\epsilon $ is a first order thickening of $X_0$ flat over the first order thickening $\mathop{\mathrm{Spec}}(k) \to \mathop{\mathrm{Spec}}(k[\epsilon ])$. Unwinding the definitions and using Lemma 99.5.8 we see that $T\mathcal{F}_{\mathcal{X}_0, k, x_0}$ is the set of lifts of $\mathcal{G}_0$ to a flat module on $X_\epsilon $. By Deformation Theory, Lemma 91.12.1 we conclude that
\[ T\mathcal{F}_{\mathcal{X}_0, k, x_0} = \mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_{X_0}}(\mathcal{G}_0, \mathcal{G}_0) \]
Here we have used the identification $\epsilon k[\epsilon ] \cong k$ of $k[\epsilon ]$-modules. Using Deformation Theory, Lemma 91.12.1 once more we see that
\[ \text{Inf}(\mathcal{F}_{\mathcal{X}, k, x_0}) = \mathop{\mathrm{Ext}}\nolimits ^0_{\mathcal{O}_{X_0}}(\mathcal{G}_0, \mathcal{G}_0) \]
These spaces are finite dimensional over $k$ as $\mathcal{G}_0$ has support proper over $\mathop{\mathrm{Spec}}(k)$. Namely, $X_0$ is of finite presentation over $\mathop{\mathrm{Spec}}(k)$, hence Noetherian. Since $\mathcal{G}_0$ is of finite presentation it is a coherent $\mathcal{O}_{X_0}$-module. Thus we may apply Derived Categories of Spaces, Lemma 75.8.4 to conclude the desired finiteness. $\square$
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