The Stacks project

Proof. Observe that by Lemma 99.16.9 our stack in groupoids $\mathcal{X}$ satisfies property (RS*) defined in Artin's Axioms, Section 98.18. 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 $\mathcal{X}_0 = \mathcal{C}\! \mathit{omplexes}_{X_0/k}$. In Remark 99.16.7 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 $\mathop{\mathrm{Spec}}(k) \to \mathop{\mathrm{Spec}}(k[\epsilon ])$. Observe that $X_0$ and $X_\epsilon $ give rise to canonically equivalent small étale topoi, see More on Morphisms of Spaces, Section 76.9. By More on Morphisms of Spaces, Lemma 76.54.4 we see that $T\mathcal{F}_{\mathcal{X}_0, k, x_0}$ is the set of isomorphism classes of lifts of $E_0$ to $X_\epsilon $ in the sense of Deformation Theory, Lemma 91.16.7. We conclude that

Here we have used the identification $\epsilon k[\epsilon ] \cong k$ of $k[\epsilon ]$-modules. Using Deformation Theory, Lemma 91.16.7 once more we see that there is a surjection

of $k$-vector spaces. As $E_0$ is pseudo-coherent it lies in $D^-_{\textit{Coh}}(\mathcal{O}_{X_0})$ by Derived Categories of Spaces, Lemma 75.13.7. Since $E_0$ locally has finite tor dimension and $X_0$ is quasi-compact we see $E_0 \in D^ b_{\textit{Coh}}(\mathcal{O}_{X_0})$. Thus the $\mathop{\mathrm{Ext}}\nolimits $s above are finite dimensional $k$-vector spaces by Derived Categories of Spaces, Lemma 75.8.4. $\square$