The Stacks project

Lemma 99.13.8. Let $k$ be a field and let $x = (X \to \mathop{\mathrm{Spec}}(k))$ be an object of $\mathcal{X} = \mathcal{S}\! \mathit{paces}'_{fp, flat, proper}$ 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. The discussion in Artin's Axioms, Section 98.8 only applies to fields of finite type over the base scheme $\mathop{\mathrm{Spec}}(\mathbf{Z})$. Our stack satisfies (RS*) by Lemma 99.13.7 and we may apply Artin's Axioms, Lemma 98.21.2 to get the vector spaces $T_ x(k)$ and $\text{Inf}_ x(k)$ mentioned in (2). Moreover, in the finite type case these spaces agree with the ones mentioned in (1) by Artin's Axioms, Remark 98.21.7. With this out of the way we can start the proof. Observe that the first order thickening $\mathop{\mathrm{Spec}}(k) \to \mathop{\mathrm{Spec}}(k[\epsilon ]) = \mathop{\mathrm{Spec}}(k[k])$ has conormal module $k$. Hence the formula in Deformation Theory, Lemma 91.14.2 describing infinitesimal deformations of $X$ and infinitesimal automorphisms of $X$ become

\[ T_ x(k) = \mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_ X}(\mathop{N\! L}\nolimits _{X/k}, \mathcal{O}_ X) \quad \text{and}\quad \text{Inf}_ x(k) = \mathop{\mathrm{Ext}}\nolimits ^0_{\mathcal{O}_ X}(\mathop{N\! L}\nolimits _{X/k}, \mathcal{O}_ X) \]

By More on Morphisms of Spaces, Lemma 76.21.5 and the fact that $X$ is Noetherian, we see that $\mathop{N\! L}\nolimits _{X/k}$ has coherent cohomology sheaves zero except in degrees $0$ and $-1$. By Derived Categories of Spaces, Lemma 75.8.4 the displayed $\mathop{\mathrm{Ext}}\nolimits $-groups are finite $k$-vector spaces and the proof is complete. $\square$

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