The Stacks project

Lemma 99.14.10. Let $k$ be a field and let $x = (X \to \mathop{\mathrm{Spec}}(k), \mathcal{L})$ be an object of $\mathcal{X} = \mathcal{P}\! \mathit{olarized}$ 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 99.14.9 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 part (1) by Artin's Axioms, Remark 98.21.7. With this out of the way we can start the proof.

One proof is to use an argument as in the proof of Lemma 99.13.8; this would require us to develop a deformation theory for pairs consisting of a scheme and a quasi-coherent module. Another proof would be the use the result from Lemma 99.13.8, the algebraicity of $\mathcal{P}\! \mathit{olarized}\to \mathcal{S}\! \mathit{paces}'_{fp, flat, proper}$, and a computation of the deformation space of an invertible module. However, what we will do instead is to translate the question into a deformation question on graded $k$-algebras and deduce the result that way.

Let $\mathcal{C}_ k$ be the category of Artinian local $k$-algebras $A$ with residue field $k$. We get a predeformation category $p : \mathcal{F} \to \mathcal{C}_ k$ from our object $x$ of $\mathcal{X}$ over $k$, see Artin's Axioms, Section 98.3. Thus $\mathcal{F}(A)$ is the category of triples $(X_ A, \mathcal{L}_ A, \alpha )$, where $(X_ A, \mathcal{L}_ A)$ is an object of $\mathcal{P}\! \mathit{olarized}$ over $A$ and $\alpha $ is an isomorphism $(X_ A, \mathcal{L}_ A) \times _{\mathop{\mathrm{Spec}}(A)} \mathop{\mathrm{Spec}}(k) \cong (X, \mathcal{L})$. On the other hand, let $q : \mathcal{G} \to \mathcal{C}_ k$ be the category cofibred in groupoids defined in Deformation Problems, Example 93.7.1. Choose $d_0 \gg 0$ (we'll see below how large). Let $P$ be the graded $k$-algebra

\[ P = k \oplus \bigoplus \nolimits _{d \geq d_0} H^0(X, \mathcal{L}^{\otimes d}) \]

Then $y = (k, P)$ is an object of $\mathcal{G}(k)$. Let $\mathcal{G}_ y$ be the predeformation category of Formal Deformation Theory, Remark 90.6.4. Given $(X_ A, \mathcal{F}_ A, \alpha )$ as above we set

\[ Q = A \oplus \bigoplus \nolimits _{d \geq d_0} H^0(X_ A, \mathcal{L}_ A^{\otimes d}) \]

The isomorphism $\alpha $ induces a map $\beta : Q \to P$. By deformation theory of projective schemes (More on Morphisms, Lemma 37.10.6) we obtain a $1$-morphism

\[ \mathcal{F} \longrightarrow \mathcal{G}_ y,\quad (X_ A, \mathcal{F}_ A, \alpha ) \longmapsto (Q, \beta : Q \to P) \]

of categories cofibred in groupoids over $\mathcal{C}_ k$. In fact, this functor is an equivalence with quasi-inverse given by $Q \mapsto \underline{\text{Proj}}_ A(Q)$. Namely, the scheme $X_ A = \underline{\text{Proj}}_ A(Q)$ is flat over $A$ by Divisors, Lemma 31.30.6. Set $\mathcal{L}_ A = \mathcal{O}_{X_ A}(1)$; this is flat over $A$ by the same lemma. We get an isomorphism $(X_ A, \mathcal{L}_ A) \times _{\mathop{\mathrm{Spec}}(A)} \mathop{\mathrm{Spec}}(k) = (X, \mathcal{L})$ from $\beta $. Then we can deduce all the desired properties of the pair $(X_ A, \mathcal{L}_ A)$ from the corresponding properties of $(X, \mathcal{L})$ using the techniques in More on Morphisms, Sections 37.3 and 37.10. Some details omitted.

In conclusion, we see that $T\mathcal{F} = T\mathcal{G}_ y = T_ y\mathcal{G}$ and $\text{Inf}(\mathcal{F}) = \text{Inf}_ y(\mathcal{G})$. These vector spaces are finite dimensional by Deformation Problems, Lemma 93.7.3 and the proof is complete. $\square$


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