The discussion in Artin's Axioms, Section \ref{artin-section-tangent-spaces} only applies to fields of finite type over the base scheme $\Spec(\mathbf{Z})$. Our stack satisfies (RS*) by Lemma \ref{lemma-polarized-RS-star} and we may apply Artin's Axioms, Lemma \ref{artin-lemma-properties-lift-RS-star} 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 \ref{artin-remark-compare-deformation-spaces}. With this out of the way we can start the proof. \medskip\noindent One proof is to use an argument as in the proof of Lemma \ref{lemma-spaces-tangent-space}; 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 \ref{lemma-spaces-tangent-space}, the algebraicity of $\Polarizedstack \to \Spacesstack'_{fp, flat, proper}$, 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. \medskip\noindent 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 \ref{artin-section-predeformation-categories}. 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 $\Polarizedstack$ over $A$ and $\alpha$ is an isomorphism $(X_A, \mathcal{L}_A) \times_{\Spec(A)} \Spec(k) \cong (X, \mathcal{L})$. On the other hand, let $q : \mathcal{G} \to \mathcal{C}_k$ be the category cofibred in groupoids defined in Deformation Problems, Example \ref{examples-defos-example-graded-algebras}. 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 \ref{formal-defos-remark-localize-cofibered-groupoid}. 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 \ref{more-morphisms-lemma-deform-projective}) 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 \ref{divisors-lemma-relative-proj-flat}. Set $\mathcal{L}_A = \mathcal{O}_{X_A}(1)$; this is flat over $A$ by the same lemma. We get an isomorphism $(X_A, \mathcal{L}_A) \times_{\Spec(A)} \Spec(k) = (X, \mathcal{L})$ 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 \ref{more-morphisms-section-morphisms-thickenings} and \ref{more-morphisms-section-deform}. Some details omitted. \medskip\noindent 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 \ref{examples-defos-lemma-graded-algebras-TI} and the proof is complete.