First-order deformations are controlled by Ext^1 of the cotangent complex

Affine case. Let $X=\mathrm{Spec}B$ be affine over $k$. By [F2] the deformations of $B$ over $k[I]$ form a torsor under ${\mathrm{Ext}}_B^1(L_{B\mathrm{/}k},B{\otimes }_{k}I)$ once nonempty, and the trivial deformation $B{\otimes }_{k}k[I]$ provides a base point; transporting the torsor structure along this base point gives a canonical bijection ${\mathrm{Def}}_X(k[I]{)}_{\mathrm{i}\mathrm{s}\mathrm{o}}\cong {\mathrm{Ext}}_B^1(L_{B\mathrm{/}k},B{\otimes }_{k}I)={\mathrm{Ext}}_{{\mathcal{O}}_X}^1(L_{X\mathrm{/}k},{\mathcal{O}}_X{\otimes }_{k}I)$ carrying the trivial class to $0$, together with the identification of the automorphism group of the trivial deformation with ${\mathrm{Ext}}^0$. The affine ring/sheaf Ext equality uses the affine derived quasi-coherent equivalence recorded in The cotangent complex of a morphism of schemes and the affine cotangent comparison. A square-zero thickening of this affine special fibre is affine by Stacks tag 04EW, and the finite-presentation assertion in Deformations of algebras: obstruction in degree two and torsor structure in degree one ensures the stipulated local finiteness.

Global case. For a general flat locally finitely presented $X$, use the ringed-space extension classification of Stacks tag 08UZ. The exact cited source theorem 08UZ applies to the thickening $\mathrm{Spec}k\subseteq \mathrm{Spec}k[I]$ with coefficient ${\mathcal{O}}_X{\otimes }_{k}I$ and its canonical map from $I$. A ringed-space solution is an extension $0\to G\to \mathcal{A}\to {\mathcal{O}}_X\to 0$ with square-zero kernel $G={\mathcal{O}}_X{\otimes }_{k}I$, which is quasi-coherent. Stacks tag 04EW (the square-zero scheme criterion) therefore makes $(X,\mathcal{A})$ a scheme, with the corresponding opens over affine charts of $X$ also affine. The prescribed map from $I$ induces the identity ${\mathcal{O}}_X{\otimes }_{k}I\to G$, so Stacks tag 063Y gives flatness over $k[I]$ and reduction exactly $X$. On affine finite-presentation charts, the finite-presentation assertion of Deformations of algebras: obstruction in degree two and torsor structure in degree one gives finite presentation of the lift. Conversely every flat lift has this canonical kernel by the same flatness criterion. Thus the ringed-space solutions and the required scheme lifts are the same groupoid. This argument applies also to infinite-dimensional $I$. The source theorem computes the extension classes as a torsor under ${\mathrm{Ext}}_{{\mathcal{O}}_X}^1(L_{X\mathrm{/}k},{\mathcal{O}}_X{\otimes }_{k}I)$; consequently the isomorphism classes of global deformations are in canonical bijection with that group, the trivial deformation supplying the distinguished origin. Naturality under isomorphisms of $X$ and covariance in $I$ follow from the functoriality of base change on augmented bases and of the extension classification. In finite-dimensional cases this includes the corresponding morphisms of small extensions. This proves the first-order classification.

Infinitesimal automorphisms and the smooth specialization: the automorphism statement is the ${\mathrm{Ext}}^0$ part of [F2] and [F3] at the trivial deformation, and ${\mathrm{Ext}}_{{\mathcal{O}}_X}^0(L_{X\mathrm{/}k},{\mathcal{O}}_X)={\mathrm{Der}}_k({\mathcal{O}}_X,{\mathcal{O}}_X)$ is the affine identification glued over the cover. If $X$ is smooth over $k$, [F4] gives $L_{X\mathrm{/}k}\simeq {\mathrm{\Omega }}_{X\mathrm{/}k}^1[0]$ and hence $T_X^1\cong {\mathrm{Ext}}^1(L_{X\mathrm{/}k},{\mathcal{O}}_X)\cong H^1(X,\mathcal{H}om({\mathrm{\Omega }}_{X\mathrm{/}k}^1,{\mathcal{O}}_X))=H^1(X,T_{X\mathrm{/}k})$, the Kodaira-Spencer description. The Axiom of Choice is used only through the declared derived-Hom and Cech suppliers.