Obstructions to deformations lie in Ext^2 of the cotangent complex

Affine case. If $X=\mathrm{Spec}B$ is affine over $A$, the relative lifting problem for this specified $A$-scheme is exactly the affine problem of [F1]; hence there is a canonical obstruction class $o(X)\in {\mathrm{Ext}}_B^2(L_{B\mathrm{/}A},B{\otimes }_{A}I)$ vanishing if and only if a lift exists, and when it does the isomorphism classes of lifts form a torsor under ${\mathrm{Ext}}^1$ with automorphism group over $X$ given by ${\mathrm{Ext}}^0={\mathrm{Der}}_A(B,B{\otimes }_{A}I)$.

Global case. For a general $X$, choose an affine open cover. On each member the specified $A$-scheme restricts to the affine lifting problem of step 1.1; the local obstruction classes transform by the canonical isomorphisms of the local cotangent complexes on overlaps. By the exact cited ringed-space extension theorem 08UZ, with $G={\mathcal{O}}_X{\otimes }_{A}I$ and the canonical map $I\to G$, the global obstruction is the single class $o(X)\in {\mathrm{Ext}}_{{\mathcal{O}}_X}^2(L_{X\mathrm{/}A},{\mathcal{O}}_X{\otimes }_{A}I)$ obtained as the boundary of the prescribed extension datum in the long exact Ext sequence of the transitivity triangle $L{f}^{\ast }{L}_{A\mathrm{/}{A}^{\mathrm{\prime }}}\to L_{X\mathrm{/}{A}^{\mathrm{\prime }}}\to L_{X\mathrm{/}A}$, and its vanishing is equivalent to the existence of a ringed-space solution. A ringed-space solution is an extension $0\to G\to \mathcal{A}\to {\mathcal{O}}_X\to 0$ with square-zero kernel $G={\mathcal{O}}_X{\otimes }_{A}I$, 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 ${\mathcal{O}}_X{\otimes }_{A}I\to G$, so Stacks tag 063Y gives flatness over $A^{\mathrm{\prime }}$ 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. The isomorphism classes of lifts and their automorphisms over $X$ glue in the same way, giving a torsor under ${\mathrm{Ext}}^1$ and automorphism group ${\mathrm{Ext}}^0={\mathrm{Der}}_A({\mathcal{O}}_X,{\mathcal{O}}_X{\otimes }_{A}I)$.

Naturality under isomorphisms of the specified $A$-scheme and functoriality in the small extension follow from the functoriality of the affine construction and of the gluing, both computed from the chart data; no functoriality for a bare morphism between underlying schemes is asserted. The boundary description is the construction in the full proof of Stacks tag 08UZ; its naturality supplies the stated compatibility for morphisms of the specified square-zero lifting data. If ${\mathrm{Ext}}^2=0$ then $o(X)=0$, so this specified $A$-scheme lifts.

For $X$ smooth over $A$, [F4] identifies ${\mathrm{Ext}}_{{\mathcal{O}}_X}^2(L_{X\mathrm{/}A},{\mathcal{O}}_X{\otimes }_{A}I)\cong H^2(X,T_{X\mathrm{/}A}{\otimes }_{A}I)$, $T_{X\mathrm{/}A}=\mathcal{H}om({\mathrm{\Omega }}_{X\mathrm{/}A}^1,{\mathcal{O}}_X)$; substituting into step 2.1 gives the smooth-case obstruction statement. The Axiom of Choice is used only through the declared derived-Hom and Cech suppliers.