Vanishing of the deformation tangent space forces rigidity of deformation classes

Base case. For $A=k$ a deformation of $X$ over $\mathrm{Spec}k$ is a scheme $X^{\mathrm{\prime }}$ flat and locally finitely presented over $k$ isomorphic to $X$; hence ${\mathrm{Def}}_X(k)$ has exactly one isomorphism class, the trivial one. [F4, given]

Inductive hypothesis. Let $n\ge 1$ and assume that for every local Artin $k$-algebra $B$ with residue field $k$ and length $n$, every deformation of $X$ over $B$ is isomorphic to the trivial deformation. [F2, given]

Inductive step. Let $A^{\mathrm{\prime }}$ have length $n+1>1$. Its maximal ideal $\mathfrak{m}$ is nonzero and nilpotent by [F2]. Choose the last nonzero power ${\mathfrak{m}}^r$ and a nonzero $v\in {\mathfrak{m}}^r$; then $\mathfrak{m}v=0$, so $I=kv$ is an ideal of $A^{\mathrm{\prime }}$ of length one. Since $I\subseteq \mathfrak{m}$ and $\mathfrak{m}I=0$, also $I^2=0$. The quotient $A=A^{\mathrm{\prime }}\mathrm{/}I$ has length $n$, and $A^{\mathrm{\prime }}\to A$ is a small extension. The reduction $\bar{\xi }$ of a deformation $\xi$ over $A^{\mathrm{\prime }}$ is trivial by step 1.2. Fix an isomorphism of this reduction with $X{\times }_k\mathrm{Spec}A$. The trivial deformation over $A^{\mathrm{\prime }}$ is one relative lift of that specified $A$-deformation, so the set of relative lifts fixing it is nonempty. For the trivial family $Y=X{\times }_k\mathrm{Spec}A$, flat base change gives $L_{Y\mathrm{/}A}\simeq L_{X\mathrm{/}k}{\otimes }_{k}A$ (The cotangent complex of a morphism of schemes). The coefficient sheaf is the pushforward of ${\mathcal{O}}_X{\otimes }_{k}I$ from the closed special fibre $i:X\to Y$. The derived pullback/pushforward adjunction, Stacks tag 079W, gives ${\mathrm{Ext}}_Y^q(L_{Y\mathrm{/}A},i_{\ast }M)\cong {\mathrm{Ext}}_X^q(L{i}^{\ast }{L}_{Y\mathrm{/}A},M)$ for every $q$. Here $R{i}_{\ast }M=i_{\ast }M$ because the closed special fibre has the same underlying topological space as $Y$, so $i_{\ast }$ is exact restriction of scalars. Also $L{i}^{\ast }{L}_{Y\mathrm{/}A}\simeq L_{X\mathrm{/}k}$ by the flat-base-change identification and its special-fibre restriction. This uses the derived adjunction; exactness of $i_{\ast }$ alone does not imply that it preserves injectives. Therefore the isomorphism classes in this relative lift fibre form a torsor under ${\mathrm{Ext}}_{{\mathcal{O}}_X}^1(L_{X\mathrm{/}k},{\mathcal{O}}_X{\otimes }_{k}I)=0$ by [F3], and hence have a single element. Forgetting the chosen reduction isomorphism shows that $\xi$ is trivial. This proves the inductive step.

Conclusion of the induction. Steps 1.1 and 2.1 show that the statement holds for local Artin $k$-algebras of residue field $k$ and every positive length $n$, hence for every local Artin $k$-algebra $A$ with residue field $k$: all deformations of $X$ over $\mathrm{Spec}A$ are isomorphic to the trivial deformation. The conclusion is about isomorphism classes; the first-order infinitesimal automorphism group is ${\mathrm{Ext}}^0$, which is not assumed to vanish, so the groupoid itself need not be trivial. The Axiom of Choice is used only through the declared deformation-theoretic supplier. [F1, F4, step 2.1]