Vanishing deformation tangent space does not force rigidity of the deformation groupoid

The affine line has vanishing deformation tangent space. The affine line is quasi-compact and separated, so it satisfies the geometric hypotheses of [F4]. By [F1], $L_{X\mathrm{/}k}\simeq {\mathrm{\Omega }}_{X\mathrm{/}k}^1[0]\cong {\mathcal{O}}_X[0]$, so [F2] and [F3] give ${\mathrm{Ext}}_{{\mathcal{O}}_X}^1(L_{X\mathrm{/}k},{\mathcal{O}}_X)\cong H^1({\mathbb{A}}_k^1,{\mathcal{O}}_{{\mathbb{A}}_k^1})=0$.

A non-identity infinitesimal automorphism. Consider the trivial deformation $\mathrm{Spec}k[ϵ][x]$ over the dual numbers and the $k[ϵ]$-algebra endomorphism $\sigma \text{\hspace{0.17em}⁣}:k[ϵ][x]\to k[ϵ][x]$, $x\mapsto x+ϵx^2$. It is a well-defined $k[ϵ]$-algebra map because $x+ϵx^2$ is a polynomial; its inverse is $\tau \text{\hspace{0.17em}⁣}:x\mapsto x-ϵx^2$, since $\sigma (\tau (x))=x+ϵx^2-ϵ(x+ϵx^2{)}^2=x+ϵx^2-ϵx^2=x$ and likewise $\tau (\sigma (x))=x$ because ${ϵ}^2=0$. Moreover $\sigma$ reduces to the identity modulo $ϵ$ but $\sigma (x)-x=ϵx^2\mathrm{\ne }0$, so $\sigma$ is a non-identity automorphism of the trivial deformation reducing to the identity.

All deformation classes are trivial. Since ${\mathrm{Ext}}_{{\mathcal{O}}_X}^1(L_{X\mathrm{/}k},{\mathcal{O}}_X)=0$ by step 1.1, [F4] gives that for every local Artin $k$-algebra $A$ with residue field $k$ every deformation of ${\mathbb{A}}_k^1$ over $\mathrm{Spec}A$ is isomorphic to the trivial deformation; this confirms the hypothesis and the ${\mathrm{Ext}}^1$-part of the refuted statement.

Identification with a derivation and conclusion. By [F5] the infinitesimal automorphism group of the trivial deformation is ${\mathrm{Ext}}_{{\mathcal{O}}_X}^0(L_{X\mathrm{/}k},{\mathcal{O}}_X)={\mathrm{Der}}_k(k[x],k[x])=k[x]{\mathrm{\partial }}_x$, which is infinite-dimensional; the automorphism $\sigma$ of step 1.2 corresponds to the derivation $x^2{\mathrm{\partial }}_x$. Hence the deformation groupoid is not trivial and the trivial deformation is not rigid, although its tangent space vanishes by step 1.1 and all deformation classes are trivial by step 2.1. The correct statement is rigidity of isomorphism classes (Vanishing of the deformation tangent space forces rigidity of deformation classes); rigidity of the groupoid would require the additional vanishing of ${\mathrm{Ext}}^0$, which fails here.