The Stacks project

Proof. Since pullback commutes with restriction, we may replace $X$ by $U$. Thus we may assume that $X$ is affine and weakly contractible and that every point of $X$ specializes to a point of $Z$. By Lemma 61.25.2 part (1) it suffices to show that $v(Z) = X$ in this case. Thus we have to show: If $A$ is a w-contractible ring, $I \subset A$ an ideal contained in the Jacobson radical of $A$ and $A \to B \to A/I$ is a factorization with $A \to B$ ind-étale, then there is a unique retraction $B \to A$ compatible with maps to $A/I$. Observe that $B/IB = A/I \times R$ as $A/I$-algebras. After replacing $B$ by a localization we may assume $B/IB = A/I$. Note that $\mathop{\mathrm{Spec}}(B) \to \mathop{\mathrm{Spec}}(A)$ is surjective as the image contains $V(I)$ and hence all closed points and is closed under specialization. Since $A$ is w-contractible there is a retraction $B \to A$. Since $B/IB = A/I$ this retraction is compatible with the map to $A/I$. We omit the proof of uniqueness (hint: use that $A$ and $B$ have isomorphic local rings at maximal ideals of $A$). $\square$