The Stacks project

Lemma 88.25.1. Let $T \subset X$ be a closed subset of a Noetherian affine scheme $X$. Let $W$ be a Noetherian affine formal algebraic space. Let $g : W \to X_{/T}$ be a rig-étale morphism. Then there exists an affine scheme $X'$ and a finite type morphism $f : X' \to X$ étale over $X \setminus T$ such that there is an isomorphism $X'_{/f^{-1}T} \cong W$ compatible with $f_{/T}$ and $g$. Moreover, if $W \to X_{/T}$ is étale, then $X' \to X$ is étale.

Proof. The existence of $X'$ is a restatement of Lemma 88.10.3. The final statement follows from More on Morphisms, Lemma 37.12.3. $\square$

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