The Stacks project

Lemma 88.24.4. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of locally Noetherian formal algebraic spaces over $S$ which is a formal modification. Then for any adic morphism $Y' \to Y$ of locally Noetherian formal algebraic spaces, the base change $f' : X \times _ Y Y' \to Y'$ is a formal modification.

Proof. The morphism $f'$ is proper by Formal Spaces, Lemma 87.31.3. The morphism $f'$ is rig-etale by Lemma 88.20.5. Then morphism $f'$ is rig-surjective by Lemma 88.21.4. Set $X' = X \times _ Y'$. The morphism $\Delta _{f'}$ is the base change of $\Delta _ f$ by the adic morphism $X' \times _{Y'} X' \to X \times _ Y X$. Hence $\Delta _{f'}$ is rig-surjective by Lemma 88.21.4. $\square$

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