In this section we define and study Artin's notion of a formal modification of locally Noetherian formal algebraic spaces. First, here is the definition.

Definition 87.24.1. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of locally Noetherian formal algebraic spaces over $S$. We say $f$ is a *formal modification* if

$f$ is a proper morphism (Formal Spaces, Definition 86.31.1),

$f$ is rig-étale,

$f$ is rig-surjective,

$\Delta _ f : X \to X \times _ Y X$ is rig-surjective.

A typical example is given in Lemma 87.24.3 and indeed we will later show that every formal modification is “formal locally” of this type, see Lemma 87.29.2. Let us compare these conditions with those in Artin's paper.

Lemma 87.24.3. Let $S$, $f : X' \to X$, $T \subset |X|$, $U \subset X$, $T' \subset |X'|$, and $U' \subset X'$ be as in Section 87.23. If $X$ is locally Noetherian, $f$ is proper, and $U' \to U$ is an isomorphism, then $f_{/T} : X'_{/T'} \to X_{/T}$ is a formal modification.

**Proof.**
By Formal Spaces, Lemmas 86.20.8 the source and target of the arrow are locally Noetherian formal algebraic spaces. The other conditions follow from Lemmas 87.23.4, 87.23.9, 87.23.10, and 87.23.11.
$\square$

Lemma 87.24.4. 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 86.31.3. The morphism $f'$ is rig-etale by Lemma 87.20.5. Then morphism $f'$ is rig-surjective by Lemma 87.21.4. 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 87.21.4.
$\square$

## Comments (0)