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.
Remark 87.24.2. In [Definition 1.7, ArtinII] a formal modification is defined as a proper morphism $f : X \to Y$ of locally Noetherian formal algebraic spaces satisfying the following three conditions1
the Cramer and Jacobian ideal of $f$ each contain an ideal of definition of $X$,
the ideal defining the diagonal map $\Delta : X \to X \times _ Y X$ is annihilated by an ideal of definition of $X \times _ Y X$, and
any adic morphism $\text{Spf}(R) \to Y$ lifts to $\text{Spf}(R) \to X$ whenever $R$ is a complete discrete valuation ring.
Let us compare these to our list of conditions above.
Ad (i). Property (i) agrees with our condition that $f$ be a rig-étale morphism: this follows from Lemma 87.8.2 part (7).
Ad (ii). Assume $f$ is rig-étale. Then $\Delta _ f : X \to X \times _ Y X$ is rig-étale as a morphism of locally Noetherian formal algebraic spaces which are rig-étale over $X$ (via $\text{id}_ X$ for the first one and via $\text{pr}_1$ for the second one). See Lemmas 87.20.5 and 87.20.7. Hence property (ii) agrees with our condition that $\Delta _ f$ be rig-surjective by Lemma 87.21.13.
Ad (iii). Property (iii) does not quite agree with our notion of a rig-surjective morphism, as Artin requires all adic morphisms $\text{Spf}(R) \to Y$ to lift to morphisms into $X$ whereas our notion of rig-surjective only asserts the existence of a lift after replacing $R$ by an extension. However, since we already have that $\Delta _ f$ is rig-étale and rig-surjective by (i) and (ii), these conditions are equivalent by Lemma 87.22.3.
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. $\square$
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