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

1. the Cramer and Jacobian ideal of $f$ each contain an ideal of definition of $X$,

2. 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

3. 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.

 We will not completely translate these conditions into the language developed in the Stacks project. We hope nonetheless the discussion here will be useful to the reader.

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