The Stacks project

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.

[1] 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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