We clarify the discussion in this post resulting in a generalization of a result of Mike Artin.
Let X be a Noetherian algebraic space. Let T ⊂ X be a closed subspace. Let us denote X/T the formal completion of X along T (Tag 0AIX). Let W —> X/T be a rig-etale morphism of formal algebraic spaces, which means that
- W —> X/T is representable by algebraic spaces, i.e., it is an adic morphism of formal algebraic spaces Tag 0AQ2),
- W —> X/T is locally of finite type, i.e., it is etale locally on affine formal algebraic pieces given by a continuous ring map A —> B which is topologically of finite type Tag 0ALL),
- these ring maps A —> B have a completed cotangent complex whose cohomology groups are annihilated by an ideal of definition of A, for more details see Tag 0ALP and Tag 0AQK.
These three conditions correspond to condition (i) of Definition 1.7 of Artin’s paper “Formal Moduli: II”. The first result is that
Given W —> X/T rig-etale there exists a morphism of algebraic spaces Y —> X which is an isomorphism over X – T and whose completion Y/T —> X/T is isomorphic to W —> X/T.
In fact, Theorem 0ARB tells us we obtain an equivalence of categories.
This theorem does not often produce separated morphisms Y —> X if we start with a random W —> X/T. A typical thing that happens can be seen by starting with X equal to the affine line, T = {0} and W two copies of the completion of X at 0. Then the resulting Y is the affine line with 0 doubled.
Thus to get Artin’s theorem on dilatations we need to impose conditions on W —> X/T guaranteeing that Y —> X is separated or even proper. To do this we will use the notion of a rig-surjective morphism W’ —> W of locally Noetherian formal algebraic spaces W, W’ defined by requiring adic morphisms Spf(R) —> W with R a cdvr to lift to Spf(R’) —> W’ for some extension of cdvr R ⊂ R’ (Tag 0AQP). Let’s say an adic morphism W —> W’ of locally Noetherian formal algebraic spaces is a rig-monomorphism if the diagonal morphism is rig-surjective. In this language the conditions (ii) and (iii) from Definition 1.7 of Artin’s paper have the following interpretations:
- If W —> X/T as above is separated and a rig-monomorphism, then Y —> X is separated (Tag 0ARW),
- If W —> X/T as above is proper, a rig-monomorphism, and rig-surjective, then Y —> X is proper (Tag 0ARX).
The second statement recovers exactly Artin’s theorem on dilatations.
One typically applies the result to construct modifications Y —> X (Tag 0AD7) by taking the complete local ring A of X at a closed point x and setting W equal to the formal completion of the blow up of A at an ideal I ⊂ A which is locally principal on the puctured spectrum of A. Here the funny situation occurs that we can first read Tag 0ARX backwards over Spec(A) to conclude that W —> X/x has the required properties and then forwards to conclude that Y —> X is proper. In other situations it may not be that easy to verify the assumptions needed for the application of the theorem and it would behoove us to prove a few lemmas that help with this task.