The Stacks project

88.1 Introduction

The main goal of this chapter is to prove Artin's theorem on dilatations, see Theorem 88.29.1; the result on contractions will be discussed in Artin's Axioms, Section 98.27. Both results use some material on formal algebraic spaces, hence in the middle part of this chapter, we continue the discussion of formal algebraic spaces from the previous chapter, see Formal Spaces, Section 87.1. The first part of this chapter is dedicated to algebraic preliminaries, mostly dealing with algebraization of rig-étale algebras.

Let $A$ be a Noetherian ring and let $I \subset A$ be an ideal. In the first part of this chapter (Sections 88.2 – 88.10) we discuss the category of $I$-adically complete algebras $B$ topologically of finite type over a Noetherian ring $A$. It is shown that $B = A\{ x_1, \ldots , x_ n\} /J$ for some (closed) ideal $J$ in the restricted power series ring (where $A$ is endowed with the $I$-adic topology). We show there is a good notion of a naive cotangent complex $\mathop{N\! L}\nolimits _{B/A}^\wedge $. If some power of $I$ annihilates $\mathop{N\! L}\nolimits _{B/A}^\wedge $, then we say $B$ is a rig-étale algebra over $(A, I)$; there is a similar notion of rig-smooth algebras. If $A$ is a G-ring, then we can show, using Popescu's theorem, that any rig-smooth algebra $B$ over $(A, I)$ is the completion of a finite type $A$-algebra; informally we say that we can “algebraize” $B$. However, the main result of the first part is that any rig-étale algebra $B$ over $(A, I)$ can be algebraized, see Lemma 88.10.2. One thing to note here is that we prove this without assuming the ring $A$ is a G-ring.

Many of the results discussed in the first part can be found in the paper [Elkik]. Other general references for this part are [EGA], [Abbes], and [Fujiwara-Kato].

In the second part of this chapter (Sections 88.12 – 88.24) we talk about types of morphisms of formal algebraic spaces in a reasonable level of generality (mostly for locally Noetherian formal algebraic spaces). The most interesting of these is the notion of a “formal modification” in the last section. We carefully check that our definition agrees with Artin's definition in [ArtinII].

Finally, in the third and last part of this chapter (Sections 88.25 – 88.30) we prove the main theorem and we give a few applications. In fact, we deduce Artin's theorem from a stronger result, namely, Theorem 88.27.4. This theorem says very roughly: if $f : \mathfrak X \to \mathfrak X'$ is a rig-étale morphism and $\mathfrak X'$ is the formal completion of a locally Noetherian algebraic space, then so is $\mathfrak X$. In Artin's work the morphism $f$ is assumed proper and rig-surjective.