The Stacks project

86.1 Introduction

The title of this chapter is a bit misleading because the most basic material on restricted power series is in the chapter on formal algebraic spaces. For example Formal Spaces, Section 85.21 defines the restricted power series ring $A\{ x_1, \ldots , x_ n\} $ given a linearly topologized ring $A$. In Formal Spaces, Section 85.22 we discuss the relationship between these restricted power series rings and morphisms of finite type between locally countably indexed formal algebraic spaces.

Let $A$ be a Noetherian ring and let $I \subset A$ be an ideal. In the first part of this chapter (Sections 86.286.8) 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 think of $\text{Spf}(B)$ as a rig-étale formal algebraic space over $A$. This leads to a definition of rig-étale morphisms of Noetherian formal algebraic spaces. After a certain amount of work we are able to prove the main result of the first part: if $\text{Spf}(B)$ is rig-étale over $A$ as above, then there exists a finite type $A$-algebra $C$ such that $B$ is isomorphic to the $I$-adic completion of $C$, see Lemma 86.7.4. One thing to note here is that we prove this without assuming the ring $A$ is excellent or even a G-ring. In the last section of the first part we show that under the assumption that $A$ is a G-ring there is a straightforward proof of the lemma based on Popescu's theorem.

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 we use the main result of the first part to prove Artin's result on dilatations from [ArtinII]. The result on contractions will be the subject of a later chapter (insert future reference here). The main existence theorem is the equivalence of categories in Theorem 86.10.9. It is more general than Artin's result in that it shows that any rig-étale morphism $f : W \to \mathop{\mathrm{Spec}}(A)_{/V(I)}$ is the completion of a morphism $Y \to \mathop{\mathrm{Spec}}(A)$ of algebraic spaces $X$ which is locally of finite type and isomorphism away from $V(I)$. In Artin's work the morphism $f$ is assumed proper and rig-surjective. A special case of this is the main lemma mentioned above and the general case follows from this by a straightforward (somewhat lengthy) glueing procedure. There are several lemmas modifying the main theorem the final one of which is (almost) exactly the statement in Artin's paper. In the last section we apply the results to modifications of $\mathop{\mathrm{Spec}}(A)$ before and after completion.

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