The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

80.1 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 79.21 defines the restricted power series ring $A\{ x_1, \ldots , x_ n\} $ given a linearly topologized ring $A$. In Formal Spaces, Section 79.22 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 80.280.8) 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 80.7.4. 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 80.10.9. 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.

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0AM8. Beware of the difference between the letter 'O' and the digit '0'.