用户: Scavenger/Recording Some Facts/A Proof of Elkik’s Algebraization Theorem

In these notes we formulate and prove a version of Elkik’s algebraization theorem, using deformation theory of animated rings.

Definition 0.1. Let $A$ be an ordinary ring, $R$ be an ordinary $A$-algebra. Let $I\subseteq A$ be a finitely generated ideal. Then $R$ is called $I$-complete etale ($I$-complete smooth) if $R$ is derived $I$-complete as an $A$-module, $R{\otimes }_A^{L}A/I$ is concerntrated in degree $0$, and $R/I$ is etale (smooth) over $A/I$.

Theorem 0.2. Let $A$ be an ordinary ring, $R$ be an ordinary $A$-algebra. Let $I\subseteq A$ be a finitely generated ideal. If $R$ is $I$-completely etale ($I$-completely smooth), then it is the nondiscrete derived $I$-completion of an etale (smooth) $A$-algebra.

Proof. We only treat the smooth case, since the proof of the etale case is similar and slightly easier.

We first fix some notations. In the following texts, ${\otimes }_A$ will always mean (derived) tensor product over the $E_{\infty }$-ring $A$, instead of the ordinary tensor product of ordinary modules.

Using [1] tag 07M8 We can choose a smooth $A$-algebra $R^{\prime }$ such that $R^{\prime }/I{R}^{\prime }\simeq R/IR$ as $A/I$-algebras. We will show that the derived $I$-completion of $R^{\prime }$ is isomorphic to $R$ as $A$-algebras. Let $f_1,...,f_r\in A$ be generators of $I$, write $A_n=\mathrm{Kos}(A;f_1^n,...,f_r^n)$ (as an animated $A$-algebra). For $n\in {\mathbb{Z}}_{\ge 1}$, the map $A_{n+1}\to A_n$ is a base change of the map $\mathrm{Kos}(\mathbb{Z}[f_1^{n+1},...,f_r^{n+1}];f_1^{n+1},...,f_r^{n+1})\to \mathrm{Kos}(\mathbb{Z}[f_1^n,...,f_r^n];f_1^n,...,f_r^n)$, which is a map between discrete animated rings. Thus $A_{n+1}\to A_n$ is a finite compisition of square zero extensions of animated rings. The map $A_1\to {\pi }_0{A}_1\simeq A/I$ is also a finite composition of square zero extensions of animated rings. The $A$-algebras $R$ and $R^{\prime }$ are isomorphic after derived base change to $A/I$, so using deformation theory of animated rings, we see that we have a compatible system of isomorphisms $R{\otimes }_A{A}_n\simeq R^{\prime }{\otimes }_A{A}_n$ of $A_n$-algebras. Since ${\lim }_n(R{\otimes }_A{A}_n)$ and ${\lim }_n(R^{\prime }{\otimes }_A{A}_n)$ (taken in the $\infty$-category $\mathrm{AniRing}$) are derived $I$-completions of $R$ and $R^{\prime }$ (when passing to underlying connective $E_{\infty }$-rings), respectively. So $R$ is equivalent to the derived $I$-completion of $R^{\prime }$, viewing $R^{\prime }$ as a connective $E_{\infty }$-ring, thus $R$ is also the derived $I$-completion of $R^{\prime }$ when viewing $R^{\prime }$ as an ordinary ring.

$\square$

References

[1]	The Stacks project authors, “The Stacks project”, https://stacks.math.columbia.edu