Lemma 88.10.1 (0ALS)—The Stacks project

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Table of contents
Part 5: Topics in Geometry
Chapter 88: Algebraization of Formal Spaces
Section 88.10: Algebraization of rig-étale algebras
Lemma 88.10.1 (cite)

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The rig-étale case of [III Theorem 7, Elkik]

Lemma 88.10.1. Let $A$ be a Noetherian ring and $I = (a)$ a principal ideal. Let $B$ be an object of (88.2.0.2) which is rig-étale over $(A, I)$ . Then there exists a finite type $A$ -algebra $C$ and an isomorphism $B \cong C^\wedge$ .

Proof. Choose a presentation $B = A[x_1, \ldots , x_ r]^\wedge /J$ . By Lemma 88.8.2 part (6) we can find $c \geq 0$ and $f_1, \ldots , f_ r \in J$ such that $\det _{1 \leq i, j \leq r}(\partial f_ j/\partial x_ i)$ divides $a^ c$ in $B$ and $a^ c J \subset (f_1, \ldots , f_ r) + J^2$ . Hence Lemma 88.7.2 applies. This finishes the proof, but we'd like to point out that in this case the use of Lemma 88.5.3 can be replaced by the much easier Lemma 88.5.5. $\square$

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