The Stacks project

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

Lemma 86.10.1. Let $A$ be a Noetherian ring and $I = (a)$ a principal ideal. Let $B$ be an object of (86.2.0.2) 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 86.8.2 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 86.7.2 applies. This finishes the proof, but we'd like to point out that in this case the use of Lemma 86.5.3 can be replaced by the much easier Lemma 86.5.5. $\square$


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