The Stacks project

Lemma 87.21.13. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of formal algebraic spaces. Assume that $X$ and $Y$ are locally Noetherian and $f$ is a closed immersion. The following are equivalent

  1. $f$ is rig-smooth and rig-surjective,

  2. $f$ is rig-├ętale and rig-surjective, and

  3. for every affine formal algebraic space $V$ and every morphism $V \to Y$ which is representable by algebraic spaces and ├ętale the morphism $X \times _ Y V \to V$ corresponds to a surjective morphism $B \to A$ in $\textit{WAdm}^{Noeth}$ whose kernel $J$ has the following property: $IJ = 0$ for some ideal of definition $I$ of $B$.

Proof. Let $I$ and $J$ be ideals of a ring $B$ such that $IJ^ n = 0$ and $I(J/J^2) = 0$. Then $I^ nJ = 0$ (proof omitted). Hence this lemma follows from a trivial combination of Lemmas 87.20.9 and 87.21.12. $\square$

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