The Stacks project

Lemma 86.17.8. 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,

  2. $f$ is rig-étale,

  3. for every affine formal algebraic space $V$ and every morphism $V \to Y$ which is representable by algebraic spaces and étale 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: $I(J/J^2) = 0$ for some ideal of definition $I$ of $B$.

Proof. Let us observe that given $V$ and $V \to Y$ as in (2) without any further assumption on $f$ we see that the morphism $X \times _ Y V \to V$ corresponds to a surjective morphism $B \to A$ in $\textit{WAdm}^{Noeth}$ by Formal Spaces, Lemma 85.24.5.

We have (2) $\Rightarrow $ (1) by Lemma 86.17.4.

Proof of (3) $\Rightarrow $ (2). Assume (3). By Lemma 86.17.2 it suffices to show that the ring maps $B \to A$ occuring in (3) are rig-étale in the sense of Definition 86.16.2. Let $I$ be as in (3). The naive cotangent complex $\mathop{N\! L}\nolimits _{A/B}^\wedge $ of $A$ over $(B, I)$ is the complex of $A$-modules given by putting $J/J^2$ in degree $-1$. Hence $A$ is rig-étale over $(B, I)$ by Definition 86.8.1.

Assume (1) and let $V$ and $B \to A$ be as in (3). By Definition 86.15.1 we see that $B \to A$ is rig-smooth. Choose any ideal of definition $I \subset B$. Then $A$ is rig-smooth over $(B, I)$. As above the complex $\mathop{N\! L}\nolimits _{A/B}^\wedge $ is given by putting $J/J^2$ in degree $-1$. Hence by Lemma 86.4.2 we see that $J/J^2$ is annihilated by a power $I^ n$ for some $n \geq 1$. Since $B$ is adic, we see that $I^ n$ is an ideal of definition of $B$ and the proof is complete. $\square$


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