In this section we use the work done in Section 88.19 to define rig-étale morphisms of locally Noetherian algebraic spaces.
Definition 88.20.1. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of locally Noetherian formal algebraic spaces over $S$. We say $f$ is rig-étale if for every commutative diagram with $U$ and $V$ affine formal algebraic spaces, $U \to X$ and $V \to Y$ representable by algebraic spaces and étale, the morphism $U \to V$ corresponds to a rig-étale map of adic Noetherian topological rings.
\[ \xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y } \]
Let us prove that we can check this condition étale locally on source and target.
Lemma 88.20.2. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of locally Noetherian formal algebraic spaces over $S$. The following are equivalent
$f$ is rig-étale,
for every commutative diagram
with $U$ and $V$ affine formal algebraic spaces, $U \to X$ and $V \to Y$ representable by algebraic spaces and étale, the morphism $U \to V$ corresponds to a rig-étale map in $\textit{WAdm}^{Noeth}$,
there exists a covering $\{ Y_ j \to Y\} $ as in Formal Spaces, Definition 87.11.1 and for each $j$ a covering $\{ X_{ji} \to Y_ j \times _ Y X\} $ as in Formal Spaces, Definition 87.11.1 such that each $X_{ji} \to Y_ j$ corresponds to a rig-étale map in $\textit{WAdm}^{Noeth}$, and
there exist a covering $\{ X_ i \to X\} $ as in Formal Spaces, Definition 87.11.1 and for each $i$ a factorization $X_ i \to Y_ i \to Y$ where $Y_ i$ is an affine formal algebraic space, $Y_ i \to Y$ is representable by algebraic spaces and étale, and $X_ i \to Y_ i$ corresponds to a rig-étale map in $\textit{WAdm}^{Noeth}$.
\[ \xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y } \]
Proof. The equivalence of (1) and (2) is Definition 88.20.1. The equivalence of (2), (3), and (4) follows from the fact that being rig-étale is a local property of arrows of $\text{WAdm}^{Noeth}$ by Lemma 88.19.3 and an application of the variant of Formal Spaces, Lemma 87.21.3 for morphisms between locally Noetherian algebraic spaces mentioned in Formal Spaces, Remark 87.21.5. $\square$
To be sure, a rig-étale morphism is locally of finite type.
Lemma 88.20.3. A rig-étale morphism of locally Noetherian formal algebraic spaces is locally of finite type.
Proof. The property $P$ in Lemma 88.19.3 implies the equivalent conditions (a), (b), (c), and (d) in Formal Spaces, Lemma 87.29.6. Hence this follows from Formal Spaces, Lemma 87.29.9. $\square$
Lemma 88.20.4. A rig-étale morphism of locally Noetherian formal algebraic spaces is rig-smooth.
Proof. Follows from the definitions and Lemma 88.8.3. $\square$
Lemma 88.20.5. Let $S$ be a scheme. Let $f : X \to Y$ and $g : Z \to Y$ be morphisms of locally Noetherian formal algebraic spaces over $S$. If $f$ is rig-étale and $g$ is adic, then the base change $X \times _ Y Z \to Z$ is rig-étale.
Proof. By Formal Spaces, Remark 87.21.10 and the discussion in Formal Spaces, Section 87.23, this follows from Lemma 88.19.4. $\square$
Lemma 88.20.6. Let $S$ be a scheme. Let $f : X \to Y$ and $g : Y \to Z$ be morphisms of locally Noetherian formal algebraic spaces over $S$. If $f$ and $g$ are rig-étale, then so is $g \circ f$.
Proof. By Formal Spaces, Remark 87.21.14 this follows from Lemma 88.19.5. $\square$
Lemma 88.20.7. Let $S$ be a scheme. Let $f : X \to Y$ and $g : Y \to Z$ be a morphism of locally Noetherian formal algebraic spaces over $S$. If $g \circ f$ and $g$ are rig-étale, then so is $f$.
Proof. By Formal Spaces, Remark 87.21.18 this follows from Lemma 88.19.6. $\square$
Lemma 88.20.8. Let $S$ be a scheme. Let $f : X \to Y$ and $g : Y \to Z$ be morphisms of locally Noetherian formal algebraic spaces over $S$. If $g \circ f$ is rig-étale and $g$ is an adic monomorphism, then $f$ is rig-étale.
Proof. Use Lemma 88.20.5 and that $f$ is the base change of $g \circ f$ by $g$. $\square$
Lemma 88.20.9. 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
$f$ is rig-smooth,
$f$ is rig-étale,
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 87.29.5.
We have (2) $\Rightarrow $ (1) by Lemma 88.20.4.
Proof of (3) $\Rightarrow $ (2). Assume (3). By Lemma 88.20.2 it suffices to show that the ring maps $B \to A$ occurring in (3) are rig-étale in the sense of Definition 88.19.2. 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 88.8.1.
Assume (1) and let $V$ and $B \to A$ be as in (3). By Definition 88.18.1 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 88.4.2 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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