In this section we use the work done in Section 88.17 to define rig-smooth morphisms of locally Noetherian algebraic spaces.
Definition 88.18.1. 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-smooth 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-smooth 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 88.18.2. 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-smooth,
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-smooth 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-smooth 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-smooth 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 88.18.1. The equivalence of (2), (3), and (4) follows from the fact that being rig-smooth is a local property of arrows of $\text{WAdm}^{Noeth}$ by Lemma 88.17.3 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. $\square$
Lemma 88.18.3. 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-smooth and $g$ is adic, then the base change $X \times _ Y Z \to Z$ is rig-smooth.
Proof. By Formal Spaces, Remark 87.21.10 and the discussion in Formal Spaces, Section 87.23, this follows from Lemma 88.17.4. $\square$
Lemma 88.18.4. 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-smooth, then so is $g \circ f$.
Proof. By Formal Spaces, Remark 87.21.14 this follows from Lemma 88.17.5. $\square$
Lemma 88.18.5. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of locally Noetherian formal algebraic spaces over $S$. If $f$ is rig-smooth, then $f$ is rig-flat.
Proof. Follows immediately from Lemma 88.17.6 and the definitions. $\square$
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