The Stacks project

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

1. $f$ is rig-smooth,

2. for every commutative diagram

   $$\xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y }$$

   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 $\textit{WAdm}^{Noeth}$,

3. 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

4. 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 $\textit{WAdm}^{Noeth}$.

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.

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 $g \circ f$.

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.