## 86.16 Rig-flat morphisms

In this section we use the work done in Section 86.15 to define rig-flat morphisms of locally Noetherian algebraic spaces.

Definition 86.16.1. 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-flat if 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-flat map of adic Noetherian topological rings.

Let us prove that we can check this condition étale locally on source and target.

Lemma 86.16.2. 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-flat,

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-flat map in $\textit{WAdm}^{Noeth}$,

3. there exists a covering $\{ Y_ j \to Y\}$ as in Formal Spaces, Definition 85.7.1 and for each $j$ a covering $\{ X_{ji} \to Y_ j \times _ Y X\}$ as in Formal Spaces, Definition 85.7.1 such that each $X_{ji} \to Y_ j$ corresponds to a rig-flat map in $\textit{WAdm}^{Noeth}$, and

4. there exist a covering $\{ X_ i \to X\}$ as in Formal Spaces, Definition 85.7.1 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-flat map in $\textit{WAdm}^{Noeth}$.

Proof. The equivalence of (1) and (2) is Definition 86.16.1. The equivalence of (2), (3), and (4) follows from the fact that being rig-flat is a local property of arrows of $\text{WAdm}^{Noeth}$ by Lemma 86.15.9 and an application of the variant of Formal Spaces, Lemma 85.17.3 for morphisms between locally Noetherian algebraic spaces mentioned in Formal Spaces, Remark 85.17.5. $\square$

Lemma 86.16.3. 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-flat and $g$ is locally of finite type, then the base change $X \times _ Y Z \to Z$ is rig-flat.

Proof. By Formal Spaces, Remark 85.17.10 and the discussion in Formal Spaces, Section 85.19, this follows from Lemma 86.15.6. $\square$

Lemma 86.16.4. 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-flat, then so is $g \circ f$.

Proof. By Formal Spaces, Remark 85.17.14 this follows from Lemma 86.15.10. $\square$

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