The Stacks project

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. $\square$


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0GGW. Beware of the difference between the letter 'O' and the digit '0'.