The Stacks project

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

  1. $f$ is rig-étale,

  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-étale 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-étale 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-étale map in $\textit{WAdm}^{Noeth}$.

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. $\square$


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Your email address will not be published.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 0GCZ. Beware of the difference between the letter 'O' and the digit '0'.