The Stacks project

Lemma 76.19.11. The property $\mathcal{P}(f) =$“$f$ is formally smooth” is fpqc local on the base.

Proof. Let $f : X \to Y$ be a morphism of algebraic spaces over a scheme $S$. Choose an index set $I$ and diagrams

\[ \xymatrix{ U_ i \ar[d] \ar[r]_{\psi _ i} & V_ i \ar[d] \\ X \ar[r]^ f & Y } \]

with étale vertical arrows and $U_ i$, $V_ i$ affine schemes. Moreover, assume that $\coprod U_ i \to X$ and $\coprod V_ i \to Y$ are surjective, see Properties of Spaces, Lemma 66.6.1. By Lemma 76.19.10 we see that $f$ is formally smooth if and only if each of the morphisms $\psi _ i$ are formally smooth. Hence we reduce to the case of a morphism of affine schemes. In this case the result follows from Algebra, Lemma 10.138.16. Some details omitted. $\square$


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Your email address will not be published.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 06CS. Beware of the difference between the letter 'O' and the digit '0'.