The Stacks project

Lemma 98.3.2. Let $S$ be a locally Noetherian scheme. Let $F : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. Assume either

  1. $F$ is formally smooth on objects (Criteria for Representability, Section 97.6),

  2. $F$ is representable by algebraic spaces and formally smooth, or

  3. $F$ is representable by algebraic spaces and smooth.

Then for every finite type field $k$ over $S$ and object $x_0$ of $\mathcal{X}$ over $k$ the functor (98.3.1.1) is smooth in the sense of Formal Deformation Theory, Definition 90.8.1.

Proof. Case (1) is a matter of unwinding the definitions. Assumption (2) implies (1) by Criteria for Representability, Lemma 97.6.3. Assumption (3) implies (2) by More on Morphisms of Spaces, Lemma 76.19.6 and the principle of Algebraic Stacks, Lemma 94.10.9. $\square$

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Your email address will not be published.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 07WK. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 98.3.2 as pdf