The Stacks project

Remark 88.8.10. Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal C_\Lambda $, and let $\xi $ be a formal object of $\mathcal{F}$. It follows from the definition of smoothness that versality of $\xi $ is equivalent to the following condition: If

\[ \xymatrix{ & y \ar[d] \\ \xi \ar[r] & x } \]

is a diagram in $\widehat{\mathcal{F}}$ such that $y \to x$ lies over a surjective map $B \to A$ of Artinian rings (we may assume it is a small extension), then there exists a morphism $\xi \to y$ such that

\[ \xymatrix{ & y \ar[d] \\ \xi \ar[r] \ar[ur] & x } \]

commutes. In particular, the condition that $\xi $ be versal does not depend on the choices of pushforwards made in the construction of $\underline{\xi } : \underline{R}|_{\mathcal{C}_\Lambda } \to \mathcal{F}$ in Remark 88.7.12.


Comments (0)

There are also:

  • 2 comment(s) on Section 88.8: Smooth morphisms

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Your email address will not be published.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