The Stacks project

Lemma 91.11.8. Let $(f, f')$ be a morphism of first order thickenings as in Situation 91.9.1. Let $\mathcal{F}$ be an $\mathcal{O}$-module. Assume $(f, f')$ is a strict morphism of thickenings and $\mathcal{F}$ flat over $\mathcal{O}_\mathcal {B}$. There exists an $\mathcal{O}'$-module $\mathcal{F}'$ flat over $\mathcal{O}_{\mathcal{B}'}$ with $i^*\mathcal{F}' \cong \mathcal{F}$, if and only if

  1. the canonical map $f^*\mathcal{J} \otimes _\mathcal {O} \mathcal{F} \to \mathcal{I} \otimes _\mathcal {O} \mathcal{F}$ is an isomorphism, and

  2. the class $o(\mathcal{F}, \mathcal{I} \otimes _\mathcal {O} \mathcal{F}, 1) \in \mathop{\mathrm{Ext}}\nolimits ^2_\mathcal {O}( \mathcal{F}, \mathcal{I} \otimes _\mathcal {O} \mathcal{F})$ of Lemma 91.10.4 is zero.

Proof. This follows immediately from the characterization of $\mathcal{O}'$-modules flat over $\mathcal{O}_{\mathcal{B}'}$ of Lemma 91.11.2 and Lemma 91.10.4. $\square$


Comments (0)

There are also:

  • 2 comment(s) on Section 91.11: Infinitesimal deformations of modules on ringed topoi

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Your email address will not be published.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 08MV. Beware of the difference between the letter 'O' and the digit '0'.