The Stacks project

Remark 91.10.6. Let $i : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}')$ be a trivial first order thickening of ringed topoi and let $\pi : (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}') \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$ be a trivialization. Then given any triple $(\mathcal{F}, \mathcal{K}, c)$ consisting of a pair of $\mathcal{O}$-modules and a map $c : \mathcal{I} \otimes _\mathcal {O} \mathcal{F} \to \mathcal{K}$ we may set

\[ \mathcal{F}'_{c, triv} = \mathcal{F} \oplus \mathcal{K} \]

and use the splitting (91.10.5.1) associated to $\pi $ and the map $c$ to define the $\mathcal{O}'$-module structure and obtain an extension (91.10.0.1). We will call $\mathcal{F}'_{c, triv}$ the trivial extension of $\mathcal{F}$ by $\mathcal{K}$ corresponding to $c$ and the trivialization $\pi $. Given any extension $\mathcal{F}'$ as in (91.10.0.1) we can use $\pi ^\sharp : \mathcal{O} \to \mathcal{O}'$ to think of $\mathcal{F}'$ as an $\mathcal{O}$-module extension, hence a class $\xi _{\mathcal{F}'}$ in $\mathop{\mathrm{Ext}}\nolimits ^1_\mathcal {O}(\mathcal{F}, \mathcal{K})$. Lemma 91.10.3 assures that $\mathcal{F}' \mapsto \xi _{\mathcal{F}'}$ induces a bijection

\[ \left\{ \begin{matrix} \text{isomorphism classes of extensions} \\ \mathcal{F}'\text{ as in (08MB) with } c = c_{\mathcal{F}'} \end{matrix} \right\} \longrightarrow \mathop{\mathrm{Ext}}\nolimits ^1_\mathcal {O}(\mathcal{F}, \mathcal{K}) \]

Moreover, the trivial extension $\mathcal{F}'_{c, triv}$ maps to the zero class.


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Your email address will not be published.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