Lemma 84.12.5. In Situation 84.3.3 let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}_{total}$. The category of cartesian $\mathcal{O}$-modules is equivalent to the category of pairs $(\mathcal{F}, \alpha )$ where $\mathcal{F}$ is a $\mathcal{O}_0$-module and

$\alpha : (f_{\delta _1^1})^*\mathcal{F} \longrightarrow (f_{\delta _0^1})^*\mathcal{F}$

is an isomorphism of $\mathcal{O}_1$-modules such that $(f_{\delta ^2_1})^*\alpha = (f_{\delta ^2_0})^*\alpha \circ (f_{\delta ^2_2})^*\alpha$ as $\mathcal{O}_2$-module maps.

Proof. The proof is identical to the proof of Lemma 84.12.4 with pullback of sheaves of abelian groups replaced by pullback of modules. $\square$

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.

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 07TH. Beware of the difference between the letter 'O' and the digit '0'.