The Stacks project

Lemma 99.7.6. In Situation 99.7.1 let $\{ X_ i \to X\} _{i \in I}$ be an fpqc covering and for each $i, j \in I$ let $\{ X_{ijk} \to X_ i \times _ X X_ j\} $ be an fpqc covering. Denote $\mathcal{F}_ i$, resp. $\mathcal{F}_{ijk}$ the pullback of $\mathcal{F}$ to $X_ i$, resp. $X_{ijk}$. For every scheme $T$ over $B$ the diagram

\[ \xymatrix{ Q_{\mathcal{F}/X/B}(T) \ar[r] & \prod \nolimits _ i Q_{\mathcal{F}_ i/X_ i/B}(T) \ar@<1ex>[r]^-{\text{pr}_0^*} \ar@<-1ex>[r]_-{\text{pr}_1^*} & \prod \nolimits _{i, j, k} Q_{\mathcal{F}_{ijk}/X_{ijk}/B}(T) } \]

presents the first arrow as the equalizer of the other two. The same is true for the functor $\text{Q}^{fp}_{\mathcal{F}/X/B}$.

Proof. Let $\mathcal{F}_{i, T} \to \mathcal{Q}_ i$ be an element in the equalizer of $\text{pr}_0^*$ and $\text{pr}_1^*$. By Remark 99.7.3 we obtain a surjection $\mathcal{F}_ T \to \mathcal{Q}$ of quasi-coherent $\mathcal{O}_{X_ T}$-modules whose restriction to $X_{i, T}$ recovers $\mathcal{F}_ i \to \mathcal{Q}_ i$. By Morphisms of Spaces, Lemma 67.31.5 we see that $\mathcal{Q}$ is flat over $T$ as desired. In the case of the functor $\text{Q}^{fp}_{\mathcal{F}/X/B}$, i.e., if $\mathcal{Q}_ i$ is of finite presentation, then $\mathcal{Q}$ is of finite presentation too by Descent on Spaces, Lemma 74.6.2. $\square$


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Your email address will not be published.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