The Stacks project

Lemma 58.39.2. Let $S$ be a scheme. Let $\tau \in \{ Zariski, {\acute{e}tale}\} $. Consider the morphism

\[ \pi _ S : (\mathit{Sch}/S)_\tau \longrightarrow S_\tau \]

of Topologies, Lemma 34.3.13 or 34.4.13. Let $\mathcal{F}$ be a sheaf on $S_\tau $. Then $\pi _ S^{-1}\mathcal{F}$ is given by the rule

\[ (\pi _ S^{-1}\mathcal{F})(T) = \Gamma (T_\tau , f_{small}^{-1}\mathcal{F}) \]

where $f : T \to S$. Moreover, $\pi _ S^{-1}\mathcal{F}$ satisfies the sheaf condition with respect to fpqc coverings.

Proof. Observe that we have a morphism $i_ f : \mathop{\mathit{Sh}}\nolimits (T_\tau ) \to \mathop{\mathit{Sh}}\nolimits (\mathit{Sch}/S)_\tau )$ such that $\pi _ S \circ i_ f = f_{small}$ as morphisms $T_\tau \to S_\tau $, see Topologies, Lemmas 34.3.12, 34.3.16, 34.4.12, and 34.4.16. Since pullback is transitive we see that $i_ f^{-1} \pi _ S^{-1}\mathcal{F} = f_{small}^{-1}\mathcal{F}$ as desired.

Let $\{ g_ i : T_ i \to T\} _{i \in I}$ be an fpqc covering. The final statement means the following: Given a sheaf $\mathcal{G}$ on $T_\tau $ and given sections $s_ i \in \Gamma (T_ i, g_{i, small}^{-1}\mathcal{G})$ whose pullbacks to $T_ i \times _ T T_ j$ agree, there is a unique section $s$ of $\mathcal{G}$ over $T$ whose pullback to $T_ i$ agrees with $s_ i$.

Let $V \to T$ be an object of $T_\tau $ and let $t \in \mathcal{G}(V)$. For every $i$ there is a largest open $W_ i \subset T_ i \times _ T V$ such that the pullbacks of $s_ i$ and $t$ agree as sections of the pullback of $\mathcal{G}$ to $W_ i \subset T_ i \times _ T V$, see Lemma 58.39.1. Because $s_ i$ and $s_ j$ agree over $T_ i \times _ T T_ j$ we find that $W_ i$ and $W_ j$ pullback to the same open over $T_ i \times _ T T_ j \times _ T V$. By Descent, Lemma 35.10.6 we find an open $W \subset V$ whose inverse image to $T_ i \times _ T V$ recovers $W_ i$.

By construction of $g_{i, small}^{-1}\mathcal{G}$ there exists a $\tau $-covering $\{ T_{ij} \to T_ i\} _{j \in J_ i}$, for each $j$ an open immersion or étale morphism $V_{ij} \to T$, a section $t_{ij} \in \mathcal{G}(V_{ij})$, and commutative diagrams

\[ \xymatrix{ T_{ij} \ar[r] \ar[d] & V_{ij} \ar[d] \\ T_ i \ar[r] & T } \]

such that $s_ i|_{T_{ij}}$ is the pullback of $t_{ij}$. In other words, after replacing the covering $\{ T_ i \to T\} $ by $\{ T_{ij} \to T\} $ we may assume there are factorizations $T_ i \to V_ i \to T$ with $V_ i \in \mathop{\mathrm{Ob}}\nolimits (T_\tau )$ and sections $t_ i \in \mathcal{G}(V_ i)$ pulling back to $s_ i$ over $T_ i$. By the result of the previous paragraph we find opens $W_ i \subset V_ i$ such that $t_ i|_{W_ i}$ “agrees with” every $s_ j$ over $T_ j \times _ T W_ i$. Note that $T_ i \to V_ i$ factors through $W_ i$. Hence $\{ W_ i \to T\} $ is a $\tau $-covering and the lemma is proven. $\square$


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 09XN. Beware of the difference between the letter 'O' and the digit '0'.