The Stacks project

Lemma 59.39.2. 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 34.3.14 or 34.4.14. 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 34.3.13, 34.3.17, 34.4.13, and 34.4.17. 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 59.39.1. 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 35.13.6 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. $\square$

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