The Stacks project

Lemma 6.29.3. In the situation described above, let $i \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{I})$ and let $\mathcal{G}$ be a sheaf on $X_ i$. For $U_ i \subset X_ i$ quasi-compact open we have

\[ p_ i^{-1}\mathcal{G}(p_ i^{-1}(U_ i)) = \mathop{\mathrm{colim}}\nolimits _{a : j \to i} f_ a^{-1}\mathcal{G}(f_ a^{-1}(U_ i)) \]

Proof. Let us prove the canonical map $\mathop{\mathrm{colim}}\nolimits _{a : j \to i} f_ a^{-1}\mathcal{G}(f_ a^{-1}(U_ i)) \to p_ i^{-1}\mathcal{G}(p_ i^{-1}(U_ i))$ is injective. Let $s, s'$ be sections of $f_ a^{-1}\mathcal{G}$ over $f_ a^{-1}(U_ i)$ for some $a : j \to i$. For $b : k \to j$ let $Z_ k \subset f_{a \circ b}^{-1}(U_ i)$ be the closed subset of points $x$ such that the image of $s$ and $s'$ in the stalk $(f_{a \circ b}^{-1}\mathcal{G})_ x$ are different. If $Z_ k$ is nonempty for all $b : k \to j$, then by Topology, Lemma 5.24.2 we see that $\mathop{\mathrm{lim}}\nolimits _{b : k \to j} Z_ k$ is nonempty too. Then for $x \in \mathop{\mathrm{lim}}\nolimits _{b : k \to j} Z_ k \subset X$ (observe that $\mathcal{I}/j \to \mathcal{I}$ is initial) we see that the image of $s$ and $s'$ in the stalk of $p_ i^{-1}\mathcal{G}$ at $x$ are different too since $(p_ i^{-1}\mathcal{G})_ x = (f_{b \circ a}^{-1}\mathcal{G})_{p_ k(x)}$ for all $b : k \to j$ as above. Thus if the images of $s$ and $s'$ in $p_ i^{-1}\mathcal{G}(p_ i^{-1}(U_ i))$ are the same, then $Z_ k$ is empty for some $b : k \to j$. This proves injectivity.

Surjectivity. Let $s$ be a section of $p_ i^{-1}\mathcal{G}$ over $p_ i^{-1}(U_ i)$. By Topology, Lemma 5.24.5 the set $p_ i^{-1}(U_ i)$ is a quasi-compact open of the spectral space $X$. By construction of the pullback sheaf, we can find an open covering $p_ i^{-1}(U_ i) = \bigcup _{l \in L} W_ l$, opens $V_{l, i} \subset X_ i$, sections $s_{l, i} \in \mathcal{G}(V_{l, i})$ such that $p_ i(W_ l) \subset V_{l, i}$ and $p_ i^{-1}s_{l, i}|_{W_ l} = s|_{W_ l}$. Because $X$ and $X_ i$ are spectral and $p_ i^{-1}(U_ i)$ is quasi-compact open, we may assume $L$ is finite and $W_ l$ and $V_{l, i}$ quasi-compact open for all $l$. Then we can apply Topology, Lemma 5.24.6 to find $a : j \to i$ and open covering $f_ a^{-1}(U_ i) = \bigcup _{l \in L} W_{l, j}$ by quasi-compact opens whose pullback to $X$ is the covering $p_ i^{-1}(U_ i) = \bigcup _{l \in L} W_ l$ and such that moreover $W_{l, j} \subset f_ a^{-1}(V_{l, i})$. Write $s_{l, j}$ the restriction of the pullback of $s_{l, i}$ by $f_ a$ to $W_{l, j}$. Then we see that $s_{l, j}$ and $s_{l', j}$ restrict to elements of $(f_ a^{-1}\mathcal{G})(W_{l, j} \cap W_{l', j})$ which pullback to the same element $(p_ i^{-1}\mathcal{G})(W_ l \cap W_{l'})$, namely, the restriction of $s$. Hence by injectivity, we can find $b : k \to j$ such that the sections $f_ b^{-1}s_{l, j}$ glue to a section over $f_{a \circ b}^{-1}(U_ i)$ as desired. $\square$

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