The Stacks project

Proof. Assume all the transition maps are injective. In this case the presheaf $\mathcal{F}' : V \mapsto \mathop{\mathrm{colim}}\nolimits _ i \mathcal{F}_ i(V)$ is separated (see Definition 6.11.2). By the discussion above we have $(\mathcal{F}')^\# = \mathop{\mathrm{colim}}\nolimits _ i \mathcal{F}_ i$. By Lemma 6.17.5 we see that $\mathcal{F}' \to (\mathcal{F}')^\#$ is injective. This proves (1).

Assume $U$ is quasi-compact. Suppose that $s \in \mathcal{F}_ i(U)$ and $s' \in \mathcal{F}_{i'}(U)$ give rise to elements on the left hand side which have the same image under $\Psi$. Since $U$ is quasi-compact this means there exists a finite open covering $U = \bigcup _{j = 1, \ldots , m} U_ j$ and for each $j$ an index $i_ j \in I$, $i_ j \geq i$, $i_ j \geq i'$ such that $\varphi _{ii_ j}(s)$ and $\varphi _{i'i_ j}(s')$ have the same restriction to $U_ j$. Let $i''\in I$ be $\geq$ than all of the $i_ j$. We conclude that $\varphi _{ii''}(s)$ and $\varphi _{i'i''}(s')$ agree on the opens $U_ j$ for all $j$ and hence that $\varphi _{ii''}(s) = \varphi _{i'i''}(s')$. This proves (2).

Assume $U$ is quasi-compact and all transition maps injective. Let $s$ be an element of the target of $\Psi$. Since $U$ is quasi-compact there exists a finite open covering $U = \bigcup _{j = 1, \ldots , m} U_ j$, for each $j$ an index $i_ j \in I$ and $s_ j \in \mathcal{F}_{i_ j}(U_ j)$ such that $s|_{U_ j}$ comes from $s_ j$ for all $j$. Pick $i \in I$ which is $\geq$ than all of the $i_ j$. By (1) the sections $\varphi _{i_ ji}(s_ j)$ agree over the overlaps $U_ j \cap U_{j'}$. Hence they glue to a section $s' \in \mathcal{F}_ i(U)$ which maps to $s$ under $\Psi$. This proves (3).

Assume the hypothesis of (4). In particular we see that $U$ is quasi-compact and hence by (2) we have injectivity of $\Psi$. Let $s$ be an element of the target of $\Psi$. By assumption there exists a finite open covering $U = \bigcup _{j = 1, \ldots , m} U_ j$, with $U_ j \cap U_{j'}$ quasi-compact for all $j, j' \in J$ and for each $j$ an index $i_ j \in I$ and $s_ j \in \mathcal{F}_{i_ j}(U_ j)$ such that $s|_{U_ j}$ is the image of $s_ j$ for all $j$. Since $U_ j \cap U_{j'}$ is quasi-compact we can apply (2) and we see that there exists an $i_{jj'} \in I$, $i_{jj'} \geq i_ j$, $i_{jj'} \geq i_{j'}$ such that $\varphi _{i_ ji_{jj'}}(s_ j)$ and $\varphi _{i_{j'}i_{jj'}}(s_{j'})$ agree over $U_ j \cap U_{j'}$. Choose an index $i \in I$ which is bigger or equal than all the $i_{jj'}$. Then we see that the sections $\varphi _{i_ ji}(s_ j)$ of $\mathcal{F}_ i$ glue to a section of $\mathcal{F}_ i$ over $U$. This section is mapped to the element $s$ as desired. $\square$

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. $\square$

Proof. Recall that $p_ i^{-1}(U_ i)$ is a quasi-compact open of the spectral space $X$, see Topology, Lemma 5.24.5. Hence Lemma 6.29.1 applies and we have

A formal argument shows that

Thus it suffices to show that

This is Lemma 6.29.3 applied to $\mathcal{F}_ j$ and the quasi-compact open $f_ a^{-1}(U_ i)$. $\square$