The Stacks project

Proof. Denote $\mathcal{G}$ the presheaf given by the formula in the lemma. Of course the pullback $f^{-1}$ in the formula denotes usual pullback of sheaves on topological spaces. It is immediate from the definitions that $\mathcal{G}$ is a sheaf for the Zar topology.

Let $Y \to X$ be a morphism in $\textit{LC}_{qc}$. Let $\mathcal{V} = \{ g_ i : Y_ i \to Y\} _{i \in I}$ be a qc covering. To prove $\mathcal{G}$ is a sheaf for the qc topology it suffices to show that $\mathcal{G}(Y) \to H^0(\mathcal{V}, \mathcal{G})$ is an isomorphism, see Sites, Section 7.10. We first point out that the map is injective as a qc covering is surjective and we can detect equality of sections at stalks (use Sheaves, Lemmas 6.11.1 and 6.21.4). Thus $\mathcal{G}$ is a separated presheaf on $\textit{LC}_{qc}$ hence it suffices to show that any element $(s_ i) \in H^0(\mathcal{V}, \mathcal{G})$ maps to an element in the image of $\mathcal{G}(Y)$ after replacing $\mathcal{V}$ by a refinement (Sites, Theorem 7.10.10).

Identifying sheaves on $Y_{i, Zar}$ and sheaves on $Y_ i$ we find that $\mathcal{G}|_{Y_{i, Zar}}$ is the pullback of $f^{-1}\mathcal{F}$ under the continuous map $g_ i : Y_ i \to Y$. Thus we can choose an open covering $Y_ i = \bigcup V_{ij}$ such that for each $j$ there is an open $W_{ij} \subset Y$ and a section $t_{ij} \in \mathcal{G}(W_{ij})$ such that $V_{ij}$ maps into $W_{ij}$ and such that $s|_{V_{ij}}$ is the pullback of $t_{ij}$. In other words, after refining the covering $\{ Y_ i \to Y\} $ we may assume there are opens $W_ i \subset Y$ such that $Y_ i \to Y$ factors through $W_ i$ and sections $t_ i$ of $\mathcal{G}$ over $W_ i$ which restrict to the given sections $s_ i$. Moreover, if $y \in Y$ is in the image of both $Y_ i \to Y$ and $Y_ j \to Y$, then the images $t_{i, y}$ and $t_{j, y}$ in the stalk $f^{-1}\mathcal{F}_ y$ agree (because $s_ i$ and $s_ j$ agree over $Y_ i \times _ Y Y_ j$). Thus for $y \in Y$ there is a well defined element $t_ y$ of $f^{-1}\mathcal{F}_ y$ agreeing with $t_{i, y}$ whenever $y$ is in the image of $Y_ i \to Y$. We will show that the element $(t_ y)$ comes from a global section of $f^{-1}\mathcal{F}$ over $Y$ which will finish the proof of the lemma.

It suffices to show that this is true locally on $Y$, see Sheaves, Section 6.17. Let $y_0 \in Y$. Pick $i_1, \ldots , i_ n \in I$ and quasi-compact subsets $E_ j \subset Y_{i_ j}$ such that $\bigcup g_{i_ j}(E_ j)$ is a neighbourhood of $y_0$. Let $V \subset Y$ be an open neighbourhood of $y_0$ contained in $\bigcup g_{i_ j}(E_ j)$ and contained in $W_{i_1} \cap \ldots \cap W_{i_ n}$. Since $t_{i_1, y_0} = \ldots = t_{i_ n, y_0}$, after shrinking $V$ we may assume the sections $t_{i_ j}|_ V$, $j = 1, \ldots , n$ of $f^{-1}\mathcal{F}$ agree. As $V \subset \bigcup g_{i_ j}(E_ j)$ we see that $(t_ y)_{y \in V}$ comes from this section.

We still have to show that $\mathcal{G}$ is equal to $\epsilon _ X^{-1}\pi _ X^{-1}\mathcal{F}$ on $\textit{LC}_{qc}$, resp. $\pi _ X^{-1}\mathcal{F}$ on $\textit{LC}_{Zar}$. In both cases the pullback is defined by taking the presheaf

and then sheafifying. Sheafifying in the Zar topology exactly produces our sheaf $\mathcal{G}$ and the fact that $\mathcal{G}$ is a qc sheaf, shows that it works as well in the qc topology. $\square$