The Stacks project

Lemma 21.31.6. Let $X$ be an object of $\textit{LC}_{qc}$. Let $\mathcal{F}$ be a sheaf on $X$. The rule

\[ \textit{LC}_{qc}/X \longrightarrow \textit{Sets},\quad (f : Y \to X) \longmapsto \Gamma (Y, f^{-1}\mathcal{F}) \]

is a sheaf and a fortiori also a sheaf on $\textit{LC}_{Zar}/X$. This sheaf is equal to $\pi _ X^{-1}\mathcal{F}$ on $\textit{LC}_{Zar}/X$ and $\epsilon _ X^{-1}\pi _ X^{-1}\mathcal{F}$ on $\textit{LC}_{qc}/X$.

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

\[ (f : Y \to X) \longmapsto \mathop{\mathrm{colim}}\nolimits _{f(Y) \subset U \subset X} \mathcal{F}(U) \]

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. $\square$

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 09X3. Beware of the difference between the letter 'O' and the digit '0'.