Lemma 21.31.9. Consider the comparison morphism $\epsilon : \textit{LC}_{qc} \to \textit{LC}_{Zar}$. Let $\mathcal{P}$ denote the class of proper maps of topological spaces. For $X$ in $\textit{LC}_{Zar}$ denote $\mathcal{A}'_ X \subset \textit{Ab}(\textit{LC}_{Zar}/X)$ the full subcategory consisting of sheaves of the form $\pi _ X^{-1}\mathcal{F}$ with $\mathcal{F}$ in $\textit{Ab}(X)$. Then (1), (2), (3), (4), and (5) of Situation 21.30.1 hold.

**Proof.**
We first show that $\mathcal{A}'_ X \subset \textit{Ab}(\textit{LC}_{Zar}/X)$ is a weak Serre subcategory by checking conditions (1), (2), (3), and (4) of Homology, Lemma 12.9.3. Parts (1), (2), (3) are immediate as $\pi _ X^{-1}$ is exact and fully faithful by Lemma 21.31.7 part (3). If $0 \to \pi _ X^{-1}\mathcal{F} \to \mathcal{G} \to \pi _ X^{-1}\mathcal{F}' \to 0$ is a short exact sequence in $\textit{Ab}(\textit{LC}_{Zar}/X)$ then $0 \to \mathcal{F} \to \pi _{X, *}\mathcal{G} \to \mathcal{F}' \to 0$ is exact by Lemma 21.31.7 part (2). Hence $\mathcal{G} = \pi _ X^{-1}\pi _{X, *}\mathcal{G}$ is in $\mathcal{A}'_ X$ which checks the final condition.

Property (1) holds by Lemma 21.31.1 and the fact that the base change of a proper map is a proper map, see Topology, Theorem 5.17.5.

Property (2) follows from the commutative diagram (5) in Lemma 21.31.7.

Property (3) is Lemma 21.31.6.

Property (4) is Lemma 21.31.7 part (7)(b).

Proof of (5). Suppose given a qc covering $\{ U_ i \to U\} $. For $u \in U$ pick $i_1, \ldots , i_ m \in I$ and quasi-compact subsets $E_ j \subset U_{i_ j}$ such that $\bigcup f_{i_ j}(E_ j)$ is a neighbourhood of $u$. Observe that $Y = \coprod _{j = 1, \ldots , m} E_ j \to U$ is proper as a continuous map from a quasi-compact space to a Hausdorff one (Topology, Lemma 5.17.7). Choose an open neighbourhood $u \in V$ contained in $\bigcup f_{i_ j}(E_ j)$. Then $Y \times _ U V \to V$ is a surjective proper morphism and hence a $qc$ covering by Lemma 21.31.4. Since we can do this for every $u \in U$ we see that (5) holds. $\square$

## 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Your email address will not be published.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.

## Comments (0)