Lemma 34.3.12. The category of sheaves on $S_{Zar}$ is equivalent to the category of sheaves on the underlying topological space of $S$.

**Proof.**
We will use repeatedly that for any object $U/S$ of $S_{Zar}$ the morphism $U \to S$ is an isomorphism onto an open subscheme. Let $\mathcal{F}$ be a sheaf on $S$. Then we define a sheaf on $S_{Zar}$ by the rule $\mathcal{F}'(U/S) = \mathcal{F}(\mathop{\mathrm{Im}}(U \to S))$. For the converse, we choose for every open subscheme $U \subset S$ an object $U'/S \in \mathop{\mathrm{Ob}}\nolimits (S_{Zar})$ with $\mathop{\mathrm{Im}}(U' \to S) = U$ (here you have to use Sets, Lemma 3.9.9). Given a sheaf $\mathcal{G}$ on $S_{Zar}$ we define a sheaf on $S$ by setting $\mathcal{G}'(U) = \mathcal{G}(U'/S)$. To see that $\mathcal{G}'$ is a sheaf we use that for any open covering $U = \bigcup _{i \in I} U_ i$ the covering $\{ U_ i \to U\} _{i \in I}$ is combinatorially equivalent to a covering $\{ U_ j' \to U'\} _{j \in J}$ in $S_{Zar}$ by Sets, Lemma 3.11.1, and we use Sites, Lemma 7.8.4. Details omitted.
$\square$

## 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Your email address will not be published.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 (2)

Comment #3068 by Herman Rohrbach on

Comment #3169 by Johan on

There are also: