Example 7.22.5. Let $\mathcal{C}$ and $\mathcal{D}$ be sites. Let $u : \mathcal{C} \to \mathcal{D}$ and $v : \mathcal{D} \to \mathcal{C}$ be functors of the underlying categories such that $v$ is right adjoint to $u$. Lemma 7.22.4 shows that if $v$ is continuous, then $u$ is cocontinous. Conversely, if $u$ is cocontinuous, then we can't conclude that $v$ is continuous. We will give an example of this phenomenon using the big étale and smooth sites of a scheme, but presumably there is an elementary example as well. Namely, consider a scheme $S$ and the sites $(\mathit{Sch}/S)_{\acute{e}tale}$ and $(\mathit{Sch}/S)_{smooth}$. We may assume these sites have the same underlying category, see Topologies, Remark 34.11.1. Let $u = v = \text{id}$. Then $u$ as a functor from $(\mathit{Sch}/S)_{\acute{e}tale}$ to $(\mathit{Sch}/S)_{smooth}$ is cocontinuous as every smooth covering of a scheme can be refined by an étale covering, see More on Morphisms, Lemma 37.38.7. Conversely, the functor $v$ from $(\mathit{Sch}/S)_{smooth}$ to $(\mathit{Sch}/S)_{\acute{e}tale}$ is not continuous as a smooth covering is not an étale covering in general.

## 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Your email address will not be published.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)