Lemma 7.22.1. Let $\mathcal{C}$ and $\mathcal{D}$ be sites. Let $u : \mathcal{C} \to \mathcal{D}$, and $v : \mathcal{D} \to \mathcal{C}$ be functors. Assume that $u$ is cocontinuous, and that $v$ is a right adjoint to $u$. Let $g : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D})$ be the morphism of topoi associated to $u$, see Lemma 7.21.1. Then $g_*\mathcal{F}$ is equal to the presheaf $v^ p\mathcal{F}$, in other words, $(g_*\mathcal{F})(V) = \mathcal{F}(v(V))$.

**Proof.**
We have $u^ ph_ V = h_{v(V)}$ by Lemma 7.19.3. By Lemma 7.20.4 this implies that $g^{-1}(h_ V^\# ) = (u^ ph_ V^\# )^\# = (u^ ph_ V)^\# = h_{v(V)}^\# $. Hence for any sheaf $\mathcal{F}$ on $\mathcal{C}$ we have

\begin{eqnarray*} (g_*\mathcal{F})(V) & = & \mathop{Mor}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{D})}(h_ V^\# , g_*\mathcal{F}) \\ & = & \mathop{Mor}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C})}(g^{-1}(h_ V^\# ), \mathcal{F}) \\ & = & \mathop{Mor}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C})}(h_{v(V)}^\# , \mathcal{F}) \\ & = & \mathcal{F}(v(V)) \end{eqnarray*}

which proves the lemma. $\square$

## 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Your email address will not be published.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)