Lemma 18.20.1. Let $(f, f^\sharp ) : (\mathcal{C}, \mathcal{O}) \longrightarrow (\mathcal{D}, \mathcal{O}')$ be a morphism of ringed sites where $f$ is given by the continuous functor $u : \mathcal{D} \to \mathcal{C}$. Let $V$ be an object of $\mathcal{D}$ and set $U = u(V)$. Then there is a canonical map of sheaves of rings $(f')^\sharp$ such that the diagram of Sites, Lemma 7.28.1 is turned into a commutative diagram of ringed topoi

$\xymatrix{ (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U), \mathcal{O}_ U) \ar[rr]_{(j_ U, j_ U^\sharp )} \ar[d]_{(f', (f')^\sharp )} & & (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \ar[d]^{(f, f^\sharp )} \\ (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}/V), \mathcal{O}'_ V) \ar[rr]^{(j_ V, j_ V^\sharp )} & & (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}'). }$

Moreover, in this situation we have $f'_*j_ U^{-1} = j_ V^{-1}f_*$ and $f'_*j_ U^* = j_ V^*f_*$.

Proof. Just take $(f')^\sharp$ to be

$(f')^{-1}\mathcal{O}'_ V = (f')^{-1}j_ V^{-1}\mathcal{O}' = j_ U^{-1}f^{-1}\mathcal{O}' \xrightarrow {j_ U^{-1}f^\sharp } j_ U^{-1}\mathcal{O} = \mathcal{O}_ U$

and everything else follows from Sites, Lemma 7.28.1. (Note that $j^{-1} = j^*$ on sheaves of modules if $j$ is a localization morphism, hence the first equality of functors implies the second.) $\square$

## 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Your email address will not be published.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.

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 04J0. Beware of the difference between the letter 'O' and the digit '0'.