The Stacks project

Remark 18.41.2. Warning! Let $u : \mathcal{C} \to \mathcal{D}$, $g$, $\mathcal{O}_\mathcal {D}$, and $\mathcal{O}_\mathcal {C}$ be as in Lemma 18.41.1. In general it is not the case that the diagram

\[ \xymatrix{ \textit{Mod}(\mathcal{O}_\mathcal {C}) \ar[r]_{g_!} \ar[d]_{forget} & \textit{Mod}(\mathcal{O}_\mathcal {D}) \ar[d]^{forget} \\ \textit{Ab}(\mathcal{C}) \ar[r]^{g^{Ab}_!} & \textit{Ab}(\mathcal{D}) } \]

commutes (here $g^{Ab}_!$ is the one from Lemma 18.16.2). There is a transformation of functors

\[ g_!^{Ab} \circ forget \longrightarrow forget \circ g_! \]

From the proof of Lemma 18.41.1 we see that this is an isomorphism if and only if $g^{Ab}_!j_{U!}\mathcal{O}_ U \to g_!j_{U!}\mathcal{O}_ U$ is an isomorphism for all objects $U$ of $\mathcal{C}$. Since we have $g_!j_{U!}\mathcal{O}_ U = j_{u(U)!}\mathcal{O}_{u(U)}$ this holds if and only if

\[ g^{Ab}_!j_{U!}\mathcal{O}_ U \longrightarrow j_{u(U)!}\mathcal{O}_{u(U)} \]

is an isomorphism for all objects $U$ of $\mathcal{C}$. Note that for such a $U$ we obtain a commutative diagram

\[ \xymatrix{ \mathcal{C}/U \ar[r]_-{u'} \ar[d]_{j_ U} & \mathcal{D}/u(U) \ar[d]^{j_{u(U)}} \\ \mathcal{C} \ar[r]^ u & \mathcal{D} } \]

of cocontinuous functors of sites, see Sites, Lemma 7.28.4 and therefore $g^{Ab}_!j_{U!} = j_{u(U)!}(g')^{Ab}_!$ where $g' : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D}/u(U))$ is the morphism of topoi induced by the cocontinuous functor $u'$. Hence we see that $g_! = g^{Ab}_!$ if the canonical map
\begin{equation} \label{sites-modules-equation-compare-on-localizations} (g')^{Ab}_!\mathcal{O}_ U \longrightarrow \mathcal{O}_{u(U)} \end{equation}

is an isomorphism for all objects $U$ of $\mathcal{C}$.

Comments (0)

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Your email address will not be published.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 0798. Beware of the difference between the letter 'O' and the digit '0'.