
Remark 107.6.1 (No map from lower shriek to pushforward). Let $U$ be an object of $\mathcal{C}$. For any abelian sheaf $\mathcal{G}$ on $\mathcal{C}/U$ one may wonder whether there is a canonical map

$c : j_{U!}\mathcal{G} \longrightarrow j_{U*}\mathcal{G}$

To construct such a thing is the same as constructing a map $j_ U^{-1}j_{U!}\mathcal{G} \to \mathcal{G}$. Note that restriction commutes with sheafification. Thus we can use the presheaf of Modules on Sites, Lemma 18.19.2. Hence it suffices to define for $V/U$ a map

$\bigoplus \nolimits _{\varphi \in \mathop{Mor}\nolimits _\mathcal {C}(V, U)} \mathcal{G}(V \xrightarrow {\varphi } U) \longrightarrow \mathcal{G}(V/U)$

compatible with restrictions. It looks like we can take the which is zero on all summands except for the one where $\varphi$ is the structure morphism $\varphi _0 : V \to U$ where we take $1$. However, this isn't compatible with restriction mappings: namely, if $\alpha : V' \to V$ is a morphism of $\mathcal{C}$, then denote $V'/U$ the object of $\mathcal{C}/U$ with structure morphism $\varphi '_0 = \varphi _0 \circ \alpha$. We need to check that the diagram

$\xymatrix{ \bigoplus \nolimits _{\varphi \in \mathop{Mor}\nolimits _\mathcal {C}(V, U)} \mathcal{G}(V \xrightarrow {\varphi } U) \ar[d] \ar[r] & \mathcal{G}(V/U) \ar[d] \\ \bigoplus \nolimits _{\varphi ' \in \mathop{Mor}\nolimits _\mathcal {C}(V', U)} \mathcal{G}(V' \xrightarrow {\varphi '} U) \ar[r] & \mathcal{G}(V'/U) }$

commutes. The problem here is that there may be a morphism $\varphi : V \to U$ different from $\varphi _0$ such that $\varphi \circ \alpha = \varphi '_0$. Thus the left vertical arrow will send the summand corresponding to $\varphi$ into the summand on which the lower horizontal arrow is equal to $1$ and almost surely the diagram doesn't commute.

Comment #3053 by anonymous on

It seems to me that the second displayed map is not compatible with restrictions.

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).