The Stacks project

115.7 Sites and sheaves

Remark 115.7.1 (No map from lower shriek to pushforward). Let $U$ be an object of a site $\mathcal{C}$. 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

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

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

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

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.

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