Remark 114.7.1 (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

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

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.

Comment #3053 by anonymous on

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

Comment #5427 by CQ on

For etale sheaves on schemes, there is such a map constructed using the $j^*,j_*$ adjoint (see p.85 Etale cohomology and the Weil conjecture). Why the proof doesn't work here. (Or the proof in loc. cit. is wrong?)

Comment #5428 by CQ on

I meant $j_!,j^*$ adjoint. And the book is the one of Freitag and Kiehl.

Comment #5655 by on

OK, the etale cohomology case is discussed in Lemma 59.70.6 but the discussion there only works if the \'etale morphism $j : U \to X$ is separated! Having this map $j_! \to j_*$ is quite rare indeed. Everybody: try to avoid using this map!

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