The Stacks project

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.

Comments (5)

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 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 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 is separated! Having this map is quite rare indeed. Everybody: try to avoid using this map!

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