\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

The Stacks project

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.

Comments (2)

Comment #3053 by anonymous on

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

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