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Tag 0931

Chapter 106: Obsolete > Section 106.6: Sites and sheaves

Remark 106.6.1 (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.

    The code snippet corresponding to this tag is a part of the file obsolete.tex and is located in lines 719–765 (see updates for more information).

    \begin{remark}[No map from lower shriek to pushforward]
    \label{remark-from-shriek-to-star}
    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 \ref{sites-modules-lemma-extension-by-zero}.
    Hence it suffices to define for $V/U$ a map
    $$
    \bigoplus\nolimits_{\varphi \in \Mor_\mathcal{C}(V, U)}
    \mathcal{G}(V \xrightarrow{\varphi} U)
    \longrightarrow
    \mathcal{G}(V/U)
    $$
    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
    $$
    \xymatrix{
    \bigoplus\nolimits_{\varphi \in \Mor_\mathcal{C}(V, U)}
    \mathcal{G}(V \xrightarrow{\varphi} U)
    \ar[d] \ar[r] &
    \mathcal{G}(V/U) \ar[d] \\
    \bigoplus\nolimits_{\varphi' \in \Mor_\mathcal{C}(V', U)}
    \mathcal{G}(V' \xrightarrow{\varphi'} U)
    \ar[r] &
    \mathcal{G}(V'/U)
    }
    $$
    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.
    \end{remark}

    Comments (2)

    Comment #3053 by anonymous on January 8, 2018 a 2:22 pm UTC

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

    Comment #3158 by Johan (site) on February 2, 2018 a 2:58 am UTC

    Indeed! Just terrible. Thanks very much. Fix is here.

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