## 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}
```

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