## Tag `04D7`

Chapter 7: Sites and Sheaves > Section 7.40: Exactness properties of pushforward

Lemma 7.40.2. Let $f : \mathcal{D} \to \mathcal{C}$ be a morphism of sites associated to the continuous functor $u : \mathcal{C} \to \mathcal{D}$. Assume that for any object $U$ of $\mathcal{C}$ and any covering $\{V_j \to u(U)\}$ in $\mathcal{D}$ there exists a covering $\{U_i \to U\}$ in $\mathcal{C}$ such that the map of sheaves $$ \coprod h_{u(U_i)}^\# \to h_{u(U)}^\# $$ factors through the map of sheaves $$ \coprod h_{V_j}^\# \to h_{u(U)}^\#. $$ Then $f_*$ transforms surjective maps of sheaves into surjective maps of sheaves.

Proof.Let $a : \mathcal{F} \to \mathcal{G}$ be a surjective map of sheaves on $\mathcal{D}$. Let $U$ be an object of $\mathcal{C}$ and let $s \in f_*\mathcal{G}(U) = \mathcal{G}(u(U))$. By assumption there exists a covering $\{V_j \to u(U)\}$ and sections $s_j \in \mathcal{F}(V_j)$ with $a(s_j) = s|_{V_j}$. Now we may think of the sections $s$, $s_j$ and $a$ as giving a commutative diagram of maps of sheaves $$ \xymatrix{ \coprod h_{V_j}^\# \ar[r]_-{\coprod s_j} \ar[d] & \mathcal{F} \ar[d]^a \\ h_{u(U)}^\# \ar[r]^s & \mathcal{G} } $$ By assumption there exists a covering $\{U_i \to U\}$ such that we can enlarge the commutative diagram above as follows $$ \xymatrix{ & \coprod h_{V_j}^\# \ar[r]_-{\coprod s_j} \ar[d] & \mathcal{F} \ar[d]^a \\ \coprod h_{u(U_i)}^\# \ar[r] \ar[ur] & h_{u(U)}^\# \ar[r]^s & \mathcal{G} } $$ Because $\mathcal{F}$ is a sheaf the map from the left lower corner to the right upper corner corresponds to a family of sections $s_i \in \mathcal{F}(u(U_i))$, i.e., sections $s_i \in f_*\mathcal{F}(U_i)$. The commutativity of the diagram implies that $a(s_i)$ is equal to the restriction of $s$ to $U_i$. In other words we have shown that $f_*a$ is a surjective map of sheaves. $\square$

The code snippet corresponding to this tag is a part of the file `sites.tex` and is located in lines 8813–8829 (see updates for more information).

```
\begin{lemma}
\label{lemma-weaker}
Let $f : \mathcal{D} \to \mathcal{C}$ be a morphism of sites associated to
the continuous functor $u : \mathcal{C} \to \mathcal{D}$.
Assume that for any object $U$ of $\mathcal{C}$ and any covering
$\{V_j \to u(U)\}$ in $\mathcal{D}$ there exists a covering $\{U_i \to U\}$
in $\mathcal{C}$ such that the map of sheaves
$$
\coprod h_{u(U_i)}^\# \to h_{u(U)}^\#
$$
factors through the map of sheaves
$$
\coprod h_{V_j}^\# \to h_{u(U)}^\#.
$$
Then $f_*$ transforms surjective maps of sheaves into surjective maps of
sheaves.
\end{lemma}
\begin{proof}
Let $a : \mathcal{F} \to \mathcal{G}$ be a surjective map of sheaves on
$\mathcal{D}$. Let $U$ be an object of $\mathcal{C}$ and let
$s \in f_*\mathcal{G}(U) = \mathcal{G}(u(U))$. By assumption there exists
a covering $\{V_j \to u(U)\}$ and sections $s_j \in \mathcal{F}(V_j)$
with $a(s_j) = s|_{V_j}$. Now we may think of the sections $s$,
$s_j$ and $a$ as giving a commutative diagram of maps of sheaves
$$
\xymatrix{
\coprod h_{V_j}^\# \ar[r]_-{\coprod s_j} \ar[d] & \mathcal{F} \ar[d]^a \\
h_{u(U)}^\# \ar[r]^s & \mathcal{G}
}
$$
By assumption there exists a covering $\{U_i \to U\}$ such that we can
enlarge the commutative diagram above as follows
$$
\xymatrix{
& \coprod h_{V_j}^\# \ar[r]_-{\coprod s_j} \ar[d] & \mathcal{F} \ar[d]^a \\
\coprod h_{u(U_i)}^\# \ar[r] \ar[ur] &
h_{u(U)}^\# \ar[r]^s & \mathcal{G}
}
$$
Because $\mathcal{F}$ is a sheaf the map from the left lower corner to
the right upper corner corresponds to a family of sections
$s_i \in \mathcal{F}(u(U_i))$, i.e., sections $s_i \in f_*\mathcal{F}(U_i)$.
The commutativity of the diagram implies that $a(s_i)$ is equal to the
restriction of $s$ to $U_i$. In other words we have shown that $f_*a$ is a
surjective map of sheaves.
\end{proof}
```

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