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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 8755–8771 (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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