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Tag 0738

Chapter 7: Sites and Sheaves > Section 7.17: Quasi-compact objects and colimits

Lemma 7.17.5. Let $\mathcal{C}$ be a site. Let $\mathcal{I} \to \mathop{\textit{Sh}}\nolimits(\mathcal{C})$, $i \mapsto \mathcal{F}_i$ be a filtered diagram of sheaves of sets. Let $U \in \mathop{\rm Ob}\nolimits(\mathcal{C})$. Consider the canonical map $$ \Psi : \mathop{\rm colim}\nolimits_i \mathcal{F}_i(U) \longrightarrow \left(\mathop{\rm colim}\nolimits_i \mathcal{F}_i\right)(U) $$ With the terminology introduced above:

  1. If all the transition maps are injective then $\Psi$ is injective for any $U$.
  2. If $U$ is quasi-compact, then $\Psi$ is injective.
  3. If $U$ is quasi-compact and all the transition maps are injective then $\Psi$ is an isomorphism.
  4. If $U$ has a cofinal system of coverings $\{U_j \to U\}_{j \in J}$ with $J$ finite and $U_j \times_U U_{j'}$ quasi-compact for all $j, j' \in J$, then $\Psi$ is bijective.

Proof. Assume all the transition maps are injective. In this case the presheaf $\mathcal{F}' : V \mapsto \mathop{\rm colim}\nolimits_i \mathcal{F}_i(V)$ is separated (see Definition 7.10.9). By Lemma 7.10.13 we have $(\mathcal{F}')^\# = \mathop{\rm colim}\nolimits_i \mathcal{F}_i$. By Theorem 7.10.10 we see that $\mathcal{F}' \to (\mathcal{F}')^\#$ is injective. This proves (1).

Assume $U$ is quasi-compact. Suppose that $s \in \mathcal{F}_i(U)$ and $s' \in \mathcal{F}_{i'}(U)$ give rise to elements on the left hand side which have the same image under $\Psi$. This means we can choose a covering $\{U_a \to U\}_{a \in A}$ and for each $a \in A$ an index $i_a \in I$, $i_a \geq i$, $i_a \geq i'$ such that $\varphi_{ii_a}(s) = \varphi_{i'i_a}(s')$. Because $U$ is quasi-compact we can choose a covering $\{V_b \to U\}_{b \in B}$, a map $\alpha : B \to A$ with finite image, and morphisms $V_b \to U_{\alpha(b)}$ over $U$. Pick $i''\in I$ to be $\geq$ than all of the $i_{\alpha(b)}$ which is possible because the image of $\alpha$ is finite. We conclude that $\varphi_{ii''}(s)$ and $\varphi_{i'i''}(s)$ agree on $V_b$ for all $b \in B$ and hence that $\varphi_{ii''}(s) = \varphi_{i'i''}(s)$. This proves (2).

Assume $U$ is quasi-compact and all transition maps injective. Let $s$ be an element of the target of $\Psi$. There exists a covering $\{U_a \to U\}_{a \in A}$ and for each $a \in A$ an index $i_a \in I$ and a section $s_a \in \mathcal{F}_{i_a}(U_a)$ such that $s|_{U_a}$ comes from $s_a$ for all $a \in A$. Because $U$ is quasi-compact we can choose a covering $\{V_b \to U\}_{b \in B}$, a map $\alpha : B \to A$ with finite image, and morphisms $V_b \to U_{\alpha(b)}$ over $U$. Pick $i \in I$ to be $\geq$ than all of the $i_{\alpha(b)}$ which is possible because the image of $\alpha$ is finite. By (1) the sections $s_b = \varphi_{i_{\alpha(b)} i}(s_{\alpha(b)})|_{V_b}$ agree over $V_b \times_U V_{b'}$. Hence they glue to a section $s' \in \mathcal{F}_i(U)$ which maps to $s$ under $\Psi$. This proves (3).

Assume the hypothesis of (4). Let $s$ be an element of the target of $\Psi$. By assumption there exists a finite covering $\{U_j \to U\}_{j = 1, \ldots, m} U_j$, with $U_j \times_U U_{j'}$ quasi-compact for all $j, j' \in J$ and for each $j$ an index $i_j \in I$ and $s_j \in \mathcal{F}_{i_j}(U_j)$ such that $s|_{U_j}$ is the image of $s_j$ for all $j$. Since $U_j \times_U U_{j'}$ is quasi-compact we can apply (2) and we see that there exists an $i_{jj'} \in I$, $i_{jj'} \geq i_j$, $i_{jj'} \geq i_{j'}$ such that $\varphi_{i_ji_{jj'}}(s_j)$ and $\varphi_{i_{j'}i_{jj'}}(s_{j'})$ agree over $U_j \times_U U_{j'}$. Choose an index $i \in I$ wich is bigger or equal than all the $i_{jj'}$. Then we see that the sections $\varphi_{i_ji}(s_j)$ of $\mathcal{F}_i$ glue to a section of $\mathcal{F}_i$ over $U$. This section is mapped to the element $s$ as desired. $\square$

    The code snippet corresponding to this tag is a part of the file sites.tex and is located in lines 3194–3219 (see updates for more information).

    \begin{lemma}
    \label{lemma-directed-colimits-sections}
    Let $\mathcal{C}$ be a site. Let
    $\mathcal{I} \to \Sh(\mathcal{C})$, $i \mapsto \mathcal{F}_i$
    be a filtered diagram of sheaves of sets.
    Let $U \in \Ob(\mathcal{C})$.
    Consider the canonical map
    $$
    \Psi :
    \colim_i \mathcal{F}_i(U)
    \longrightarrow
    \left(\colim_i \mathcal{F}_i\right)(U)
    $$
    With the terminology introduced above:
    \begin{enumerate}
    \item If all the transition maps are injective then
    $\Psi$ is injective for any $U$.
    \item If $U$ is quasi-compact, then $\Psi$ is injective.
    \item If $U$ is quasi-compact and all the transition maps are injective
    then $\Psi$ is an isomorphism.
    \item If $U$ has a cofinal system of coverings
    $\{U_j \to U\}_{j \in J}$ with
    $J$ finite and $U_j \times_U U_{j'}$ quasi-compact
    for all $j, j' \in J$, then $\Psi$ is bijective.
    \end{enumerate}
    \end{lemma}
    
    \begin{proof}
    Assume all the transition maps are injective. In this case the presheaf
    $\mathcal{F}' : V \mapsto \colim_i \mathcal{F}_i(V)$ is
    separated (see Definition \ref{definition-separated}).
    By Lemma \ref{lemma-colimit-sheaves}
    we have
    $(\mathcal{F}')^\# = \colim_i \mathcal{F}_i$.
    By Theorem \ref{theorem-plus}
    we see that $\mathcal{F}' \to (\mathcal{F}')^\#$ is injective.
    This proves (1).
    
    \medskip\noindent
    Assume $U$ is quasi-compact. Suppose that $s \in \mathcal{F}_i(U)$ and
    $s' \in \mathcal{F}_{i'}(U)$ give rise to elements on
    the left hand side which have the same image under $\Psi$.
    This means we can choose a covering $\{U_a \to U\}_{a \in A}$
    and for each $a \in A$ an index $i_a \in I$, $i_a \geq i$, $i_a \geq i'$
    such that $\varphi_{ii_a}(s) = \varphi_{i'i_a}(s')$.
    Because $U$ is quasi-compact we can choose a covering
    $\{V_b \to U\}_{b \in B}$, a map $\alpha : B \to A$ with finite image,
    and morphisms $V_b \to U_{\alpha(b)}$ over $U$.
    Pick $i''\in I$ to be $\geq$ than all of the $i_{\alpha(b)}$
    which is possible because the image of $\alpha$ is finite.
    We conclude that $\varphi_{ii''}(s)$ and $\varphi_{i'i''}(s)$
    agree on $V_b$ for all $b \in B$ and hence that
    $\varphi_{ii''}(s) = \varphi_{i'i''}(s)$. This proves (2).
    
    \medskip\noindent
    Assume $U$ is quasi-compact and all transition maps injective.
    Let $s$ be an element of the target of $\Psi$. There exists a covering
    $\{U_a \to U\}_{a \in A}$ and for each $a \in A$ an index $i_a \in I$
    and a section $s_a \in \mathcal{F}_{i_a}(U_a)$
    such that $s|_{U_a}$ comes from $s_a$ for all $a \in A$.
    Because $U$ is quasi-compact we can choose a covering
    $\{V_b \to U\}_{b \in B}$, a map $\alpha : B \to A$ with finite image,
    and morphisms $V_b \to U_{\alpha(b)}$ over $U$.
    Pick $i \in I$ to be $\geq$ than all of the $i_{\alpha(b)}$
    which is possible because the image of $\alpha$ is finite.
    By (1) the sections
    $s_b = \varphi_{i_{\alpha(b)} i}(s_{\alpha(b)})|_{V_b}$
    agree over $V_b \times_U V_{b'}$.
    Hence they glue to a section
    $s' \in \mathcal{F}_i(U)$ which maps to $s$ under $\Psi$.
    This proves (3).
    
    \medskip\noindent
    Assume the hypothesis of (4).
    Let $s$ be an element of the target of $\Psi$.
    By assumption there  exists a finite covering
    $\{U_j \to U\}_{j = 1, \ldots, m} U_j$, with $U_j \times_U U_{j'}$
    quasi-compact for all $j, j' \in J$ and
    for each $j$ an index $i_j \in I$ and $s_j \in \mathcal{F}_{i_j}(U_j)$
    such that $s|_{U_j}$ is the image of $s_j$ for all $j$.
    Since $U_j \times_U U_{j'}$ is quasi-compact we can apply (2)
    and we see that there exists an $i_{jj'} \in I$,
    $i_{jj'} \geq i_j$, $i_{jj'} \geq i_{j'}$ such that
    $\varphi_{i_ji_{jj'}}(s_j)$ and $\varphi_{i_{j'}i_{jj'}}(s_{j'})$
    agree over $U_j \times_U U_{j'}$. Choose an index $i \in I$
    wich is bigger or equal than all the $i_{jj'}$. Then we see that
    the sections $\varphi_{i_ji}(s_j)$ of $\mathcal{F}_i$ glue
    to a section of $\mathcal{F}_i$ over $U$. This section is mapped
    to the element $s$ as desired.
    \end{proof}

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