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Tag 0A33

Chapter 6: Sheaves on Spaces > Section 6.29: Limits and colimits of sheaves

Lemma 6.29.4. In the situation described above, let $i \in \mathop{\rm Ob}\nolimits(\mathcal{I})$ and let $U_i \subset X_i$ be a quasi-compact open. Then $$ \mathop{\rm colim}\nolimits_{a : j \to i} \mathcal{F}_j(f_a^{-1}(U_i)) = \mathcal{F}(p_i^{-1}(U_i)) $$

Proof. Recall that $p_i^{-1}(U_i)$ is a quasi-compact open of the spectral space $X$, see Topology, Lemma 5.24.5. Hence Lemma 6.29.1 applies and we have $$ \mathcal{F}(p_i^{-1}(U_i)) = \mathop{\rm colim}\nolimits_{a : j \to i} p_j^{-1}\mathcal{F}_j(p_i^{-1}(U_i)). $$ A formal argument shows that $$ \mathop{\rm colim}\nolimits_{a : j \to i} \mathcal{F}_j(f_a^{-1}(U_i)) = \mathop{\rm colim}\nolimits_{a : j \to i} \mathop{\rm colim}\nolimits_{b : k \to j} f_b^{-1}\mathcal{F}_j(f_{a \circ b}^{-1}(U_i)) $$ Thus it suffices to show that $$ p_j^{-1}\mathcal{F}_j(p_i^{-1}(U_i)) = \mathop{\rm colim}\nolimits_{b : k \to j} f_b^{-1}\mathcal{F}_j(f_{a \circ b}^{-1}(U_i)) $$ This is Lemma 6.29.3 applied to $\mathcal{F}_j$ and the quasi-compact open $f_a^{-1}(U_i)$. $\square$

    The code snippet corresponding to this tag is a part of the file sheaves.tex and is located in lines 3565–3572 (see updates for more information).

    \begin{lemma}
    \label{lemma-descend-opens}
    In the situation described above, let $i \in \Ob(\mathcal{I})$ and let
    $U_i \subset X_i$ be a quasi-compact open. Then
    $$
    \colim_{a : j \to i} \mathcal{F}_j(f_a^{-1}(U_i)) = \mathcal{F}(p_i^{-1}(U_i))
    $$
    \end{lemma}
    
    \begin{proof}
    Recall that $p_i^{-1}(U_i)$ is a quasi-compact open of the spectral space
    $X$, see
    Topology, Lemma \ref{topology-lemma-directed-inverse-limit-spectral-spaces}.
    Hence Lemma \ref{lemma-directed-colimits-sections} applies and we have
    $$
    \mathcal{F}(p_i^{-1}(U_i)) =
    \colim_{a : j \to i} p_j^{-1}\mathcal{F}_j(p_i^{-1}(U_i)).
    $$
    A formal argument shows that
    $$
    \colim_{a : j \to i} \mathcal{F}_j(f_a^{-1}(U_i)) =
    \colim_{a : j \to i} \colim_{b : k \to j}
    f_b^{-1}\mathcal{F}_j(f_{a \circ b}^{-1}(U_i))
    $$
    Thus it suffices to show that
    $$
    p_j^{-1}\mathcal{F}_j(p_i^{-1}(U_i)) =
    \colim_{b : k \to j} f_b^{-1}\mathcal{F}_j(f_{a \circ b}^{-1}(U_i))
    $$
    This is Lemma \ref{lemma-compute-pullback-to-limit}
    applied to $\mathcal{F}_j$ and the quasi-compact open $f_a^{-1}(U_i)$.
    \end{proof}

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