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Tag 0A33

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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