Lemma 20.19.2. Let $f : X \to Y$ be a continuous map of topological spaces. Let $(\mathcal{F}_ i, \varphi _{ii'})$ be a system of abelian sheaves on $X$. Set $\mathcal{F} = \mathop{\mathrm{colim}}\nolimits \mathcal{F}_ i$. Let $p \geq 0$ be an integer. Assume the set of opens $V \subset Y$ such that $H^ p(f^{-1}(V), \mathcal{F}) = \mathop{\mathrm{colim}}\nolimits H^ p(f^{-1}(V), \mathcal{F}_ i)$ is a basis for the topology on $Y$. Then $R^ pf_*\mathcal{F} = \mathop{\mathrm{colim}}\nolimits R^ pf_*\mathcal{F}_ i$.
Proof. Recall that $R^ pf_*\mathcal{F}$ is the sheafification of the presheaf $\mathcal{G}$ sending $V$ to $H^ p(f^{-1}(V), \mathcal{F})$, see Lemma 20.7.3. Similarly, $R^ pf_*\mathcal{F}_ i$ is the sheafification of the presheaf $\mathcal{G}_ i$ sending $V$ to $H^ p(f^{-1}(V), \mathcal{F}_ i)$. Recall that sheafification is the left adjoint to the inclusion from sheaves to presheaves, see Sheaves, Section 6.17. Hence sheafification commutes with colimits, see Categories, Lemma 4.24.5. Hence it suffices to show that the map of presheaves (with colimit in the category of presheaves)
\[ \mathop{\mathrm{colim}}\nolimits \mathcal{G}_ i \longrightarrow \mathcal{G} \]
induces an isomorphism on sheafifications. For this it suffices to show that the presheaves $\mathcal{G}$ and $\mathop{\mathrm{colim}}\nolimits \mathcal{G}_ i$ agree on a basis for the topology of $Y$. Namely, in this case the stalks of their sheafifications, which can be computed directly from the presheaf values on elements of the basis, agree. The required agreement is exactly the assumption of the lemma. $\square$
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