The Stacks project

Lemma 20.19.2. In the situation discussed above. Let $i \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{I})$ and let $U_ i \subset X_ i$ be quasi-compact open. Then

\[ \mathop{\mathrm{colim}}\nolimits _{a : j \to i} H^ p(f_ a^{-1}(U_ i), \mathcal{F}_ j) = H^ p(p_ i^{-1}(U_ i), \mathcal{F}) \]

for all $p \geq 0$. In particular we have $H^ p(X, \mathcal{F}) = \mathop{\mathrm{colim}}\nolimits H^ p(X_ i, \mathcal{F}_ i)$.

Proof. The case $p = 0$ is Sheaves, Lemma 6.29.4.

In this paragraph we show that we can find a map of systems $(\gamma _ i) : (\mathcal{F}_ i, \varphi _ a) \to (\mathcal{G}_ i, \psi _ a)$ with $\mathcal{G}_ i$ an injective abelian sheaf and $\gamma _ i$ injective. For each $i$ we pick an injection $\mathcal{F}_ i \to \mathcal{I}_ i$ where $\mathcal{I}_ i$ is an injective abelian sheaf on $X_ i$. Then we can consider the family of maps

\[ \gamma _ i : \mathcal{F}_ i \longrightarrow \prod \nolimits _{b : k \to i} f_{b, *}\mathcal{I}_ k = \mathcal{G}_ i \]

where the component maps are the maps adjoint to the maps $f_ b^{-1}\mathcal{F}_ i \to \mathcal{F}_ k \to \mathcal{I}_ k$. For $a : j \to i$ in $\mathcal{I}$ there is a canonical map

\[ \psi _ a : f_ a^{-1}\mathcal{G}_ i \to \mathcal{G}_ j \]

whose components are the canonical maps $f_ b^{-1}f_{a \circ b, *}\mathcal{I}_ k \to f_{b, *}\mathcal{I}_ k$ for $b : k \to j$. Thus we find an injection $\{ \gamma _ i\} : \{ \mathcal{F}_ i, \varphi _ a) \to (\mathcal{G}_ i, \psi _ a)$ of systems of abelian sheaves. Note that $\mathcal{G}_ i$ is an injective sheaf of abelian groups on $X_ i$, see Lemma 20.11.11 and Homology, Lemma 12.27.3. This finishes the construction.

Arguing exactly as in the proof of Lemma 20.19.1 we see that it suffices to prove that $H^ p(X, \mathop{\mathrm{colim}}\nolimits f_ i^{-1}\mathcal{G}_ i) = 0$ for $p > 0$.

Set $\mathcal{G} = \mathop{\mathrm{colim}}\nolimits f_ i^{-1}\mathcal{G}_ i$. To show vanishing of cohomology of $\mathcal{G}$ on every quasi-compact open of $X$, it suffices to show that the Čech cohomology of $\mathcal{G}$ for any covering $\mathcal{U}$ of a quasi-compact open of $X$ by finitely many quasi-compact opens is zero, see Lemma 20.11.9. Such a covering is the inverse by $p_ i$ of such a covering $\mathcal{U}_ i$ on the space $X_ i$ for some $i$ by Topology, Lemma 5.24.6. We have

\[ \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{G}) = \mathop{\mathrm{colim}}\nolimits _{a : j \to i} \check{\mathcal{C}}^\bullet (f_ a^{-1}(\mathcal{U}_ i), \mathcal{G}_ j) \]

by the case $p = 0$. The right hand side is a filtered colimit of complexes each of which is acyclic in positive degrees by Lemma 20.11.1. Thus we conclude by Algebra, Lemma 10.8.8. $\square$

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