The Stacks project

Lemma 21.16.6. In the situation of Lemma 21.16.5 set $\mathcal{F} = \mathop{\mathrm{colim}}\nolimits f_ i^{-1}\mathcal{F}_ i$. Let $i \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{I})$, $X_ i \in \text{Ob}(\mathcal{C}_ i)$. Then

\[ \mathop{\mathrm{colim}}\nolimits _{a : j \to i} H^ p(u_ a(X_ i), \mathcal{F}_ j) = H^ p(u_ i(X_ i), \mathcal{F}) \]

for all $p \geq 0$.

Proof. The case $p = 0$ is Sites, Lemma 7.18.4.

Choose $(\mathcal{F}_ i, \varphi _ a) \to (\mathcal{G}_ i, \psi _ a)$ as in Lemma 21.16.5. Arguing exactly as in the proof of Lemma 21.16.1 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 object of $\mathcal{C}$ we show that the Čech cohomology of $\mathcal{G}$ for any covering $\mathcal{U}$ of $\mathcal{C}$ is zero (Lemma 21.10.9). The covering $\mathcal{U}$ comes from a covering $\mathcal{U}_ i$ of $\mathcal{C}_ i$ for some $i$. We have

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

by the case $p = 0$. The right hand side is acyclic in positive degrees as a filtered colimit of acyclic complexes by Lemma 21.10.2. See Algebra, Lemma 10.8.8. $\square$

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