The Stacks project

Lemma 21.13.1. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $K$ be a presheaf of sets on $\mathcal{C}$. Let $\mathcal{F}$ be an $\mathcal{O}$-module and denote $\mathcal{F}_{ab}$ the underlying sheaf of abelian groups. Then $H^ p(K, \mathcal{F}) = H^ p(K, \mathcal{F}_{ab})$.

Proof. We may replace $K$ by its sheafification and assume $K$ is a sheaf. Note that both $H^ p(K, \mathcal{F})$ and $H^ p(K, \mathcal{F}_{ab})$ depend only on the topos, not on the underlying site. Hence by Sites, Lemma 7.29.5 we may replace $\mathcal{C}$ by a “larger” site such that $K = h_ U$ for some object $U$ of $\mathcal{C}$. In this case the result follows from Lemma 21.12.4. $\square$

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