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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