The Stacks project

Lemma 21.31.12. With $X \in \mathop{\mathrm{Ob}}\nolimits (\textit{LC}_{qc})$ and $a_ X : \mathop{\mathit{Sh}}\nolimits (\textit{LC}_{qc}/X) \to \mathop{\mathit{Sh}}\nolimits (X)$ as above:

1. for an abelian sheaf $\mathcal{F}$ on $X$ we have $H^ n(X, \mathcal{F}) = H^ n_{qc}(X, a_ X^{-1}\mathcal{F})$,

2. for $K \in D^+(X)$ we have $H^ n(X, K) = H^ n_{qc}(X, a_ X^{-1}K)$.

For example, if $A$ is an abelian group, then we have $H^ n(X, \underline{A}) = H^ n_{qc}(X, \underline{A})$.