Lemma 21.23.12. Let (\mathcal{C}, \mathcal{O}) be a ringed site. Let (K_ n) be an inverse system of objects of D(\mathcal{O}). Let \mathcal{B} \subset \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}) be a subset. Assume
every object of \mathcal{C} has a covering whose members are elements of \mathcal{B},
for all U \in \mathcal{B} and all q \in \mathbf{Z} we have
H^ p(U, H^ q(K_ n)) = 0 for p > 0,
the inverse system H^0(U, H^ q(K_ n)) has vanishing R^1\mathop{\mathrm{lim}}\nolimits .
Then H^ q(R\mathop{\mathrm{lim}}\nolimits K_ n) = \mathop{\mathrm{lim}}\nolimits H^ q(K_ n) for q \in \mathbf{Z}.
Proof.
Set K = R\mathop{\mathrm{lim}}\nolimits K_ n. We will use notation as in Remark 21.23.4. Let U \in \mathcal{B}. By Lemma 21.23.11 and (2)(a) we have H^ q(U, K_ n) = H^0(U, H^ q(K_ n)). Using that the functor R\Gamma (U, -) commutes with derived limits we have
H^ q(U, K) = H^ q(R\mathop{\mathrm{lim}}\nolimits R\Gamma (U, K_ n)) = \mathop{\mathrm{lim}}\nolimits H^0(U, H^ q(K_ n))
where the final equality follows from More on Algebra, Remark 15.86.10 and assumption (2)(b). Thus H^ q(U, K) is the inverse limit the sections of the sheaves H^ q(K_ n) over U. Since \mathop{\mathrm{lim}}\nolimits H^ q(K_ n) is a sheaf we find using assumption (1) that H^ q(K), which is the sheafification of the presheaf U \mapsto H^ q(U, K), is equal to \mathop{\mathrm{lim}}\nolimits H^ q(K_ n). This proves the lemma.
\square
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