Lemma 21.22.2. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. The functors $R\Gamma (\mathcal{C}, -)$ and $R\Gamma (U, -)$ for $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ commute with $R\mathop{\mathrm{lim}}\nolimits$. Moreover, there are short exact sequences

$0 \to R^1\mathop{\mathrm{lim}}\nolimits H^{m - 1}(U, K_ n) \to H^ m(U, R\mathop{\mathrm{lim}}\nolimits K_ n) \to \mathop{\mathrm{lim}}\nolimits H^ m(U, K_ n) \to 0$

for any inverse system $(K_ n)$ in $D(\mathcal{O})$ and $m \in \mathbf{Z}$. Similar for $H^ m(\mathcal{C}, R\mathop{\mathrm{lim}}\nolimits K_ n)$.

Proof. The first statement follows from Injectives, Lemma 19.13.6. Then we may apply More on Algebra, Remark 15.85.9 to $R\mathop{\mathrm{lim}}\nolimits R\Gamma (U, K_ n) = R\Gamma (U, R\mathop{\mathrm{lim}}\nolimits K_ n)$ to get the short exact sequences. $\square$

Comment #3274 by Kevin Carlson on

Suggested slogan: Cohomology commutes with derived limits

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