Lemma 21.23.2. 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 15.86.10 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.
$\square$

## Comments (2)

Comment #3274 by Kevin Carlson on

Comment #3366 by Johan on