Lemma 20.37.11. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $(K_ n)$ be an inverse system of objects of $D(\mathcal{O}_ X)$. Let $\mathcal{B}$ be a set of opens of $X$. Assume
every open of $X$ 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 $K = R\mathop{\mathrm{lim}}\nolimits K_ n$. Let $U \in \mathcal{B}$. By Lemma 20.37.10 and (2)(a) we have $H^ q(U, K_ n) = H^0(U, H^ q(K_ n))$. By Lemma 20.37.1 and (2)(b) we have $H^ q(U, K) = \mathop{\mathrm{lim}}\nolimits H^0(U, H^ q(K_ n))$. 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.
$\square$
Comments (2)
Comment #11607 by Alex Scheffelin on
Comment #11647 by Stacks Project on