The Stacks project

Lemma 74.5.4. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $(K_ n)$ be an inverse system of $D_\mathit{QCoh}(\mathcal{O}_ X)$ with derived limit $K = R\mathop{\mathrm{lim}}\nolimits K_ n$ in $D(\mathcal{O}_ X)$. Assume $H^ q(K_{n + 1}) \to H^ q(K_ n)$ is surjective for all $q \in \mathbf{Z}$ and $n \geq 1$. Then

  1. $H^ q(K) = \mathop{\mathrm{lim}}\nolimits H^ q(K_ n)$,

  2. $R\mathop{\mathrm{lim}}\nolimits H^ q(K_ n) = \mathop{\mathrm{lim}}\nolimits H^ q(K_ n)$, and

  3. for every affine open $U \subset X$ we have $H^ p(U, \mathop{\mathrm{lim}}\nolimits H^ q(K_ n)) = 0$ for $p > 0$.

Proof. Let $\mathcal{B} \subset \mathop{\mathrm{Ob}}\nolimits (X_{\acute{e}tale})$ be the set of affine objects. Since $H^ q(K_ n)$ is quasi-coherent we have $H^ p(U, H^ q(K_ n)) = 0$ for $U \in \mathcal{B}$ by the discussion in Cohomology of Spaces, Section 68.3 and Cohomology of Schemes, Lemma 30.2.2. Moreover, the maps $H^0(U, H^ q(K_{n + 1})) \to H^0(U, H^ q(K_ n))$ are surjective for $U \in \mathcal{B}$ by similar reasoning. Part (1) follows from Cohomology on Sites, Lemma 21.23.12 whose conditions we have just verified. Parts (2) and (3) follow from Cohomology on Sites, Lemma 21.23.5. $\square$

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