The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

Lemma 35.3.2. Let $X$ be a scheme. 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 $\mathcal{B}$ be the set of affine opens of $X$. Since $H^ q(K_ n)$ is quasi-coherent we have $H^ p(U, H^ q(K_ n)) = 0$ for $U \in \mathcal{B}$ by Cohomology of Schemes, Lemma 29.2.2. 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 Schemes, Lemma 25.7.5. Part (1) follows from Cohomology, Lemma 20.32.11 whose conditions we have just verified. Parts (2) and (3) follow from Cohomology, Lemma 20.32.4. $\square$


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