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 21.48.4. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $U$ be an object of $\mathcal{C}$. The $\mathcal{O}$-module $j_!\mathcal{O}_ U$ is a compact object of $D(\mathcal{O})$ if there exists an integer $d$ such that

  1. $H^ p(U, \mathcal{F}) = 0$ for all $p > d$, and

  2. the functors $\mathcal{F} \mapsto H^ p(U, \mathcal{F})$ commute with direct sums.

Proof. Assume (1) and (2). The first means that the functor $F = H^0(U, -)$ has finite cohomological dimension. Moreover, any direct sum of injective modules is acyclic for $F$ by (2). Since we may compute $RF$ by applying $F$ to any complex of acyclics (Derived Categories, Lemma 13.30.2). Thus, if $K_ i$ be a family of objects of $D(\mathcal{O})$, then we can choose K-injective representatives $I_ i^\bullet $ and we see that $\bigoplus K_ i$ is represented by $\bigoplus I_ i^\bullet $. Thus $H^0(U, -)$ commutes with direct sums. $\square$


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