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*}

20.44 Compact objects

In this section we study compact objects in the derived category of modules on a ringed space. We recall that compact objects are defined in Derived Categories, Definition 13.34.1. On suitable ringed spaces the perfect objects are compact.

Lemma 20.44.1. Let $X$ be a ringed space. Assume that the underlying topological space of $X$ has the following properties:

  1. $X$ is quasi-compact,

  2. there exists a basis of quasi-compact open subsets, and

  3. the intersection of any two quasi-compact opens is quasi-compact.

Then any perfect object of $D(\mathcal{O}_ X)$ is compact.

Proof. Let $K$ be a perfect object and let $K^\vee $ be its dual, see Lemma 20.43.11. Then we have

\[ \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ X)}(K, M) = H^0(X, K^\vee \otimes _{\mathcal{O}_ X}^\mathbf {L} M) \]

functorially in $M$ in $D(\mathcal{O}_ X)$. Since $K^\vee \otimes _{\mathcal{O}_ X}^\mathbf {L} -$ commutes with direct sums (by construction) and $H^0$ does by Lemma 20.20.1 and the construction of direct sums in Injectives, Lemma 19.13.4 we obtain the result of the lemma. $\square$

Comments (2)

Comment #2116 by BB on

This is just a suggestion: the notation of Lemma 20.41.11 is \vee for the dual, so it would be better to harmonize than to use \hat for the dual (unless there's a good reason to do so).

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