
## 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. $\square$

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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