The Stacks project

Proposition 75.16.1. Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. An object of $D_\mathit{QCoh}(\mathcal{O}_ X)$ is compact if and only if it is perfect.

Proof. If $K$ is a perfect object of $D(\mathcal{O}_ X)$ with dual $K^\vee $ (Cohomology on Sites, Lemma 21.48.4) 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 $M$. Since $K^\vee \otimes _{\mathcal{O}_ X}^\mathbf {L} -$ commutes with direct sums and since $H^0(X, -)$ commutes with direct sums on $D_\mathit{QCoh}(\mathcal{O}_ X)$ by Lemma 75.6.2 we conclude that $K$ is compact in $D_\mathit{QCoh}(\mathcal{O}_ X)$.

Conversely, let $K$ be a compact object of $D_\mathit{QCoh}(\mathcal{O}_ X)$. To show that $K$ is perfect, it suffices to show that $K|_ U$ is perfect for every affine scheme $U$ ├ętale over $X$, see Cohomology on Sites, Lemma 21.47.2. Observe that $j : U \to X$ is a quasi-compact and separated morphism. Hence $Rj_* : D_\mathit{QCoh}(\mathcal{O}_ U) \to D_\mathit{QCoh}(\mathcal{O}_ X)$ commutes with direct sums, see Lemma 75.6.2. Thus the adjointness of restriction to $U$ and $Rj_*$ implies that $K|_ U$ is a perfect object of $D_\mathit{QCoh}(\mathcal{O}_ U)$. Hence we reduce to the case that $X$ is affine, in particular a quasi-compact and quasi-separated scheme. Via Lemma 75.4.2 and 75.13.5 we reduce to the case of schemes, i.e., to Derived Categories of Schemes, Proposition 36.17.1. $\square$

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