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The Stacks project

Proposition 36.17.1. Let X be a quasi-compact and quasi-separated scheme. 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, Lemma 20.50.5) 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 36.4.5 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 open U \subset X, see Cohomology, Lemma 20.49.2. 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 36.4.5. Thus the adjointness of restriction to U and Rj_* implies that K|_ U is a compact object of D_\mathit{QCoh}(\mathcal{O}_ U). Hence we reduce to the case that X is affine.

Assume X = \mathop{\mathrm{Spec}}(A) is affine. By Lemma 36.3.5 the problem is translated into the same problem for D(A). For D(A) the result is More on Algebra, Proposition 15.78.3. \square


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