The Stacks project

Proposition 36.37.5. Let $X$ be a quasi-compact and quasi-separated scheme with the resolution property. Denote

  1. $\mathcal{A}$ the additive category of finite locally free $\mathcal{O}_ X$-modules,

  2. $K^ b(\mathcal{A})$ the homotopy category of bounded complexes in $\mathcal{A}$, see Derived Categories, Section 13.8, and

  3. $D_{perf}(\mathcal{O}_ X)$ the strictly full, saturated, triangulated subcategory of $D(\mathcal{O}_ X)$ consisting of perfect objects.

With this notation the obvious functor

\[ K^ b(\mathcal{A}) \longrightarrow D_{perf}(\mathcal{O}_ X) \]

is an exact functor of trianglated categories which factors through an equivalence $S^{-1}K^ b(\mathcal{A}) \to D_{perf}(\mathcal{O}_ X)$ of triangulated categories where $S$ is the saturated multiplicative system of quasi-isomorphisms in $K^ b(\mathcal{A})$.

Proof. If you can parse the statement of the proposition, then please skip this first paragraph. For some of the definitions used, please see Derived Categories, Definition 13.3.4 (triangulated subcategory), Derived Categories, Definition 13.6.1 (saturated triangulated subcategory), Derived Categories, Definition 13.5.1 (multiplicative system compatible with the triangulated structure), and Categories, Definition 4.27.20 (saturated multiplicative system). Observe that $D_{perf}(\mathcal{O}_ X)$ is a saturated triangulated subcategory of $D(\mathcal{O}_ X)$ by Cohomology, Lemmas 20.49.7 and 20.49.9. Also, note that $K^ b(\mathcal{A})$ is a triangulated category, see Derived Categories, Lemma 13.10.5.

It is clear that the functor sends distinguished triangles to distinguished triangles, i.e., is exact. Then $S$ is a saturated multiplicative system compatible with the triangulated structure on $K^ b(\mathcal{A})$ by Derived Categories, Lemma 13.5.4. Hence the localization $S^{-1}K^ b(\mathcal{A})$ exists and is a triangulated category by Derived Categories, Proposition 13.5.6. We get an exact factorization $S^{-1}K^ b(\mathcal{A}) \to D_{perf}(\mathcal{O}_ X)$ by Derived Categories, Lemma 13.5.7. By Lemmas 36.37.2, 36.37.3, and 36.37.4 this functor is an equivalence. Then finally the functor $S^{-1}K^ b(\mathcal{A}) \to D_{perf}(\mathcal{O}_ X)$ is an equivalence of triangulated categories (in the sense that distinguished triangles correspond) by Derived Categories, Lemma 13.4.18. $\square$

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