Theorem 36.18.2. Let X be a quasi-compact and quasi-separated scheme. Then there exist a differential graded algebra (E, \text{d}) with only a finite number of nonzero cohomology groups H^ i(E) such that D_\mathit{QCoh}(\mathcal{O}_ X) is equivalent to D(E, \text{d}).
Proof. Let K^\bullet be a K-injective complex of \mathcal{O}-modules which is perfect and generates D_\mathit{QCoh}(\mathcal{O}_ X). Such a thing exists by Theorem 36.15.3 and the existence of K-injective resolutions. We will show the theorem holds with
(E, \text{d}) = \mathop{\mathrm{Hom}}\nolimits _{\text{Comp}^{dg}(\mathcal{O}_ X)}(K^\bullet , K^\bullet )
where \text{Comp}^{dg}(\mathcal{O}_ X) is the differential graded category of complexes of \mathcal{O}-modules. Please see Differential Graded Algebra, Section 22.35. Since K^\bullet is K-injective we have
36.18.2.1
\begin{equation} \label{perfect-equation-E-is-OK} H^ n(E) = \mathop{\mathrm{Ext}}\nolimits ^ n_{D(\mathcal{O}_ X)}(K^\bullet , K^\bullet ) \end{equation}
for all n \in \mathbf{Z}. Only a finite number of these Exts are nonzero by Lemma 36.13.5. Consider the functor
- \otimes _ E^\mathbf {L} K^\bullet : D(E, \text{d}) \longrightarrow D(\mathcal{O}_ X)
of Differential Graded Algebra, Lemma 22.35.3. Since K^\bullet is perfect, it defines a compact object of D(\mathcal{O}_ X), see Proposition 36.17.1. Combined with (36.18.2.1) the functor above is fully faithful as follows from Differential Graded Algebra, Lemmas 22.35.6. It has a right adjoint
R\mathop{\mathrm{Hom}}\nolimits (K^\bullet , - ) : D(\mathcal{O}_ X) \longrightarrow D(E, \text{d})
by Differential Graded Algebra, Lemmas 22.35.5 which is a left quasi-inverse functor by generalities on adjoint functors. On the other hand, it follows from Lemma 36.18.1 that we obtain
- \otimes _ E^\mathbf {L} K^\bullet : D(E, \text{d}) \longrightarrow D_\mathit{QCoh}(\mathcal{O}_ X)
and by our choice of K^\bullet as a generator of D_\mathit{QCoh}(\mathcal{O}_ X) the kernel of the adjoint restricted to D_\mathit{QCoh}(\mathcal{O}_ X) is zero. A formal argument shows that we obtain the desired equivalence, see Derived Categories, Lemma 13.7.2. \square
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