The Stacks project

Theorem 73.17.3. Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. 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 73.15.4 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

73.17.3.1
\begin{equation} \label{spaces-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 73.17.2. 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 73.16.1. Combined with (73.17.3.1) 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 73.17.1 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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