36.18 Derived categories as module categories
In this section we draw some conclusions of what has gone before. Before we do so we need a couple more lemmas.
Lemma 36.18.1. Let $X$ be a scheme. Let $K^\bullet $ be a complex of $\mathcal{O}_ X$-modules whose cohomology sheaves are quasi-coherent. Let $(E, d) = \mathop{\mathrm{Hom}}\nolimits _{\text{Comp}^{dg}(\mathcal{O}_ X)}(K^\bullet , K^\bullet )$ be the endomorphism differential graded algebra. Then 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 has image contained in $D_\mathit{QCoh}(\mathcal{O}_ X)$.
Proof.
Let $P$ be a differential graded $E$-module with property (P) and let $F_\bullet $ be a filtration on $P$ as in Differential Graded Algebra, Section 22.20. Then we have
\[ P \otimes _ E K^\bullet = \text{hocolim}\ F_ iP \otimes _ E K^\bullet \]
Each of the $F_ iP$ has a finite filtration whose graded pieces are direct sums of $E[k]$. The result follows easily.
$\square$
The following lemma can be strengthened (there is a uniformity in the vanishing over all $L$ with nonzero cohomology sheaves only in a fixed range).
Lemma 36.18.2. Let $X$ be a quasi-compact and quasi-separated scheme. Let $K$ be a perfect object of $D(\mathcal{O}_ X)$. Then
there exist integers $a \leq b$ such that $\mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ X)}(K, L) = 0$ for $L \in D_\mathit{QCoh}(\mathcal{O}_ X)$ with $H^ i(L) = 0$ for $i \in [a, b]$, and
if $L$ is bounded, then $\mathop{\mathrm{Ext}}\nolimits ^ n_{D(\mathcal{O}_ X)}(K, L)$ is zero for all but finitely many $n$.
Proof.
Part (2) follows from (1) as $\mathop{\mathrm{Ext}}\nolimits ^ n_{D(\mathcal{O}_ X)}(K, L) = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ X)}(K, L[n])$. We prove (1). Since $K$ is perfect we have
\[ \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ X)}(K, L) = H^0(X, K^\vee \otimes _{\mathcal{O}_ X}^\mathbf {L} L) \]
where $K^\vee $ is the “dual” perfect complex to $K$, see Cohomology, Lemma 20.50.5. Note that $K^\vee \otimes _{\mathcal{O}_ X}^\mathbf {L} L$ is in $D_\mathit{QCoh}(X)$ by Lemmas 36.3.9 and 36.10.1 (to see that a perfect complex has quasi-coherent cohomology sheaves). Say $K^\vee $ has tor amplitude in $[a, b]$. Then the spectral sequence
\[ E_1^{p, q} = H^ p(K^\vee \otimes _{\mathcal{O}_ X}^\mathbf {L} H^ q(L)) \Rightarrow H^{p + q}(K^\vee \otimes _{\mathcal{O}_ X}^\mathbf {L} L) \]
shows that $H^ j(K^\vee \otimes _{\mathcal{O}_ X}^\mathbf {L} L)$ is zero if $H^ q(L) = 0$ for $q \in [j - b, j - a]$. Let $N$ be the integer $d$ of Cohomology of Schemes, Lemma 30.4.4. Then $H^0(X, K^\vee \otimes _{\mathcal{O}_ X}^\mathbf {L} L)$ vanishes if the cohomology sheaves
\[ H^{-N}(K^\vee \otimes _{\mathcal{O}_ X}^\mathbf {L} L), \ H^{-N + 1}(K^\vee \otimes _{\mathcal{O}_ X}^\mathbf {L} L), \ \ldots , \ H^0(K^\vee \otimes _{\mathcal{O}_ X}^\mathbf {L} L) \]
are zero. Namely, by the lemma cited and Lemma 36.3.4, we have
\[ H^0(X, K^\vee \otimes _{\mathcal{O}_ X}^\mathbf {L} L) = H^0(X, \tau _{\geq -N}(K^\vee \otimes _{\mathcal{O}_ X}^\mathbf {L} L)) \]
and by the vanishing of cohomology sheaves, this is equal to $H^0(X, \tau _{\geq 1}(K^\vee \otimes _{\mathcal{O}_ X}^\mathbf {L} L))$ which is zero by Derived Categories, Lemma 13.16.1. It follows that $\mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ X)}(K, L)$ is zero if $H^ i(L) = 0$ for $i \in [-b - N, -a]$.
$\square$
The following result is taken from [BvdB].
Theorem 36.18.3. 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
36.18.3.1
\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 36.18.2. 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 36.17.1. Combined with (36.18.3.1) 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 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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