Section 36.18 (09M2): Derived categories as module categories

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 result is taken from [BvdB].

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. $\square$

Remark 36.18.3 (Variant with support). Let $X$ be a quasi-compact and quasi-separated scheme. Let $T \subset X$ be a closed subset such that $X \setminus T$ is quasi-compact. The analogue of Theorem 36.18.2 holds for $D_{\mathit{QCoh}, T}(\mathcal{O}_ X)$ . This follows from the exact same argument as in the proof of the theorem, using Lemmas 36.15.4 and 36.17.3 and a variant of Lemma 36.18.1 with supports. If we ever need this, we will precisely state the result here and give a detailed proof.

Remark 36.18.4 (Uniqueness of dga). Let $X$ be a quasi-compact and quasi-separated scheme over a ring $R$ . By the construction of the proof of Theorem 36.18.2 there exists a differential graded algebra $(A, \text{d})$ over $R$ such that $D_\mathit{QCoh}(X)$ is $R$ -linearly equivalent to $D(A, \text{d})$ as a triangulated category. One may ask: how unique is $(A, \text{d})$ ? The answer is (only) slightly better than just saying that $(A, \text{d})$ is well defined up to derived equivalence. Namely, suppose that $(B, \text{d})$ is a second such pair. Then we have

$(A, \text{d}) = \mathop{\mathrm{Hom}}\nolimits _{\text{Comp}^{dg}(\mathcal{O}_ X)}(K^\bullet , K^\bullet )$

and

$(B, \text{d}) = \mathop{\mathrm{Hom}}\nolimits _{\text{Comp}^{dg}(\mathcal{O}_ X)}(L^\bullet , L^\bullet )$

for some K-injective complexes $K^\bullet$ and $L^\bullet$ of $\mathcal{O}_ X$ -modules corresponding to perfect generators of $D_\mathit{QCoh}(\mathcal{O}_ X)$ . Set

$\Omega = \mathop{\mathrm{Hom}}\nolimits _{\text{Comp}^{dg}(\mathcal{O}_ X)}(K^\bullet , L^\bullet ) \quad \Omega ' = \mathop{\mathrm{Hom}}\nolimits _{\text{Comp}^{dg}(\mathcal{O}_ X)}(L^\bullet , K^\bullet )$

Then $\Omega$ is a differential graded $B^{opp} \otimes _ R A$ -module and $\Omega '$ is a differential graded $A^{opp} \otimes _ R B$ -module. Moreover, the equivalence

$D(A, \text{d}) \to D_\mathit{QCoh}(\mathcal{O}_ X) \to D(B, \text{d})$

is given by the functor $- \otimes _ A^\mathbf {L} \Omega '$ and similarly for the quasi-inverse. Thus we are in the situation of Differential Graded Algebra, Remark 22.37.10. If we ever need this remark we will provide a precise statement with a detailed proof here.

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36.18 Derived categories as module categories

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