The Stacks project

36.17 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 36.17.1. 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 36.17.2. Let $X$ be a quasi-compact and quasi-separated scheme. Let $K$ be a perfect object of $D(\mathcal{O}_ X)$. Then

  1. 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

  2. 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 20.46.4. Note that $K^\vee \otimes _{\mathcal{O}_ X}^\mathbf {L} L$ is in $D_\mathit{QCoh}(X)$ by Lemmas 36.3.9 and 36.9.1 (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 36.17.3. 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 36.14.3 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

36.17.3.1
\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 36.17.2. 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 36.16.1. Combined with (36.17.3.1) 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 36.17.1 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. $\square$

Remark 36.17.4 (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 36.17.3 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 36.14.4 and 36.16.2 and a variant of Lemma 36.17.1 with supports. If we ever need this, we will precisely state the result here and give a detailed proof.

Remark 36.17.5 (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 36.17.3 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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