The Stacks project

72.17 Derived categories as module categories

The section is the analogue of Derived Categories of Schemes, Section 36.17.

Lemma 72.17.1. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. 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 $P$. 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 72.17.2. Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. 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

\[ \mathop{\mathrm{Ext}}\nolimits ^ i_{D(\mathcal{O}_ X)}(K, L) = H^ i(X, K^\vee \otimes _{\mathcal{O}_ X}^\mathbf {L} L) \]

where $K^\vee $ is the “dual” perfect complex to $K$, see Cohomology on Sites, Lemma 21.46.4. Note that $P = K^\vee \otimes _{\mathcal{O}_ X}^\mathbf {L} L$ is in $D_\mathit{QCoh}(X)$ by Lemmas 72.5.6 and 72.13.6 (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 $\max (d_ p + p)$ of Cohomology of Spaces, Lemma 66.7.3. 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 72.5.8, 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 is the analogue of Derived Categories of Schemes, Theorem 36.17.3.

Theorem 72.17.3. 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 72.15.4 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

72.17.3.1
\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 72.17.2. 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 72.16.1. Combined with (72.17.3.1) 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 72.17.1 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 72.17.4 (Variant with support). Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space. Let $T \subset |X|$ be a closed subset such that $|X| \setminus T$ is quasi-compact. The analogue of Theorem 72.17.3 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 72.15.6 and 72.16.2 and a variant of Lemma 72.17.1 with supports. If we ever need this, we will precisely state the result here and give a detailed proof.

Remark 72.17.5 (Uniqueness of dga). Let $X$ be a quasi-compact and quasi-separated algebraic space over a ring $R$. By the construction of the proof of Theorem 72.17.3 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.


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 09M9. Beware of the difference between the letter 'O' and the digit '0'.