The Stacks project

74.16 Compact and perfect objects

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

Proposition 74.16.1. Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. An object of $D_\mathit{QCoh}(\mathcal{O}_ X)$ is compact if and only if it is perfect.

Proof. If $K$ is a perfect object of $D(\mathcal{O}_ X)$ with dual $K^\vee $ (Cohomology on Sites, Lemma 21.48.4) we have

\[ \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ X)}(K, M) = H^0(X, K^\vee \otimes _{\mathcal{O}_ X}^\mathbf {L} M) \]

functorially in $M$. Since $K^\vee \otimes _{\mathcal{O}_ X}^\mathbf {L} -$ commutes with direct sums and since $H^0(X, -)$ commutes with direct sums on $D_\mathit{QCoh}(\mathcal{O}_ X)$ by Lemma 74.6.2 we conclude that $K$ is compact in $D_\mathit{QCoh}(\mathcal{O}_ X)$.

Conversely, let $K$ be a compact object of $D_\mathit{QCoh}(\mathcal{O}_ X)$. To show that $K$ is perfect, it suffices to show that $K|_ U$ is perfect for every affine scheme $U$ ├ętale over $X$, see Cohomology on Sites, Lemma 21.47.2. Observe that $j : U \to X$ is a quasi-compact and separated morphism. Hence $Rj_* : D_\mathit{QCoh}(\mathcal{O}_ U) \to D_\mathit{QCoh}(\mathcal{O}_ X)$ commutes with direct sums, see Lemma 74.6.2. Thus the adjointness of restriction to $U$ and $Rj_*$ implies that $K|_ U$ is a perfect object of $D_\mathit{QCoh}(\mathcal{O}_ U)$. Hence we reduce to the case that $X$ is affine, in particular a quasi-compact and quasi-separated scheme. Via Lemma 74.4.2 and 74.13.5 we reduce to the case of schemes, i.e., to Derived Categories of Schemes, Proposition 36.17.1. $\square$

Remark 74.16.2. Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $G$ be a perfect object of $D(\mathcal{O}_ X)$ which is a generator for $D_\mathit{QCoh}(\mathcal{O}_ X)$. By Theorem 74.15.4 there is at least one of these. Combining Lemma 74.5.3 with Proposition 74.16.1 and with Derived Categories, Proposition 13.37.6 we see that $G$ is a classical generator for $D_{perf}(\mathcal{O}_ X)$.

The following result is a strengthening of Proposition 74.16.1. Let $T \subset |X|$ be a closed subset where $X$ is an algebraic space. As before $D_ T(\mathcal{O}_ X)$ denotes the strictly full, saturated, triangulated subcategory consisting of complexes whose cohomology sheaves are supported on $T$. Since taking direct sums commutes with taking cohomology sheaves, it follows that $D_ T(\mathcal{O}_ X)$ has direct sums and that they are equal to direct sums in $D(\mathcal{O}_ X)$.

Lemma 74.16.3. Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $T \subset |X|$ be a closed subset such that $|X| \setminus T$ is quasi-compact. An object of $D_{\mathit{QCoh}, T}(\mathcal{O}_ X)$ is compact if and only if it is perfect as an object of $D(\mathcal{O}_ X)$.

Proof. We observe that $D_{\mathit{QCoh}, T}(\mathcal{O}_ X)$ is a triangulated category with direct sums by the remark preceding the lemma. By Proposition 74.16.1 the perfect objects define compact objects of $D(\mathcal{O}_ X)$ hence a fortiori of any subcategory preserved under taking direct sums. For the converse we will use there exists a generator $E \in D_{\mathit{QCoh}, T}(\mathcal{O}_ X)$ which is a perfect complex of $\mathcal{O}_ X$-modules, see Lemma 74.15.6. Hence by the above, $E$ is compact. Then it follows from Derived Categories, Proposition 13.37.6 that $E$ is a classical generator of the full subcategory of compact objects of $D_{\mathit{QCoh}, T}(\mathcal{O}_ X)$. Thus any compact object can be constructed out of $E$ by a finite sequence of operations consisting of (a) taking shifts, (b) taking finite direct sums, (c) taking cones, and (d) taking direct summands. Each of these operations preserves the property of being perfect and the result follows. $\square$

Remark 74.16.4. Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $T \subset |X|$ be a closed subset such that $|X| \setminus T$ is quasi-compact. Let $G$ be a perfect object of $D_{\mathit{QCoh}, T}(\mathcal{O}_ X)$ which is a generator for $D_{\mathit{QCoh}, T}(\mathcal{O}_ X)$. By Lemma 74.15.6 there is at least one of these. Combining the fact that $D_{\mathit{QCoh}, T}(\mathcal{O}_ X)$ has direct sums with Lemma 74.16.3 and with Derived Categories, Proposition 13.37.6 we see that $G$ is a classical generator for $D_{perf, T}(\mathcal{O}_ X)$.

The following lemma is an application of the ideas that go into the proof of the preceding lemma.

Lemma 74.16.5. Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $T \subset |X|$ be a closed subset such that the complement $U \subset X$ is quasi-compact. Let $\alpha : P \to E$ be a morphism of $D_\mathit{QCoh}(\mathcal{O}_ X)$ with either

  1. $P$ is perfect and $E$ supported on $T$, or

  2. $P$ pseudo-coherent, $E$ supported on $T$, and $E$ bounded below.

Then there exists a perfect complex of $\mathcal{O}_ X$-modules $I$ and a map $I \to \mathcal{O}_ X[0]$ such that $I \otimes ^\mathbf {L} P \to E$ is zero and such that $I|_ U \to \mathcal{O}_ U[0]$ is an isomorphism.

Proof. Set $\mathcal{D} = D_{\mathit{QCoh}, T}(\mathcal{O}_ X)$. In both cases the complex $K = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (P, E)$ is an object of $\mathcal{D}$. See Lemma 74.13.10 for quasi-coherence. It is clear that $K$ is supported on $T$ as formation of $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits $ commutes with restriction to opens. The map $\alpha $ defines an element of $H^0(K) = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ X)}(\mathcal{O}_ X[0], K)$. Then it suffices to prove the result for the map $\alpha : \mathcal{O}_ X[0] \to K$.

Let $E \in \mathcal{D}$ be a perfect generator, see Lemma 74.15.6. Write

\[ K = \text{hocolim} K_ n \]

as in Derived Categories, Lemma 13.37.3 using the generator $E$. Since the functor $\mathcal{D} \to D(\mathcal{O}_ X)$ commutes with direct sums, we see that $K = \text{hocolim} K_ n$ holds in $D(\mathcal{O}_ X)$. Since $\mathcal{O}_ X$ is a compact object of $D(\mathcal{O}_ X)$ we find an $n$ and a morphism $\alpha _ n : \mathcal{O}_ X \to K_ n$ which gives rise to $\alpha $, see Derived Categories, Lemma 13.33.9. By Derived Categories, Lemma 13.37.4 applied to the morphism $\mathcal{O}_ X[0] \to K_ n$ in the ambient category $D(\mathcal{O}_ X)$ we see that $\alpha _ n$ factors as $\mathcal{O}_ X[0] \to Q \to K_ n$ where $Q$ is an object of $\langle E \rangle $. We conclude that $Q$ is a perfect complex supported on $T$.

Choose a distinguished triangle

\[ I \to \mathcal{O}_ X[0] \to Q \to I[1] \]

By construction $I$ is perfect, the map $I \to \mathcal{O}_ X[0]$ restricts to an isomorphism over $U$, and the composition $I \to K$ is zero as $\alpha $ factors through $Q$. This proves the lemma. $\square$


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