The Stacks project

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21.48 Compact objects

In this section we study compact objects in the derived category of modules on a ringed site. We recall that compact objects are defined in Derived Categories, Definition 13.34.1.

Lemma 21.48.1. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Assume $\mathcal{C}$ has the following properties

  1. $\mathcal{C}$ has a quasi-compact final object $X$,

  2. every quasi-compact object of $\mathcal{C}$ has a cofinal system of coverings which are finite and consist of quasi-compact objects,

  3. for a finite covering $\{ U_ i \to U\} _{i \in I}$ with $U$, $U_ i$ quasi-compact the fibre products $U_ i \times _ U U_ j$ are quasi-compact.

Then any perfect object of $D(\mathcal{O})$ is compact.

Proof. Let $K$ be a perfect object and let $K^\vee $ be its dual, see Lemma 21.45.9. Then 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 $M$ in $D(\mathcal{O}_ X)$. Since $K^\vee \otimes _{\mathcal{O}_ X}^\mathbf {L} -$ commutes with direct sums (by construction) and $H^0$ does by Lemma 21.17.1 and the construction of direct sums in Injectives, Lemma 19.13.4 we obtain the result of the lemma. $\square$

Lemma 21.48.2. Let $\mathcal{A}$ be a Grothendieck abelian category. Let $S \subset \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$ be a set of objects such that

  1. any object of $\mathcal{A}$ is a quotient of a direct sum of elements of $S$, and

  2. for any $E \in S$ the functor $\mathop{\mathrm{Hom}}\nolimits _\mathcal {A}(E, -)$ commutes with direct sums.

Then every compact object of $D(\mathcal{A})$ is a direct summand in $D(\mathcal{A})$ of a finite complex of finite direct sums of elements of $S$.

Proof. Assume $K \in D(\mathcal{A})$ is a compact object. Represent $K$ by a complex $K^\bullet $ and consider the map

\[ K^\bullet \longrightarrow \bigoplus \nolimits _{n \geq 0} \tau _{\geq n} K^\bullet \]

where we have used the canonical truncations, see Homology, Section 12.14. This makes sense as in each degree the direct sum on the right is finite. By assumption this map factors through a finite direct sum. We conclude that $K \to \tau _{\geq n} K$ is zero for at least one $n$, i.e., $K$ is in $D^{-}(R)$.

We may represent $K$ by a bounded above complex $K^\bullet $ each of whose terms is a direct sum of objects from $S$, see Derived Categories, Lemma 13.16.5. Note that we have

\[ K^\bullet = \bigcup \nolimits _{n \leq 0} \sigma _{\geq n}K^\bullet \]

where we have used the stupid truncations, see Homology, Section 12.14. Hence by Derived Categories, Lemmas 13.31.7 and 13.31.9 we see that $1 : K^\bullet \to K^\bullet $ factors through $\sigma _{\geq n}K^\bullet \to K^\bullet $ in $D(R)$. Thus we see that $1 : K^\bullet \to K^\bullet $ factors as

\[ K^\bullet \xrightarrow {\varphi } L^\bullet \xrightarrow {\psi } K^\bullet \]

in $D(\mathcal{A})$ for some complex $L^\bullet $ which is bounded and whose terms are direct sums of elements of $S$. Say $L^ i$ is zero for $i \not\in [a, b]$. Let $c$ be the largest integer $\leq b + 1$ such that $L^ i$ a finite direct sum of elements of $S$ for $i < c$. Claim: if $c < b + 1$, then we can modify $L^\bullet $ to increase $c$. By induction this claim will show we have a factorization of $1_ K$ as

\[ K \xrightarrow {\varphi } L \xrightarrow {\psi } K \]

in $D(\mathcal{A})$ where $L$ can be represented by a finite complex of finite direct sums of elements of $S$. Note that $e = \varphi \circ \psi \in \text{End}_{D(\mathcal{A})}(L)$ is an idempotent. By Derived Categories, Lemma 13.4.13 we see that $L = \mathop{\mathrm{Ker}}(e) \oplus \mathop{\mathrm{Ker}}(1 - e)$. The map $\varphi : K \to L$ induces an isomorphism with $\mathop{\mathrm{Ker}}(1 - e)$ in $D(R)$ and we conclude.

Proof of the claim. Write $L^ c = \bigoplus _{\lambda \in \Lambda } E_\lambda $. Since $L^{c - 1}$ is a finite direct sum of elements of $S$ we can by assumption (2) find a finite subset $\Lambda ' \subset \Lambda $ such that $L^{c - 1} \to L^ c$ factors through $\bigoplus _{\lambda \in \Lambda '} E_\lambda \subset L^ c$. Consider the map of complexes

\[ \pi : L^\bullet \longrightarrow (\bigoplus \nolimits _{\lambda \in \Lambda \setminus \Lambda '} E_\lambda )[-i] \]

given by the projection onto the factors corresponding to $\Lambda \setminus \Lambda '$ in degree $i$. By our assumption on $K$ we see that, after possibly replacing $\Lambda '$ by a larger finite subset, we may assume that $\pi \circ \varphi = 0$ in $D(\mathcal{A})$. Let $(L')^\bullet \subset L^\bullet $ be the kernel of $\pi $. Since $\pi $ is surjective we get a short exact sequence of complexes, which gives a distinguished triangle in $D(\mathcal{A})$ (see Derived Categories, Lemma 13.12.1). Since $\mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{A})}(K, -)$ is homological (see Derived Categories, Lemma 13.4.2) and $\pi \circ \varphi = 0$, we can find a morphism $\varphi ' : K^\bullet \to (L')^\bullet $ in $D(\mathcal{A})$ whose composition with $(L')^\bullet \to L^\bullet $ gives $\varphi $. Setting $\psi '$ equal to the composition of $\psi $ with $(L')^\bullet \to L^\bullet $ we obtain a new factorization. Since $(L')^\bullet $ agrees with $L^\bullet $ except in degree $c$ and since $(L')^ c = \bigoplus _{\lambda \in \Lambda '} E_\lambda $ the claim is proved. $\square$

Lemma 21.48.3. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Assume every object of $\mathcal{C}$ has a covering by quasi-compact objects. Then every compact object of $D(\mathcal{O})$ is a direct summand in $D(\mathcal{O})$ of a finite complex whose terms are finite direct sums of $\mathcal{O}$-modules of the form $j_!\mathcal{O}_ U$ where $U$ is a quasi-compact object of $\mathcal{C}$.

Proof. Apply Lemma 21.48.2 where $S \subset \mathop{\mathrm{Ob}}\nolimits (\textit{Mod}(\mathcal{O}))$ is the set of modules of the form $j_!\mathcal{O}_ U$ with $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ quasi-compact. Assumption (1) holds by Modules on Sites, Lemma 18.28.7 and the assumption that every $U$ can be covered by quasi-compact objects. Assumption (2) follows as

\[ \mathop{\mathrm{Hom}}\nolimits _\mathcal {O}(j_!\mathcal{O}_ U, \mathcal{F}) = \mathcal{F}(U) \]

which commutes with direct sums by Sites, Lemma 7.17.5. $\square$

In the situation of the lemma above it is not always true that the modules $j_!\mathcal{O}_ U$ are compact objects of $D(\mathcal{O})$ (even if $U$ is a quasi-compact object of $\mathcal{C}$). Here is a criterion.

Lemma 21.48.4. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $U$ be an object of $\mathcal{C}$. The $\mathcal{O}$-module $j_!\mathcal{O}_ U$ is a compact object of $D(\mathcal{O})$ if there exists an integer $d$ such that

  1. $H^ p(U, \mathcal{F}) = 0$ for all $p > d$, and

  2. the functors $\mathcal{F} \mapsto H^ p(U, \mathcal{F})$ commute with direct sums.

Proof. Assume (1) and (2). The first means that the functor $F = H^0(U, -)$ has finite cohomological dimension. Moreover, any direct sum of injective modules is acyclic for $F$ by (2). Since we may compute $RF$ by applying $F$ to any complex of acyclics (Derived Categories, Lemma 13.30.2). Thus, if $K_ i$ be a family of objects of $D(\mathcal{O})$, then we can choose K-injective representatives $I_ i^\bullet $ and we see that $\bigoplus K_ i$ is represented by $\bigoplus I_ i^\bullet $. Thus $H^0(U, -)$ commutes with direct sums. $\square$

Lemma 21.48.5. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $U$ be an object of $\mathcal{C}$ which is quasi-compact and weakly contractible. Then $j_!\mathcal{O}_ U$ is a compact object of $D(\mathcal{O})$.

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