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 13.37.1.
Lemma 21.52.1. Let $\mathcal{A}$ be a Grothendieck abelian category. Let $S \subset \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$ be a set of objects such that
any object of $\mathcal{A}$ is a quotient of a direct sum of elements of $S$, and
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
where we have used the canonical truncations, see Homology, Section 12.15. 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 13.15.4. Note that we have
where we have used the stupid truncations, see Homology, Section 12.15. Hence by Derived Categories, Lemmas 13.33.7 and 13.33.9 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
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
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 13.4.14 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
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 21.52.2. 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 21.52.1 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 18.28.8 and the assumption that every $U$ can be covered by quasi-compact objects. Assumption (2) follows as
which commutes with direct sums by Sites, Lemma 7.17.7. $\square$
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 are two lemmas addressing this issue.
Lemma 21.52.3. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $U$ be an object of $\mathcal{C}$. Assume the functors $\mathcal{F} \mapsto H^ p(U, \mathcal{F})$ commute with direct sums. Then $\mathcal{O}$-module $j_!\mathcal{O}_ U$ is a compact object of $D^+(\mathcal{O})$ in the following sense: if $M = \bigoplus _{i \in I} M_ i$ in $D(\mathcal{O})$ is bounded below, then $\mathop{\mathrm{Hom}}\nolimits (j_{U!}\mathcal{O}_ U, M) = \bigoplus _{i \in I} \mathop{\mathrm{Hom}}\nolimits (j_{U!}\mathcal{O}_ U, M_ i)$.
Proof. Since $\mathop{\mathrm{Hom}}\nolimits (j_{U!}\mathcal{O}_ U, -)$ is the same as the functor $\mathcal{F} \mapsto \mathcal{F}(U)$ by Modules on Sites, Equation (18.19.2.1) it suffices to prove that $H^ p(U, M) = \bigoplus H^ p(U, M_ i)$. Let $\mathcal{I}_ i$, $i \in I$ be a collection of injective $\mathcal{O}$-modules. By assumption we have
for all $p$. Since $M = \bigoplus M_ i$ is bounded below, we see that there exists an $a \in \mathbf{Z}$ such that $H^ n(M_ i) = 0$ for $n < a$. Thus we can choose complexes of injective $\mathcal{O}$-modues $\mathcal{I}_ i^\bullet $ representing $M_ i$ with $\mathcal{I}_ i^ n = 0$ for $n < a$, see Derived Categories, Lemma 13.18.3. By Injectives, Lemma 19.13.4 we see that the direct sum complex $\bigoplus \mathcal{I}_ i^\bullet $ represents $M$. By Leray acyclicity (Derived Categories, Lemma 13.16.7) we see that
as desired. $\square$
Lemma 21.52.4. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site with set of coverings $\text{Cov}_\mathcal {C}$. Let $\mathcal{B} \subset \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$, and $\text{Cov} \subset \text{Cov}_\mathcal {C}$ be subsets. Assume that
For every $\mathcal{U} \in \text{Cov}$ we have $\mathcal{U} = \{ U_ i \to U\} _{i \in I}$ with $I$ finite, $U, U_ i \in \mathcal{B}$ and every $U_{i_0} \times _ U \ldots \times _ U U_{i_ p} \in \mathcal{B}$.
For every $U \in \mathcal{B}$ the coverings of $U$ occurring in $\text{Cov}$ is a cofinal system of coverings of $U$.
Then for $U \in \mathcal{B}$ the object $j_{U!}\mathcal{O}_ U$ is a compact object of $D^+(\mathcal{O})$ in the following sense: if $M = \bigoplus _{i \in I} M_ i$ in $D(\mathcal{O})$ is bounded below, then $\mathop{\mathrm{Hom}}\nolimits (j_{U!}\mathcal{O}_ U, M) = \bigoplus _{i \in I} \mathop{\mathrm{Hom}}\nolimits (j_{U!}\mathcal{O}_ U, M_ i)$.
Proof. This follows from Lemma 21.52.3 and Lemma 21.16.1. $\square$
Lemma 21.52.5. 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
$H^ p(U, \mathcal{F}) = 0$ for all $p > d$, and
the functors $\mathcal{F} \mapsto H^ p(U, \mathcal{F})$ commute with direct sums.
Proof. Assume (1) and (2). Recall that $\mathop{\mathrm{Hom}}\nolimits (j_!\mathcal{O}_ U, K) = R\Gamma (U, K)$ for $K$ in $D(\mathcal{O})$. Thus we have to show that $R\Gamma (U, -)$ commutes with direct sums. The first assumption means that the functor $F = H^0(U, -)$ has finite cohomological dimension. Moreover, the second assumption implies any direct sum of injective modules is acyclic for $F$. Let $K_ i$ be a family of objects of $D(\mathcal{O})$. Choose K-injective representatives $I_ i^\bullet $ with injective terms representing $K_ i$, see Injectives, Theorem 19.12.6. Since we may compute $RF$ by applying $F$ to any complex of acyclics (Derived Categories, Lemma 13.32.2) and since $\bigoplus K_ i$ is represented by $\bigoplus I_ i^\bullet $ (Injectives, Lemma 19.13.4) we conclude that $R\Gamma (U, \bigoplus K_ i)$ is represented by $\bigoplus H^0(U, I_ i^\bullet )$. Hence $R\Gamma (U, -)$ commutes with direct sums as desired. $\square$
Lemma 21.52.6. 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 $D(\mathcal{O})$.
Proof. Combine Lemmas 21.52.5 and 21.51.1 with Modules on Sites, Lemma 18.30.3. $\square$
Lemma 21.52.7. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Assume $\mathcal{C}$ has the following properties
$\mathcal{C}$ has a quasi-compact final object $X$,
every quasi-compact object of $\mathcal{C}$ has a cofinal system of coverings which are finite and consist of quasi-compact objects,
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.
Let $K$ be a perfect object of $D(\mathcal{O})$. Then
$K$ is a compact object of $D^+(\mathcal{O})$ in the following sense: if $M = \bigoplus _{i \in I} M_ i$ is bounded below, then $\mathop{\mathrm{Hom}}\nolimits (K, M) = \bigoplus _{i \in I} \mathop{\mathrm{Hom}}\nolimits (K, M_ i)$.
If $(\mathcal{C}, \mathcal{O})$ has finite cohomological dimension, i.e., if there exists a $d$ such that $H^ i(X, \mathcal{F}) = 0$ for $i > d$ for any $\mathcal{O}$-module $\mathcal{F}$, then $K$ is a compact object of $D(\mathcal{O})$.
Proof. Let $K^\vee $ be the dual of $K$, see Lemma 21.48.4. Then we have
functorially in $M$ in $D(\mathcal{O})$. Since $K^\vee \otimes _\mathcal {O}^\mathbf {L} -$ commutes with direct sums it suffices to show that $R\Gamma (X, -)$ commutes with the relevant direct sums.
Proof of (a). After reformulation as above this is a special case of Lemma 21.52.4 with $U = X$.
Proof of (b). Since $R\Gamma (X, K) = R\mathop{\mathrm{Hom}}\nolimits (\mathcal{O}, K)$ and since $H^ p(X, -)$ commutes with direct sums by Lemma 21.16.1 this is a special case of Lemma 21.52.5. $\square$
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