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20.48 Compact objects

In this section we study compact objects in the derived category of modules on a ringed space. We recall that compact objects are defined in Derived Categories, Definition 13.37.1. On suitable ringed spaces the perfect objects are compact.

Lemma 20.48.1. Let $X$ be a ringed space. Let $j : U \to X$ be the inclusion of an open. The $\mathcal{O}_ X$-module $j_!\mathcal{O}_ U$ is a compact object of $D(\mathcal{O}_ X)$ 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). Since $\mathop{\mathrm{Hom}}\nolimits (j_!\mathcal{O}_ U, \mathcal{F}) = \mathcal{F}(U)$ by Sheaves, Lemma 6.31.8 we have $\mathop{\mathrm{Hom}}\nolimits (j_!\mathcal{O}_ U, K) = R\Gamma (U, K)$ for $K$ in $D(\mathcal{O}_ X)$. 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 $D(\mathcal{O}_ X)$. 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 20.48.2. Let $X$ be a ringed space. Assume that the underlying topological space of $X$ has the following properties:

  1. $X$ is quasi-compact,

  2. there exists a basis of quasi-compact open subsets, and

  3. the intersection of any two quasi-compact opens is quasi-compact.

Let $K$ be a perfect object of $D(\mathcal{O}_ X)$. Then

  1. $K$ is a compact object of $D^+(\mathcal{O}_ X)$ in the following sense: if $M = \bigoplus _{i \in I} M_ i$ is bounded below, then $\mathop{\mathrm{Hom}}\nolimits (K, M) = \bigoplus _{i \in I} \mathop{\mathrm{Hom}}\nolimits (K, M_ i)$.

  2. If $X$ has finite cohomological dimension, i.e., if there exists a $d$ such that $H^ i(X, \mathcal{F}) = 0$ for $i > d$, then $K$ is a compact object of $D(\mathcal{O}_ X)$.

Proof. Let $K^\vee $ be the dual of $K$, see Lemma 20.46.4. 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 it suffices to show that $R\Gamma (X, -)$ commutes with the relevant direct sums.

Proof of (b). Since $R\Gamma (X, K) = R\mathop{\mathrm{Hom}}\nolimits (\mathcal{O}_ X, K)$ and since $H^ p(X, -)$ commutes with direct sums by Lemma 20.19.1 this is a special case of Lemma 20.48.1

Proof of (a). Let $\mathcal{I}_ i$, $i \in I$ be a collection of injective $\mathcal{O}_ X$-modules. By Lemma 20.19.1 we see that

\[ H^ p(X, \bigoplus \nolimits _{i \in I} \mathcal{I}_ i) = \bigoplus \nolimits _{i \in I} H^ p(X, \mathcal{I}_ i) = 0 \]

for all $p$. Now if $M = \bigoplus M_ i$ is as in (a), then 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 $\mathcal{O}_ X$-modules $\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

\[ R\Gamma (X, M) = \Gamma (X, \bigoplus \mathcal{I}_ i^\bullet ) = \bigoplus \Gamma (X, \bigoplus \mathcal{I}_ i^\bullet ) = \bigoplus R\Gamma (X, M_ i) \]

as desired. $\square$


Comments (2)

Comment #2116 by BB on

This is just a suggestion: the notation of Lemma 20.41.11 is \vee for the dual, so it would be better to harmonize than to use \hat for the dual (unless there's a good reason to do so).


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