The Stacks project

20.53 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 20.53.1. 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 20.53.2. 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 20.50.5. 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 20.53.1

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).

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 09J6. Beware of the difference between the letter 'O' and the digit '0'.