The Stacks project

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. $\square$

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