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
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