The Stacks project

Lemma 75.5.3. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Then $D_\mathit{QCoh}(\mathcal{O}_ X)$ has direct sums.

Proof. By Injectives, Lemma 19.13.4 the derived category $D(\mathcal{O}_ X)$ has direct sums and they are computed by taking termwise direct sums of any representatives. Thus it is clear that the cohomology sheaf of a direct sum is the direct sum of the cohomology sheaves as taking direct sums is an exact functor (in any Grothendieck abelian category). The lemma follows as the direct sum of quasi-coherent sheaves is quasi-coherent, see Properties of Spaces, Lemma 66.29.7. $\square$

Comments (2)

Comment #1102 by on

I think the convention (in the Stacks project, and in English in general) is to write Grothendieck abelian category, not grothendieck abelian category.

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