The Stacks project

Proposition 74.11.5. Let $S$ be a scheme. Let $X$ be a quasi-compact algebraic space over $S$ with affine diagonal over $\mathbf{Z}$ (as in Properties of Spaces, Definition 65.3.1). Then the functor (74.5.1.1)

\[ D(\mathit{QCoh}(\mathcal{O}_ X)) \longrightarrow D_\mathit{QCoh}(\mathcal{O}_ X) \]

is an equivalence with quasi-inverse given by $RQ_ X$.

Proof. Let $V \to W$ be an étale morphism with $V$ affine and $W$ a quasi-compact open subspace of $X$. Then the morphism $V \to W$ is affine as $W$ has affine diagonal over $\mathbf{Z}$ and $V$ is affine (Morphisms of Spaces, Lemma 66.20.11). Lemma 74.11.1 then guarantees that the assumption of Lemma 74.11.4 holds. Hence we conclude. $\square$

Comments (2)

Comment #450 by on

It's a silly remark, but there is no reference made to in the remainder of the statement, probably you wish to say is over . Moreover, the first line has "faithfu" instead of "faithful".

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