The Stacks project

Proposition 72.11.5. Let $S$ be a scheme. Let $X$ be a quasi-compact algebraic space over $S$ with affine diagonal. Then the functor (72.5.1.1)

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

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

Proof. Let $V \to W$ be an ├ętale morphism with $V$ affine and $W$ a quasi-compact open subspace of $X$. Then the morphism $V \to W$ is affine as $V$ is affine and $W$ has affine diagonal (Morphisms of Spaces, Lemma 64.20.11). Lemma 72.11.1 then guarantees that the assumption of Lemma 72.11.4 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".


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Your email address will not be published.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 08H1. Beware of the difference between the letter 'O' and the digit '0'.