The Stacks project

Lemma 36.3.1. Let $X$ be a scheme. 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 Schemes, Section 26.24. $\square$

Comments (0)

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Your email address will not be published.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



View Lemma 36.3.1 as pdf