The Stacks project

Lemma 12.8.2. Let $\mathcal{C}$ be an additive category. Let $S$ be a left or right multiplicative system. Then $S^{-1}\mathcal{C}$ is an additive category and the localization functor $Q : \mathcal{C} \to S^{-1}\mathcal{C}$ is additive.

Proof. By Lemma 12.8.1 we see that $S^{-1}\mathcal{C}$ is preadditive and that $Q$ is additive. Recall that the functor $Q$ commutes with finite colimits (resp. finite limits), see Categories, Lemmas 4.27.9 and 4.27.17. We conclude that $S^{-1}\mathcal{C}$ has a zero object and direct sums, see Lemmas 12.3.2 and 12.3.4. $\square$


Comments (5)

Comment #8867 by on

I think this proof is a little tricky whenever is not locally small (for non-small ) because of the size issues I discuss here. Nevertheless, one can avoid these size issues and show existence of finite products via this argument.

Comment #9219 by on

All categories are small unless mentioned otherwise... discussed elsewhere...

Comment #9222 by on

@#9219 I am aware about the SP convention that everything is a small category (unless otherwise stated). My point is that inside the SP itself there may be an application of this result to some non-small category, like, say, the derived category of some non-small abelian category (like -modules or some category of modules over a scheme). It is not far-fetched that something like this might already appear (although I don't know any instance throughout the SP now).

Comment #9224 by on

That's not a comment on this lemma however, so I am going to leave this lemma as is.

There are also:

  • 3 comment(s) on Section 12.8: Localization

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Your email address will not be published.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 05QE. Beware of the difference between the letter 'O' and the digit '0'.