The Stacks project

Remark 4.27.3. The motivation for the construction of $S^{-1} \mathcal{C}$ is to “force” the morphisms in $S$ to be invertible by artificially creating inverses to them (at the cost of some existing morphisms possibly becoming identified with each other). This is similar to the localization of a commutative ring at a multiplicative subset, and more generally to the localization of a noncommutative ring at a right denominator set (see [Section 10A, Lam]). This is more than just a similarity: The construction of $S^{-1} \mathcal{C}$ (or, more precisely, its version for additive categories $\mathcal{C}$) actually generalizes the latter type of localization. Namely, a noncommutative ring can be viewed as a pre-additive category with a single object (the morphisms being the elements of the ring); a multiplicative subset of this ring then becomes a set $S$ of morphisms satisfying LMS1 (aka RMS1). Then, the conditions RMS2 and RMS3 for this category and this subset $S$ translate into the two conditions (“right permutable” and “right reversible”) of a right denominator set (and similarly for LMS and left denominator sets), and $S^{-1} \mathcal{C}$ (with a properly defined additive structure) is the one-object category corresponding to the localization of the ring.


Comments (0)

There are also:

  • 12 comment(s) on Section 4.27: Localization in categories

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Your email address will not be published.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 0BM1. Beware of the difference between the letter 'O' and the digit '0'.