Timeline for Are there any nontrivial abelian categories with only finitely many objects?
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
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| Nov 12, 2012 at 21:41 | vote | accept | user27976 | ||
| Nov 11, 2012 at 19:04 | comment | added | François Brunault | Nice construction. I wonder if there is an example which is a full subcategory of $R-\mathrm{Mod}$ where the base ring $R$ is commutative. Is the ring you construct isomorphic to $\mathrm{End}(M)$ for some module $M$ over a commutative ring? | |
| Nov 11, 2012 at 11:48 | comment | added | Jeremy Rickard | In this particular example, I think the localization at the Serre subcategory is the same as factoring out the finite rank maps, since any linear map defined on a subspace of finite codimension can be extended to the whole space, and the dual statement is alao true, so any map in the localization is induced by a genuine linear map. So yes, that gives a more elementary construction, although saying it in terms of localzation means that less explanation is needed for why you get an abelian category. | |
| Nov 10, 2012 at 20:10 | comment | added | Manny Reyes | Just to clarify, while this category is the localization at a Serre subcategory, the endomorphism ring of a countably-infinite dimensional vector space is factored out by the set (ideal) of endomorphisms with finite rank. Correct? | |
| Nov 10, 2012 at 14:48 | comment | added | Todd Trimble | Ah, this looks good! I hadn't hit on the idea of quotienting by the Serre subcategory. | |
| Nov 10, 2012 at 14:33 | history | answered | Jeremy Rickard | CC BY-SA 3.0 |