Timeline for Applications of Jordan-Holder theorem in an abelian category
Current License: CC BY-SA 4.0
12 events
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| Oct 5, 2019 at 7:54 | comment | added | Jeremy Rickard | @GTA A full exact embedding of abelian categories might not send simple objects to simple objects, so it might not preserve length. In fact, it might not even send finite length objects to finite length objects. | |
| Oct 5, 2019 at 7:34 | history | edited | less |
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| Oct 4, 2019 at 17:22 | comment | added | less | @GTA Of course it does not mean that, so what? Length is not necessarily preserved by an embedding precisely because an embedding is not at the level of objects. Length is defined in terms of equality of subobjects, so its not at all clear that it should be preserved by a full and faithfull exact functor. Thx for the downvote and your thoughtful comment.. | |
| Oct 4, 2019 at 16:39 | comment | added | GTA | Length does not change via embedding. Remember embedding does not just mean embedding at the level of objects. | |
| Oct 4, 2019 at 15:07 | history | edited | less | CC BY-SA 4.0 |
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| Oct 4, 2019 at 14:58 | comment | added | less | @TimCampion Thx for the link. If you use F-M, you prove JH embedding your finite length obj. in a category of modules over some ring which changes every time, for each finite length object. So you don't get a homogeneous notion of length to which your JH theorem refers, since a module can have different lengths depending on the base ring. This may be a problem because I think that length doesn't behave well under equivalence of categories. So it's not even clear to me that the length in the cat. of modules you embed the obj into will be the same as the length in the general abelian cat. | |
| Oct 4, 2019 at 13:39 | comment | added | Tim Campion | Related The thing is, since the Jordan-Holder theorem doesn't do anything infinitary, you can deduce the theorem for any abelian category from the version for modules plus the Freyd-Mitchell embedding theorem. So it's hard to separate the general version from the version for modules. | |
| Oct 4, 2019 at 9:11 | comment | added | less | @YCor Yes, I'm trying to prove this fact also in the case of an abelian category. I think it does still hold at this generality, doesn't it? | |
| Oct 4, 2019 at 9:09 | comment | added | YCor | In the category of modules the JH theorem also says that the simple factors are isomorphic up to permutation. | |
| Oct 4, 2019 at 9:07 | history | edited | YCor | CC BY-SA 4.0 |
fixed typo
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| Oct 4, 2019 at 8:50 | review | First posts | |||
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| Oct 4, 2019 at 8:48 | history | asked | less | CC BY-SA 4.0 |