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The Jordan-Holder theorem says that any chain of subobjects of a finite length object can be refined to a composition series, and that any composition series has the same length.

This theorem holds for any abelian category, and a notable example is the case of modules over some ring. While I do not need an example of the usefulness of JH theorem in the context of modules, I would like to ask:

What are applications of the JH theorem in a general abelian category, which is not (or not easily proven to be) a category of modules?

[originally asked on mathstackexchange]

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  • $\begingroup$ In the category of modules the JH theorem also says that the simple factors are isomorphic up to permutation. $\endgroup$
    – YCor
    Commented Oct 4, 2019 at 9:09
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    $\begingroup$ 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. $\endgroup$
    – Tim Campion
    Commented Oct 4, 2019 at 13:39
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    $\begingroup$ @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. $\endgroup$
    – less
    Commented Oct 4, 2019 at 14:58
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    $\begingroup$ @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.. $\endgroup$
    – less
    Commented Oct 4, 2019 at 17:22
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    $\begingroup$ @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. $\endgroup$
    – Jeremy Rickard
    Commented Oct 5, 2019 at 7:54

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