Talk:Kaplansky's theorem on projective modules

Comments

I have two comments: the proof of the theorem in the article currently does not highlight the point where the assumption that the ring is local is used. This could be improved. Secondly, WP is not a textbook, and this article is practically an excerpt of a textbook. I think the article would gain much from rather highlighting applications of the theorem and also from not addressing the reader directly. Jakob.scholbach (talk) 08:23, 19 December 2019 (UTC)[reply]

The assumption on “local” is needed in the second lemma; I’m preparing the proof and so this should be fixed. (What is crucial is on the behavior of indecomposable decompositions not local; “local” is used to make modules indecomposable, and the discussion of this crucial point is currently missing.) As for the second point, on theorem articles, sometimes what is interesting is the proof (here, for instance the use of countably generated modules) rather than the applications. I personally don’t know interesting applications since often one works with finitely generated projective modules. In this article, I think the focus should be on the proof, which involves many ideas that are of independent interest. —— Taku (talk) 09:37, 19 December 2019 (UTC)[reply]

I have added a quote explaining the significance. If you know any applications, I’m interested to know. — Taku (talk) 09:51, 19 December 2019 (UTC)[reply]

I have also added the formulation of the theorem that characterizes a local ring (this should also address the first comment). -- Taku (talk) 02:44, 20 December 2019 (UTC)[reply]

OK, these steps go in the right direction, I think -- thanks! I am by no means an expert on this topic, but searching around a bit reveals for example a paper by Bass " Big projective modules are free. " where he states it relies on Kaplansky's theorem. More systematically, it may also be instructive to check out the list of papers referring to Kaplansky's paper such as this list of papers whose MR review refers to Kaplansky's paper is also insightful, I guess. Jakob.scholbach (talk) 08:27, 20 December 2019 (UTC)[reply]

Thank you for the info about Bass's result (which I didn't know). I have added it to the article. I think, while I understand the importance of mentioning applications in general, that this theorem (as well as Bass's result) says that big (=not finitely generated) modules are somehow boring as far as the subtleties like a distinction between free and projective modules are concerned. This is really not surprising if you remember algebraic topology: stable problems and situations tend to be easy (or equivalently boring for mathematicians). But this type of a general discussion probably belongs to the main article "projective module". -- Taku (talk) 01:54, 22 December 2019 (UTC)[reply]