Talk:Krull–Akizuki theorem
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missing hypothesis
[edit]The statement of the theorem is incorrect. For a counterexample, take A = , K = , L = , and B = . Then is an infinite ascending chain of ideals in B. I believe the best way to fix it would be to add the requirement that B is also reduced. Jimenfous (talk) 23:52, 6 July 2023 (UTC)
![{\displaystyle \mathbb {Q} [x]/(x^{2})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/29a2b31db445c1002a3eef4a463095b13fdafde0)
- As the main uses of this theorem are when K and L are fields, I guess that the missing hypothesis is that L should be reduced. In other words, "finite extension" should be replaced with "finite reduced ring extension". Unfortunately, I have no textbook under hand that contains this general form of the theorem, and I cannot verify my guess. You or anybody else can do the change after verifying in Bourbaki or in another source that it is correct. D.Lazard (talk) 09:35, 7 July 2023 (UTC)
- I'm not sure what the conventions are for articles of this kind. There are many published proofs of the classical statement in which A is a domain and K and L are fields (and this is the version proved in the Bourbaki reference). There's a proof assuming only that A is reduced (not necessarily a domain), but with L = K = Q(A) (which obviously implies B reduced) in Integral Closure of Ideals, Rings, and Modules in the LMS lecture note series. (Unfortunately, this doesn't directly cover the most important application, to number fields.) I can't find any published proof of the more general form, assuming that L or B is reduced (it doesn't much matter where you impose the condition - they're more or less equivalent), but the proof isn't difficult. First reduce L to KB, then replace A with a finite extension to reduce to the case K = L (i.e., L = Q(A)). Once you've reduced to this case, the existing article outlines a completion of the proof. I'm tempted not only to correct the statement but to add a paragraph outlining the reduction to the case K = L, but I'm not sure this is appropriate. Jimenfous (talk) 16:39, 7 July 2023 (UTC)
- For the general form of the theorem, my first search would be in Kaplanski's and Matsumura's books.
- For a proof for which no reference is known, I consider that WP:CALC applies, if the proof is easy to verify for people accustomed with the field (here commutative algebra). Indeed, WP:Verifiability generally requires a citation, but, in mathematics, it can be obtained by checking a proof.
- For adding the lacking proof, I suggest to rename the section § Proof as "Proof of the case K = L", and to add a section "Reduction to the case K = L". D.Lazard (talk) 17:42, 7 July 2023 (UTC)
- I'm not sure what the conventions are for articles of this kind. There are many published proofs of the classical statement in which A is a domain and K and L are fields (and this is the version proved in the Bourbaki reference). There's a proof assuming only that A is reduced (not necessarily a domain), but with L = K = Q(A) (which obviously implies B reduced) in Integral Closure of Ideals, Rings, and Modules in the LMS lecture note series. (Unfortunately, this doesn't directly cover the most important application, to number fields.) I can't find any published proof of the more general form, assuming that L or B is reduced (it doesn't much matter where you impose the condition - they're more or less equivalent), but the proof isn't difficult. First reduce L to KB, then replace A with a finite extension to reduce to the case K = L (i.e., L = Q(A)). Once you've reduced to this case, the existing article outlines a completion of the proof. I'm tempted not only to correct the statement but to add a paragraph outlining the reduction to the case K = L, but I'm not sure this is appropriate. Jimenfous (talk) 16:39, 7 July 2023 (UTC)