Talk:Nakayama lemma

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Matsumura (Commutative Ring Theory, Cambridge, 1996) mentions that Nakayama believed that this lemma be attributed to Azumaya and Krull; Matsumura uses NAK to denote this lemma. I think this should be mentioned. Kummini 20:59, 13 October 2006 (UTC)

Does a proof of the remark on coherent sheaves require NAK to prove? Take the case of an affine scheme X = SpecR for some commutative ring R, F = \tilde{M} for some finitely generated R-module M and x corresponds to a prime ideal \mathfrak p \in \mathrm{Spec} R. Then the remark can be translated into the following: M_\mathfrak p = 0 if and only if there exists an ideal I \not \subseteq \mathfrak p such that \forall \mathfrak q \not \supseteq I, M_\mathfrak q = 0. One direction is clear. Conversely, if M_\mathfrak p = 0, then there exists f \not \in \mathfrak p, fM = 0 (since M is finitely generated). Now set I = (f). Where did we use NAK? Kummini 21:22, 13 October 2006 (UTC)

The hypothesis is not that Mp = 0 but that Mp / pMp = 0, which only implies Mp = 0 by Nakayama. --128.252.164.145 (talk) 16:17, 30 April 2008 (UTC)

[edit] Who was Nakayama ?

The article doesn't state who the lemma was named after or rather has no details about him apart from his name. DFH 20:45, 27 January 2007 (UTC)

[edit] proof

it says that it's a corrolary of Cramer's rule, but i'm not sure that would be a short proof. anyways, it is a direct corrolary of the generalized version of Cayley-Hamilton theorem for modules. choosing fi=(identity of M), the characteristic polinomial coefficients are: 1, P1, ... Pn, where Pi in I. then 1+P1+...+Pn = 1 (mod I). --itaj 15:15, 11 March 2007 (UTC)