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In commutative algebra, sometimes people consider the polynomial ring over a field as a local ring and they uses the Nakayama's lemma to get some informations about the generators of a finitely generated module over that polynomial ring.

My question is : Why can they do that ?

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    This is pretty unclear. Do you mean: why is Nakayama's lemma true? Why does it apply to polynomial rings over fields? Why can we localize polynomial rings over fields? Why does this give us information about finitely generated modules?2012-10-21

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Nobody in his right mind considers a polynomial ring over anything as a local ring because , well, it just isn't one!
However there are similarities with local rings and indeed there is the following analogue of Nakayama's "lemma":

Graded Nakayama
Let $R=\oplus_{n\in \mathbb N} R_n$ be a positively graded ring and $M=\oplus_{n\in \mathbb Z} M_n$ a graded $R$-module.
Then a (maybe infinite) family $(m_i)$ of homogeneous elements of $M$ generate $M$ as an $R$-module if and only if the residue classes $\bar m_i\in M/R^+M$ generate $ M/R^+M$ as an $R/R^+=R_0$-module.
(As usual $R^+=\oplus_{n \geq 1} R_n$)
[Notice that there is absolutely no finiteness hypothesis on the $R$-module $M$]

The algebraic geometric explanation of the similarity is that a graded ring gives rise to a cone and there is a strong interaction between that cone and the local ring of its vertex.