4
$\begingroup$

I have read that Nakayama's Lemma has a nice consequence for local rings. If $R$ is a local ring with a finitely generated module $M$ and maximal ideal $\mathfrak{m}$, then we see $M/\mathfrak{m}M$ is a vector space over the field $R/\mathfrak{m}$. If a set $\{b_1,\dots,b_n\}$ is a basis of $M/\mathfrak{m}M$, then pulling the $b_i$ back to $M$ gives a set of generators of $M$.

What's the reason for this?

  • 6
    Consider the submodule of $M$ generated by the pulled-back elements and.. use Nakayama's lemma!2011-10-22
  • 1
    Read Theorem 2.3 in Matsumura's *Commutative ring theory*.2011-11-28

1 Answers 1