Let $A$ be a local ring with nilpotent maximal ideal $\mathfrak{m}$ (i.e., some power of $\mathfrak{m}$ vanishes), and $M$ an $A$-module (not necessarily finitely generated). Let $\bar{S}\subset M/\mathfrak{m}M$ be a set of generators and $S$ a set of representatives of $\bar{S}$ in $M$. Then is it true that $S$ is a set of generators of $M$? This is a common form of Nakayama's lemma with the assumption of finite generation of $M$ replacing the nilpotence of $\mathfrak{m}$. A passage in Matsumura's book "Commutative Ring Theory" (see Theorem 7.10) seems to imply this result, and I can't figure out why.
[Math] Non-finite version of Nakayama’s lemma
ac.commutative-algebramodules
Best Answer
Dear Kwan,
Let $N$ be the submodule of $M$ generated by $S$. Then by assumption $M = N +\mathfrak m M.$ Iterating this, we find that $$M = N + \mathfrak m (N + \mathfrak m M) = N + \mathfrak m^2 M = \cdots = N + \mathfrak m^n M$$ for any $n > 0.$ If we take $n$ large enough then $\mathfrak m^n = 0$ (by hypothesis). Thus $M = N,$ as desired.
P.S. I've found this to be quite a useful fact!