Nakayama's Lemma

Statement

Let $(R,\mathfrak m)$ be a local ring with maximal ideal $\mathfrak m$, and let $M$ be a finitely generated R-module (see finitely generated module ).

Nakayama’s Lemma says:

1. If

   $$\mathfrak m M = M,$$

   then $M=0$.

2. More generally, if $N\subseteq M$ and

   $$M = N + \mathfrak m M,$$

   then $M=N$.

A common general form replaces $\mathfrak m$ by any ideal $I\subseteq Jac(R)$.

Cross-links

• Local rings and maximal ideals: local ring , maximal ideal of a local ring
• Residue field: residue field
• Modules: module , finitely generated module
• Common consequence packaged separately: Nakayama corollary

Examples

1. Minimal number of generators via $M/\mathfrak m M$.
   Let $R = k[x,y]_{(x,y)}$, so $\mathfrak m=(x,y)$. Consider $M=\mathfrak m$ as an $R$-module.
   The quotient $\mathfrak m/\mathfrak m^2$ is a 2-dimensional vector space over the residue field $k$, with basis given by the classes of $x$ and $y$.
   Nakayama implies $\mathfrak m$ cannot be generated by 1 element; it needs at least 2 (and in fact $x,y$ generate it).

2. A cyclic module detected mod $\mathfrak m$.
   Let $R=\mathbb Z_{(p)}$ (localization at $(p)$), $\mathfrak m=(p)$, and $M=R/(p^2)$.
   Then $M/\mathfrak m M \cong R/(p)$ is 1-dimensional over $\mathbb F_p$, so Nakayama implies $M$ is generated by one element (indeed, by the class of $1$).

3. “Generating modulo $\mathfrak m$ generates.”
   In any local ring $(R,\mathfrak m)$, if elements $m_1,\dots,m_r\in M$ map to generators of the vector space $M/\mathfrak m M$ over $R/\mathfrak m$, then $m_1,\dots,m_r$ generate $M$.
   This is a direct application of the form $M=N+\mathfrak m M \Rightarrow M=N$ with $N=Rm_1+\cdots+Rm_r$.