Nakayama corollary: generators mod the maximal ideal lift

Let $(R,\mathfrak m)$ be a local ring, let $M$ be a finitely generated $R$ -module, and let $k=R/\mathfrak m$ be the residue field. The following are consequences of Nakayama's lemma:

Generators lift. If $m_1,\dots,m_r \in M$ map to elements whose images generate $M/\mathfrak m M$ as a $k$ -vector space, then $m_1,\dots,m_r$ generate $M$ as an $R$ -module.

Vanishing test. If $\mathfrak m M = M$ , then $M=0$ . Equivalently, $M=0$ if and only if $M/\mathfrak m M=0$ .

Minimal number of generators. The minimal number of $R$ -module generators of $M$ equals $\dim_k(M/\mathfrak m M)$ . In particular, a nonzero $M$ is cyclic if and only if $\dim_k(M/\mathfrak m M)=1$ .

Interpretation

These statements convert information modulo the maximal ideal into information about the whole module.

Examples

Free modules over a DVR-like local ring. Let $R=k[[t]]$ with $\mathfrak m=(t)$ , and take $M=R^n$ . Then $M/\mathfrak m M \cong k^n$ has $k$ -dimension $n$ , so $R^n$ needs (and has) exactly $n$ generators.

A cyclic module detected mod $\mathfrak m$ . With $R=k[[t]]$ and $\mathfrak m=(t)$ , let $M=R/(t^2)$ . Then $M/\mathfrak m M \cong k$ , so $\dim_k(M/\mathfrak m M)=1$ , and the corollary says $M$ is cyclic (generated by the class of $1$ ).

Localizing the integers. Let $R=\mathbb Z_{(p)}$ with maximal ideal $\mathfrak m=(p)$ , and let $M=(p^2)$ viewed as an $R$ -module (an ideal). Then $M/\mathfrak m M = (p^2)/(p^3)$ is $1$ -dimensional over $R/\mathfrak m \cong \mathbb F_p$ , so $M$ is generated by a single element (namely $p^2$ ).