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Nakayama's lemma

Idea

Nakayama’s lemma is a simple but fundamental result of commutative algebra frequently used to lift information from the fiber of a sheaf over a point (as for example a coherent sheaf over a scheme) to give information on the stalk at that point.

Statement and consequences

Nakayama’s lemma is frequently stated in a general but slightly unilluminating form. We begin with an easier and more intuitive form. In this article, all rings are assumed to be commutative.

Proposition

Let $R$ be a local ring, with maximal ideal $𝔪$ and residue field $k=R/𝔪$. Let $M$ be a finitely generated $R$-module. Then $M\cong 0$ if and only if $k{\otimes }_{R}M\cong 0$.

Here is a sample application. Suppose $f:N\to M$ is an $R$-module map, giving rise to an exact sequence

$$N\stackrel{f}{\to }M\stackrel{p}{\to }M/N\to 0.$$N \stackrel{f}{\to} M \stackrel{p}{\to} M/N \to 0.

Tensoring with $k$ is a right exact functor, so we have an exact sequence

$$k{\otimes }_{R}N\stackrel{k{\otimes }_{R}f}{\to }k{\otimes }_{R}M\to k{\otimes }_{R}M/N\to 0.$$k \otimes_R N \stackrel{k \otimes_R f}{\to} k \otimes_R M \to k \otimes_R M/N \to 0.

Nakayama’s lemma says that if $k{\otimes }_{R}M/N\cong 0$, then $M/N\cong 0$. Equivalently, that if $k{\otimes }_{R}f$ is epic, then $f$ is epic. In particular, to check whether a finite set of elements $v_1,\dots ,v_n$ generates $M$, it suffices to check whether the residue classes $v_i\mathrm{mod}𝔪M$ generate the vector space $M/𝔪M$, which is a linear algebra calculation.

Example

Suppose $O$ is a Noetherian local ring. A typical example is the stalk at a point $p$ of a Noetherian scheme as locally ringed space, and we will write as if we were in that situation. Being Noetherian, its maximal ideal $𝔪$ is finitely generated. Suppose $k{\otimes }_O𝔪\cong 𝔪/{𝔪}^2$ – the cotangent space – is a vector space of dimension $n$. We would like to know whether a collection of functions $f_1,\dots ,f_n$ that vanish at $p$ form a local coordinate system.

For this, it suffices to check whether the differentials $d{f}_1,\dots ,d{f}_n$ at $p$, belonging to the cotangent space $𝔪/{𝔪}^2$, are linearly independent. (For then they span the cotangent space, and one concludes from Nakayama that the $f_i$ generate $𝔪$ as an $O$-module, thereby forming a local coordinate system at $p$.) In this way, Nakayama’s lemma operates as a kind of “inverse function theorem”.

To cement this further, the following statement is offered in Harris as a corollary of Nakayama’s lemma (corollary 14.10, page 179):

Proposition

(Inverse Function Theorem) A map between complex projective varieties of dimension $n$ which is a bijection and has injective derivative at every point is an isomorphism.

We turn now to a general statement of Nakayama’s lemma.

(To be continued)