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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 RR be a local ring, with maximal ideal 𝔪\mathfrak{m} and residue field k=R/𝔪k = R/\mathfrak{m}. Let MM be a finitely generated RR-module. Then M0M \cong 0 if and only if k RM0k \otimes_R M \cong 0.

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

NfMpM/N0.N \stackrel{f}{\to} M \stackrel{p}{\to} M/N \to 0.

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

k RNk Rfk RMk RM/N0.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 RM/N0k \otimes_R M/N \cong 0, then M/N0M/N \cong 0. Equivalently, that if k Rfk \otimes_R f is epic, then ff is epic. In particular, to check whether a finite set of elements v 1,,v nv_1, \ldots, v_n generates MM, it suffices to check whether the residue classes v imod𝔪Mv_i mod \mathfrak{m}M generate the vector space M/𝔪MM/\mathfrak{m}M, which is a linear algebra calculation.

Example

Suppose OO is a Noetherian local ring. A typical example is the stalk at a point pp 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\mathfrak{m} is finitely generated. Suppose k O𝔪𝔪/𝔪 2k \otimes_O \mathfrak{m} \cong \mathfrak{m}/\mathfrak{m}^2 – the cotangent space – is a vector space of dimension nn. We would like to know whether a collection of functions f 1,,f nf_1, \ldots, f_n that vanish at pp form a local coordinate system.

For this, it suffices to check whether the differentials df 1,,df nd f_1, \ldots, d f_n at pp, belonging to the cotangent space 𝔪/𝔪 2\mathfrak{m}/\mathfrak{m}^2, are linearly independent. (For then they span the cotangent space, and one concludes from Nakayama that the f if_i generate 𝔪\mathfrak{m} as an OO-module, thereby forming a local coordinate system at pp.) 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 nn 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)

Created on February 19, 2012 08:47:02 by Todd Trimble (74.88.146.52)