Nakayama Says It’s Enough To Know The Derivative

This is a quick note on an idea that I think is worth pursuing, but I’ve mentioned in a previous post (although I was sort of blasé about it.) There is nothing really to prove here, but I feel that Nakayama’s Lemma is hard to conceptualize, and the Jacobson Radial generalization makes an intuitive result  obscure. All modules in this post are finitely generated.

Nakayama’s Lemma: If M is an R module with (R,\mathfrak{m}) local, then M=N+\mathfrac{m}M implies that N=M.

The idea here is that finitely generated modules are closely related to vector spaces in a sense that can be made precise:

Corollary:  a_1, \dots,a_n generate M as a module if and only if their images in the quotient generate M/\mathfrac{m}M as an A/\mathrfac{m}-vector space.

The forward direction is easy (every element is a linear combination, so pass to the quotient) while the backward direction is deduced from the fact that if N=\langle a_1, \dots a_n\rangle is chosen so that \tilde{a_1}, \dots \tilde{a_n} span said vector space, then we know that M=N+\mathfrak{m}M, so that Nakayama implies that N=M.

 

This has a very nice geometric interpretation. If \mathcal{O}_p:=(K[x_1,\dots x_n]/I)_{\mathfrak{p} is considered as a module over itself, then \mathfrak{m}/\mathfrak{m}^2 is the Zariski cotangent space (see here )  and can be made a vector space over the base field. The above Lemma says that knowing a spanning set for for the (co)tangent space recovers all the information required to generate the local ring, or “knowing the linear approximation is enough to know the functions on V locally.”

There are other analogues that are known by Nakayama’s Lemma:

Corollary 2: If  \phi:M \to N is surjective, it is also injective (en epi is an isomorphism for finitely generated modules.)

Corollary 3: if  \phi: M  \to N is an R-linear map (Geometrically respects the structure of a variety) with \phi* surjective on the cotangent space, then \phi was surjective to start with.