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 is an module with local, then implies that .
The idea here is that finitely generated modules are closely related to vector spaces in a sense that can be made precise:
Corollary: generate as a module if and only if their images in the quotient generate as an -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 is chosen so that span said vector space, then we know that , so that Nakayama implies that .
This has a very nice geometric interpretation. If is considered as a module over itself, then 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 locally.”
There are other analogues that are known by Nakayama’s Lemma:
Corollary 2: If is surjective, it is also injective (en epi is an isomorphism for finitely generated modules.)
Corollary 3: if is an -linear map (Geometrically respects the structure of a variety) with surjective on the cotangent space, then was surjective to start with.