I promised a minipost on Nakayama when I talked about Flattening Stratifications, and I’ve got a moment now, so I’ll do it quickly. This post is all commutative algebra. So we’ll quickly state Nakyama:
Nakayama’s Lemma: Let be an ideal contained in the Jacobson radical of a ring and let be a finitely generated -module. Then if we have and if have images in which generate it as an -module, then generate as an -module.
First up, the Jacobson radical is just the intersection of all the maximal ideals of a ring. So, for instance, in , the Jacobson radical is zero. Also, this says that we really want to work over a local ring, so for geometry, we want to work at a point and use Nakayama there.
We’ll prove it as a corollary of the following:
Cayley-Hamilton Theorem: Let be a finitely generated -module, and let be an ideal of . Let be an endomorphism with . Then satisfies an equation of the form where the are in .
Proof: Let be generators of . Then each , so we have for and . That is, we have . By multiplying on the left by the adjoint matrix of , we get that the determinant of annihilates each , and so is zero. Expanding the determinant, we get the desired equation. QED
The first corollary of Cayley-Hamilton is that if is a finitely generated -module and is an ideal with , we have such that , by taking to be the identity and .
Now we prove Nakayama.
- We apply the corollary above to get with . Since is in every maximal ideal, is in none of them, so it’s a unit. Thus .
- Now let . We have . So . Now the first part says that , so , and so is generated by the .
That’s it. Nakayama has a nice short proof. Now, here’s a corollary. We define the annihilator of a modules to be . Now if and are two finitely generated modules over , and then . Now, if is local, this reduces to or being zero in the first place.
First, we reduce to the case of a local ring, because if the sum of the annihilators wasn’t , we localize at a prime containing both annihilators, and apply the local result to get a contradiction. Now, we assume that is nonzero and is the unique maximal ideal of . Nakayama says that Since this is a vector space, it projects to , so there is a surjection . Thus surjects to . By Nakayama, we have .
So this should hint that Nakayama is helpful in general, and especially when dealing with tensor products and flatness, as we have recently. It’s a great tool, and we’ll probably be using it in the future a bit as well.