Cayley-Hamilton Theorem II

This post proves the Cayley-Hamilton for finite -modules, and generalize it to Nakayama’s lemma.

It is written in the wikipedia page that Cayley-Hamilton actually holds in any commutative ring with unity as well. Here I will present a proof which should be the same as the one in wikipedia.

In the old proof, we considered the vector space , and our argument used the concept of basis. We cannot copy our proof to the new case because the basis of modules does not behave that nicely. Anyway we can at least consider the matrix as a -module homomorphism from to itself, i.e. we are trying to regard as a -module.

Second proof

By definition, we have that , this means that

Multiplying both sides by the classical adjoint of the square matrix on the left. This immediately implies that

i.e. is the zero endomorphism, which is precisely what we want.

Generalization

This proof can be easily adapted to the following theorem that can be found in Chapter 2 of Atiyah and MacDonald’s book.

Theorem 1 Let be a commutative ring with unity, be a finitely generated module, and be an ideal. Let be a -module homomorphism such that
. Suppose that can be generated by elements. Then satisfies a monic polynomial of degree with coefficients, except the leading one, in .

Applying the above theorem to the identity map, we get

Corollary 1 Let be a commutative ring with unity, be a finitely generated module, and be an ideal such that . Then there exists such that and .

Recall that the Jacobson radical of a commutative ring is the intersection of all its maximal ideals. In particular, if then must be a unit. We then obtain the following version of Nakayama’s lemma,

Corollary 2 (Nakayama’s lemma) Let be a commutative ring with unity, be a finitely generated module, and be an ideal contained in such that . Then .

Reference

Atiyah, M. F.; MacDonald, I. G. (1969), Introduction to Commutative Algebra