A few days ago, some classmates and I were thinking about about whether or not morphisms in were well defined. In particular, given a map of -algebras , we consider the induced morphism given by . We all knew that pullbacks of maximal ideals need not be maximal for ring morphisms, which is precisely what led to a great deal of confusion– thankfully this was all fixed as soon as we were reminded that we were considering morphisms of -algebras. While this was funny, I think the proof for this fact is both an elementary and interesting check up in commutative algebra:
Theorem 1: let be finitely generated -algebras. Then the pullback of any maximal ideal in a -algebra homomorphism is again maximal
We will nee two lemmas:
Lemma 1: Let be a finite extension of rings with a domain. Then is a field if and only if is a field.
Proof: Suppose that is a field. Then is a vector space over and (multiplication by a non-zero element) is an -linear endomorphism with trivial kernel ( is a domain!), and so it must be surjective as well. But then there exists some element so that .
Now suppose is a field. Choose to be nonzero, implying that . Then, by Cayley-Hamilton, there exist so that . We clear denominators and see that .
Nice, so we have some good stuff. The next result is important and there are many proofs of it, so I won’t include any here, but here is the statement and a reference.
Lemma 2 (Zariski’s Lemma): if is a field, and a finite integral domain over and also a field, then is an algebraic extension of .
Proof: see here for a one liner.
In particular, this means that for all maximal ideals , (with a f.g -algebra) we know that is a finite field extension of . From this, the main theorem follows readily:
Proof of the main theorem: let be finitely generated -algebras with a domain and a -algebra homomorphism. Then for each maximal ideal , we have by Zariski’s lemma that is a finite field extension of . This in turn implies that is also a finite field extension since . But then by Lemma 1, it is a field, so must have been maximal.
So, is a contravariant functor.
Dig the title of this post. Now peek at the title of this blog. The little things matter :).