For a general ring homomorphism, the preimage of a prime ideal stays prime. The situation is more complicated for maximal ideals: it’s not true in general that a maximal ideal will stay maximal when pulled back! For a cheap example, look at the injection , for which the (maximal) ideal of gets pulled back to the minimal ideal of . This technical advantage of prime ideals over maximal ideals was one of the driving forces behind scheme theory.
In this post, we will explore one particular situation where maximal ideals are pullback-friendly: when the rings are -algebras, and the target is of finite type.
The notation will always denote some (any) field. Rings are commutative with unity.
Before we begin, we need a small technical lemma. Interesting in its own right, we will use this result to conclude that an integral domain is a field as soon as it admits a structure of a finite-dimensional vector space.
Lemma. If is an integral domain that is also a finite algebra over , then is, in fact, a field.
Proof. Let be some non-zero element of . The main idea is to show that the function is a -linear automorphism. To do this, it suffices to show that is an injection, because is a finite-dimensional vector space. But this is clear, since is an integral domain.
Parenthetically, notice that any integral domain admits a trivial vector space structure (multiplication by any scalar does nothing). Hence, if the integral domain has a finite number of elements, it is trivially a finite-dimensional vector space: by the lemma, it is a field!
Now we state the main point of this post:
Proposition. Let be a morphism of algebras over , and suppose further that is of finite type over . Then the preimage by of any maximal ideal of is a maximal ideal of .
In schematic terms, the induced morphism maps closed points to closed points, so it corresponds more closely to one’s intuition of what a morphism of varieties “should” be!
Proof. Suppose is a maximal ideal of . We want to show that the quotient ring is a field. In general, the preimage of any maximal ideal is prime, so we already know is an integral domain. We’ll show that it is also a finite algebra over , and then apply the lemma.
The morphism induces a canonical map . It fits in the following commutative diagram of -algebras and -algebra homomorphisms: Let’s investigate the kernel of . Pick an arbitrary element in (where and the overline denotes passing to the quotient, as usual). We have Hence the kernel is trivial, that is, is an injective morphism.
As a last step, recall Zariski’s lemma (sometimes also called the Nullstellensatz, although it seems to me like this term is overloaded!): if a field is of finite type as an algebra over some other field , then is in fact finite over , i.e. . In our case, applying gives that is actually a finite-dimensional vector space over . But now is isomorphic to a subspace of via the injection , so it is also a finite-dimensional vector space over .