Preimage of Maximal Ideal of Finitely Generated Algebra is Maximal

Theorem
Let $k$ be a field.

Let $A$ and $B$ be $k$-algebras.

Let $f: A \to B$ be a $k$-algebra homomorphism.

Let $B$ be finitely generated over $k$.

Let $\mathfrak m$ be a maximal ideal of $B$.

Then its preimage $\map {f^{-1} } {\mathfrak m}$ is a maximal ideal of $A$.

Proof
We have an injective morphism:


 * $\dfrac A {\map {f^{-1} } {\mathfrak m} } \to \dfrac B {\mathfrak m}$

We have that $\dfrac B {\mathfrak m}$ is a field extension of $k$ which is finitely generated.

Thus, by Zariski's Lemma, $\dfrac B {\mathfrak m}$ is a finite field extension.

By Subalgebra of Finite Field Extension is Field, $\dfrac A {\map {f^{-1} } {\mathfrak m} }$ is a field.

Thus $\map {f^{-1} } {\mathfrak m}$ is a maximal ideal.

Also see

 * Definition:Spectrum of k-Algebra Functor