Preimage of Maximal Ideal of Finitely Generated Algebra is Maximal
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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:
- $\dfrac A {\map {f^{-1} } {\mathfrak m} } \to \dfrac B {\mathfrak m}$
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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.
$\blacksquare$