Definition:Induced Mapping on Maximal Spectra

Definition

Let $k$ be a field.

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

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

The induced mapping on spectra by $f$ is the mapping $f^* : \operatorname{Max} B \to \operatorname{Max} A$ between their maximal spectra with:

$f^* (\mathfrak m) = f^{-1}(\mathfrak m)$,

the preimage of a maximal ideal $\mathfrak m \in \operatorname{Max} B$.

Induced morphism of locally ringed spaces

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Also see

• Preimage of Maximal Ideal of Finitely Generated Algebra is Maximal
• Definition:Maximal Spectrum Functor
• Definition:Induced Mapping on Spectra of Rings