Finite algebra

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An ${\displaystyle R}$-algebra ${\displaystyle A}$ is finite if it is finitely generated as an ${\displaystyle R}$-module. An ${\displaystyle R}$-algebra can be thought as a homomorphism of rings ${\displaystyle f\colon R\to A}$, in this case ${\displaystyle f}$ is called a finite morphism if ${\displaystyle A}$ is a finite ${\displaystyle R}$-algebra.^[1]

The definition of finite algebra is related to that of algebras of finite type.

Finite morphisms in algebraic geometry[]

This concept is closely related to that of finite morphism in algebraic geometry; in the simplest case of affine varieties, given two affine varieties ${\displaystyle V\subset \mathbb {A} ^{n}}$, ${\displaystyle W\subset \mathbb {A} ^{m}}$ and a dominant regular map ${\displaystyle \phi \colon V\to W}$, the induced homomorphism of ${\displaystyle \Bbbk }$-algebras ${\displaystyle \phi ^{*}\colon \Gamma (W)\to \Gamma (V)}$ defined by ${\displaystyle \phi ^{*}f=f\circ \phi }$ turns ${\displaystyle \Gamma (V)}$ into a ${\displaystyle \Gamma (W)}$-algebra:

${\displaystyle \phi }$ is a finite morphism of affine varieties if ${\displaystyle \phi ^{*}\colon \Gamma (W)\to \Gamma (V)}$ is a finite morphism of ${\displaystyle \Bbbk }$-algebras.^[2]

The generalisation to schemes can be found in the article on finite morphisms.

References[]

See also[]

• Finite morphism
• Finitely generated algebra
• Finitely generated module

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