Finite algebra

In abstract algebra, an associative algebra $A$ over a ring $R$ is called finite if it is finitely generated as an $R$-module. An $R$-algebra can be thought as a homomorphism of rings $f\colon R\to A$, in this case $f$ is called a finite morphism if $A$ is a finite $R$-algebra.

Being a finite algebra is a stronger condition than being an algebra 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 $V\subseteq \mathbb {A} ^{n}$, $W\subseteq \mathbb {A} ^{m}$ and a dominant regular map $\phi \colon V\to W$, the induced homomorphism of $\Bbbk$-algebras $\phi ^{*}\colon \Gamma (W)\to \Gamma (V)$ defined by $\phi ^{*}f=f\circ \phi$ turns $\Gamma (V)$ into a $\Gamma (W)$-algebra:

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

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

See also

• Finite morphism
• Finitely generated algebra
• Finitely generated module