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 $f:R\to A$, 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 $\varphi :V\to W$, the induced homomorphism of $\mathbb{k}$-algebras ${\varphi }^{*}:\mathrm{\Gamma }(W)\to \mathrm{\Gamma }(V)$ defined by ${\varphi }^{*}f=f\circ \varphi$ turns $\mathrm{\Gamma }(V)$ into a $\mathrm{\Gamma }(W)$-algebra:

$\varphi$ is a finite morphism of affine varieties if ${\varphi }^{*}:\mathrm{\Gamma }(W)\to \mathrm{\Gamma }(V)$ is a finite morphism of $\mathbb{k}$-algebras.

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

See also

• Finite morphism
• Finitely generated algebra
• Finitely generated module