Finite algebra

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

Being a finite algebra is a stronger condition than being an algebra of finite type.

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

${\displaystyle \varphi }$ is a finite morphism of affine varieties if ${\displaystyle {\varphi }^{*}:\mathrm{\Gamma }(W)\to \mathrm{\Gamma }(V)}$ is a finite morphism of ${\displaystyle \mathbb{k}}$-algebras.^[2]

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

• Finite morphism
• Finitely generated algebra
• Finitely generated module

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