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.Template:Sfnp

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 $V\subseteq {𝔸}^n$, $W\subseteq {𝔸}^m$ and a dominant regular map $\varphi :V\to W$, the induced homomorphism of $k$-algebras ${\varphi }^{\ast }:\mathrm{\Gamma }(W)\to \mathrm{\Gamma }(V)$ defined by ${\varphi }^{\ast }f=f\circ \varphi$ turns $\mathrm{\Gamma }(V)$ into a $\mathrm{\Gamma }(W)$-algebra:

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

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

See also

• Finite morphism
• Finitely generated algebra
• Finitely generated module

References

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