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.{{cite book|url=https://www.crcpress.com/Introduction-To-Commutative-Algebra/Atiyah/p/book/9780201407518|title=Introduction to commutative algebra|last1=Atiyah|first1=Michael Francis|author1link = Michael Atiyah|last2=Macdonald|first2=Ian Grant|author2link = Ian G. Macdonald|date=1994|publisher=CRC Press|isbn=9780201407518|page=30}}

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.{{cite book|url=https://www.springer.com/gp/book/9781848000551|title=Algebraic Geometry An Introduction|last=Perrin|first=Daniel|date=2008|publisher=Springer|isbn=978-1-84800-056-8|page=82}}

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

finite algebra

Finite morphisms in algebraic geometry

References

See also