Finite algebra explained

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.^[1]

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 \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.^[2]

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

See also

• Finite morphism
• Finitely generated algebra
• Finitely generated module

Notes and References

1. Book: Introduction to commutative algebra. Atiyah. Michael Francis. Michael Atiyah. Macdonald. Ian Grant. Ian G. Macdonald. 1994. CRC Press. 9780201407518. 30.
2. Book: Perrin, Daniel. Algebraic Geometry An Introduction. 2008. Springer. 978-1-84800-056-8. 82.