In abstract algebra, an associative algebra over a ring is called finite if it is finitely generated as an -module. An -algebra can be thought as a homomorphism of rings , in this case is called a finite morphism if is a finite -algebra.[1]

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

Finite morphisms in algebraic geometry

edit

This concept is closely related to that of finite morphism in algebraic geometry; in the simplest case of affine varieties, given two affine varieties  ,   and a dominant regular map  , the induced homomorphism of  -algebras   defined by   turns   into a  -algebra:

  is a finite morphism of affine varieties if   is a finite morphism of  -algebras.[2]

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

References

edit
  1. ^ Atiyah, Michael Francis; Macdonald, Ian Grant (1994). Introduction to commutative algebra. CRC Press. p. 30. ISBN 9780201407518.
  2. ^ Perrin, Daniel (2008). Algebraic Geometry An Introduction. Springer. p. 82. ISBN 978-1-84800-056-8.

See also

edit