Finite algebra

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

The definition of finite algebra is related to that of algebras 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 [math]\displaystyle{ V\subset\mathbb{A}^n }[/math], [math]\displaystyle{ W\subset\mathbb{A}^m }[/math] and a dominant regular map [math]\displaystyle{ \phi\colon V\to W }[/math], the induced homomorphism of [math]\displaystyle{ \Bbbk }[/math]-algebras [math]\displaystyle{ \phi^*\colon\Gamma(W)\to\Gamma(V) }[/math] defined by [math]\displaystyle{ \phi^*f=f\circ\phi }[/math] turns [math]\displaystyle{ \Gamma(V) }[/math] into a [math]\displaystyle{ \Gamma(W) }[/math]-algebra:

[math]\displaystyle{ \phi }[/math] is a finite morphism of affine varieties if [math]\displaystyle{ \phi^*\colon\Gamma(W)\to\Gamma(V) }[/math] is a finite morphism of [math]\displaystyle{ \Bbbk }[/math]-algebras.[2]

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

## References

1. ↑ Atiyah, Michael Francis; MacDonald, Ian Grant (1994). Introduction to commutative algebra. CRC Press. p. 30. ISBN 9780201407518. https://www.crcpress.com/Introduction-To-Commutative-Algebra/Atiyah/p/book/9780201407518.
2. ↑ Perrin, Daniel (2008). Algebraic Geometry An Introduction. Springer. p. 82. ISBN 978-1-84800-056-8. https://www.springer.com/gp/book/9781848000551.

## See also

* Finite morphism
* Finitely generated algebra
* Finitely generated module

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