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

See also

• Finite morphism
• Finitely generated algebra
• Finitely generated module

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