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 $f : R \to A$ , in this case $f$ is called a finite morphism if $A$ is a finite $R$ -algebra.

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 $ϕ : V \to W$ , the induced homomorphism of $k$ -algebras $ϕ^{*} : Γ (W) \to Γ (V)$ defined by $ϕ^{*} f = f \circ ϕ$ turns $Γ (V)$ into a $Γ (W)$ -algebra:

ϕ

is a finite morphism of affine varieties if

ϕ^{*} : Γ (W) \to Γ (V)

is a finite morphism of

k

-algebras.

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

References

↑ Atiyah & Macdonald (1969), p. 30.
↑ Perrin (2008), p. 82.

Atiyah, M. F.; Macdonald, I. G. (1969). Introduction to Commutative Algebra. Reading, MA: Addison-Wesley. ISBN 0-201-00361-9.

Perrin, Daniel (2008). Algebraic Geometry: An Introduction. London: Springer. ISBN 978-1-84800-055-1. Zbl 1132.14001.

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[FOOTNOTEAtiyahMacdonald196930-1] Atiyah & Macdonald (1969), p. 30.

[FOOTNOTEPerrin200882-2] Perrin (2008), p. 82.

Finite algebra

Finite morphisms in algebraic geometry

See also

References