Английская Википедия:Finite algebra

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Шаблон:Jargon In abstract algebra, an associative algebra <math>A</math> over a ring <math>R</math> is called finite if it is finitely generated as an <math>R</math>-module. An <math>R</math>-algebra can be thought as a homomorphism of rings <math>f\colon R \to A</math>, in this case <math>f</math> is called a finite morphism if <math>A</math> is a finite <math>R</math>-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 <math>V\subseteq\mathbb{A}^n</math>, <math>W\subseteq\mathbb{A}^m</math> and a dominant regular map <math>\phi\colon V\to W</math>, the induced homomorphism of <math>\Bbbk</math>-algebras <math>\phi^*\colon\Gamma(W)\to\Gamma(V)</math> defined by <math>\phi^*f=f\circ\phi</math> turns <math>\Gamma(V)</math> into a <math>\Gamma(W)</math>-algebra:

<math>\phi</math> is a finite morphism of affine varieties if <math>\phi^*\colon\Gamma(W)\to\Gamma(V)</math> is a finite morphism of <math>\Bbbk</math>-algebras.

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

See also

• Finite morphism
• Finitely generated algebra
• Finitely generated module

Шаблон:Algebraic-geometry-stub Шаблон:Commutative-algebra-stub