nLab finitely generated algebra

Definition

Definition

Given a commutative ring $R$ and an $R$-algebra $A$, this algebra is finitely generated over $R$ if it is a quotient of a polynomial ring $R[x_1, \cdots, x_n]$ on finitely many variables.

If moreover $A = R[x_1, \cdots, x_n]/(f_1, \cdots, f_k)$ for a finite number of polynomials $f_i$, then $A$ is called finitely presented.

Examples

Example

A morphism of finite presentation between schemes is one which is dually locally given by finitely presented algebras.

Example

A ring is an associative algebra over the integers, hence a $\mathbb{Z}$-ring. Accordingly a finitely generated ring is a finitely generated $\mathbb{Z}$-algebra, and similarly for finitely presented ring.

For rings every finitely generated ring is already also finitely presented.

Related concepts

• morphism of finite type, morphism of finite presentation

• Zariski site

• finitely generated object

  • finitely generated module

  • finitely generated group

References

Finite generation of algebras plays a role in the choice of geometry (for structured (infinity,1)-toposes) in

• Jacob Lurie, section 2.5 of Structured Spaces