nLab
finitely generated algebra

Context

Algebra

higher algebra

universal algebra

Algebraic theories

Algebras and modules

Higher algebras

Model category presentations

Geometry on formal duals of algebras

Theorems

Contents

Definition

Definition

Given a commutative ring RR and an RR-algebra AA, this algebra is finitely generated over RR if it is a quotient of a polynomial ring R[x 1,,x n]R[x_1, \cdots, x_n] on finitely many variables.

If moreover A=R[x 1,,x n]/(f 1,,f k)A = R[x_1, \cdots, x_n]/(f_1, \cdots, f_k) for a finite number of polynomials f if_i, then AA is called finiteley 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.

References

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

Revised on November 26, 2013 09:21:20 by Urs Schreiber (145.116.131.59)