nLab
finitely generated algebra

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)