# nLab finitely generated algebra

Contents

### Context

#### Algebra

higher algebra

universal algebra

# Contents

## 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.

## References

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

Last revised on April 17, 2021 at 10:34:52. See the history of this page for a list of all contributions to it.