# nLab finitely generated algebra

### Context

#### Algebra

higher algebra

universal algebra

## Higher algebras

• monoidal (?,1)-category?

• symmetric monoidal (?,1)-category?

• monoid in an (?,1)-category?

• commutative monoid in an (?,1)-category?

• symmetric monoidal (∞,1)-category of spectra

• A-? algebra?

• A-? ring?, A-? space?
• C-? algebra?

• E-? ring?, E-? algebra?

• L-? 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 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)