# nLab Artin-Tate lemma

Contents

### Context

#### Algebra

higher algebra

universal algebra

# Contents

## Statement

###### Proposition

(Artin & Tate 1951)
Given a Noetherian commutative ring $A$, a commutative algebra $B$ over $A$, and a commutative algebra $C$ over $B$ (and thus a commutative algebra over $A$), if $C$ is a finitely generated algebra over $A$ and $C$ is a finitely generated module over $B$, then $B$ is a finitely generated algebra over $A$. (If $A$ is a commutative subring of commutative ring $B$, then $B$ is a commutative algebra over $A$.)

• Michael Atiyah, Ian G. Macdonald, Introduction to commutative algebra, (1969, 1994) $[$pdf, ISBN:9780201407518$]$