nLab Artin-Tate lemma

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 $A$.)

See also

• Noetherian ring

• commutative algebra

References

The original article:

• Emil Artin, John T. Tate, A note on finite ring extensions, J. Math. Soc Japan, Volume 3 (1951) 74–77 $[$doi:10.2969/jmsj/00310074, pdf$]$

Review:

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

• David Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry, Graduate Texts in Mathematics, 150, Springer-Verlag, 1995, ISBN:0-387-94268-8