# Artin–Tate lemma

In algebra, the **Artin–Tate lemma**, named after Emil Artin and John Tate, states:[1]

- Let
*A*be a commutative Noetherian ring and algebras over*A*. If*C*is of finite type over*A*and if*C*is finite over*B*, then*B*is of finite type over*A*.

(Here, "of finite type" means "finitely generated algebra" and "finite" means "finitely generated module".) The lemma was introduced by E. Artin and J. Tate in 1951[2] to give a proof of Hilbert's Nullstellensatz.

## Proof

The following proof can be found in Atiyah–MacDonald.[3] Let generate as an -algebra and let generate as a -module. Then we can write

with . Then is finite over the -algebra generated by the . Using that and hence is Noetherian, also is finite over . Since is a finitely generated -algebra, also is a finitely generated -algebra.

## Noetherian necessary

Without the assumption that *A* is Noetherian, the statement of the Artin-Tate lemma is no longer true. Indeed, for any non-Noetherian ring *A* we can define an *A*-algebra structure on by declaring . Then for any ideal which is not finitely generated, is not of finite type over *A*, but all conditions as in the lemma are satisfied.

## References

- Eisenbud, Exercise 4.32
- E Artin, J.T Tate, "A note on finite ring extensions," J. Math. Soc Japan, Volume 3, 1951, pp. 74–77
- Atiyah–MacDonald 1969, Proposition 7.8

- Eisenbud, David,
*Commutative Algebra with a View Toward Algebraic Geometry*, Graduate Texts in Mathematics, 150, Springer-Verlag, 1995, ISBN 0-387-94268-8. - M. Atiyah, I.G. Macdonald,
*Introduction to Commutative Algebra*, Addison–Wesley, 1994. ISBN 0-201-40751-5