# 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 ${\displaystyle B\subset C}$ 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 ${\displaystyle x_{1},\ldots ,x_{m}}$ generate ${\displaystyle C}$ as an ${\displaystyle A}$-algebra and let ${\displaystyle y_{1},\ldots ,y_{n}}$ generate ${\displaystyle C}$ as a ${\displaystyle B}$-module. Then we can write

${\displaystyle x_{i}=\sum _{j}b_{ij}y_{j}\quad {\text{and}}\quad y_{i}y_{j}=\sum _{k}b_{ijk}y_{k}}$

with ${\displaystyle b_{ij},b_{ijk}\in B}$. Then ${\displaystyle C}$ is finite over the ${\displaystyle A}$-algebra ${\displaystyle B_{0}}$ generated by the ${\displaystyle b_{ij},b_{ijk}}$. Using that ${\displaystyle A}$ and hence ${\displaystyle B_{0}}$ is Noetherian, also ${\displaystyle B}$ is finite over ${\displaystyle B_{0}}$. Since ${\displaystyle B_{0}}$ is a finitely generated ${\displaystyle A}$-algebra, also ${\displaystyle B}$ is a finitely generated ${\displaystyle A}$-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 ${\displaystyle C=A\oplus A}$ by declaring ${\displaystyle (a,x)(b,y)=(ab,bx+ay)}$. Then for any ideal ${\displaystyle I\subset A}$ which is not finitely generated, ${\displaystyle B=A\oplus I\subset C}$ is not of finite type over A, but all conditions as in the lemma are satisfied.

## References

1. Eisenbud, Exercise 4.32
2. E Artin, J.T Tate, "A note on finite ring extensions," J. Math. Soc Japan, Volume 3, 1951, pp. 74–77
3. 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