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In algebra, the Artin–Tate lemma, named after John Tate and his former advisor Emil Artin, states:

Let A be a commutative Noetherian ring and ${\displaystyle B\subset C}$ commutative 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 to give a proof of Hilbert's Nullstellensatz.

The lemma is similar to the Eakin–Nagata theorem, which says: if C is finite over B and C is a Noetherian ring, then B is a Noetherian ring.

Proof

The following proof can be found in Atiyah–MacDonald. 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, David, Commutative Algebra with a View Toward Algebraic Geometry, Graduate Texts in Mathematics, 150, Springer-Verlag, 1995, ISBN 0-387-94268-8, 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. M. Atiyah, I.G. Macdonald, Introduction to Commutative Algebra, Addison–Wesley, 1994. ISBN 0-201-40751-5. Proposition 7.8

External links

• http://commalg.subwiki.org/Artin-Tate_lemma

• Theorems about algebras
• Lemmas in algebra
• Commutative algebra