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 ${\displaystyle B\subset C}$ 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^[2] 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.^[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, 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/wiki/Artin-Tate_lemma