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

  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


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