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Artin–Tate lemma

From Wikipedia, the free encyclopedia

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

Let A be a commutative Noetherian ring and 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 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.

Notes

  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

References

  • 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

External links

This page was last edited on 1 November 2020, at 23:29
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