Artin–Tate lemma


In algebra, the Artin–Tate lemma, named after Emil Artin and John Tate, states:
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 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.