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From Wikipedia, the free encyclopedia
In algebra, the Artin–Tate lemma, named after John Tate and his former advisor Emil Artin, states:[1]
(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.
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.
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.
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