Artin–Tate lemma
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In algebra, the Artin–Tate lemma, named after John Tate and his father Emil Artin, 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.