Artin–Tate lemma

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.

generate

generate

Then we can write with

{\displaystyle b_{ij},b_{ijk}\in B}

generated by the

{\displaystyle b_{ij},b_{ijk}}

is a finitely generated

is a finitely generated

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

{\displaystyle (a,x)(b,y)=(ab,bx+ay)}

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.