Artin-Tate lemma

Statement

Suppose $A,B,C$ are commutative unital rings, such that:

• $A$ is Noetherian
• $B$ and $C$ are $A$-algebras, with $B$ a subalgebra of $C$
• $C$ is finitely generated as an $A$-algebra (i.e. $C$ is an algebra of finite type over $A$)
• $C$ is finitely generated as a $B$-module (i.e. $C$ is [[[finiteness as an algebra|finite as an algebra]] over $B$)

Then: $B$ is finitely generated as an $A$-algebra

Explanation

The Artin-Tate lemma is related to the general philosophy that if an object is finitely generated, then any subobject of it is also finitely generated provided the subobject is almost the whole object. In our case, $C$ is finitely generated as an $A$-algebra, but it is not necessary that all subalgebras of $C$ are finitely generated as $A$-algebras. However, if the subalgebra is large enough inside $C$, i.e. it is not too deep, then it behaves like $C$, and we can try to show it is finitely generated.

An analogous result in group theory is the statement that any subgroup of finite index in a finitely generated group, is also finitely generated.