 Artin–Tate lemma

Let $A\subseteq B\subseteq C$ be rings. If $A$ is Noetherian, $C$ is a finitely generated $A$-algebra, and $C$ is a finite $B$-module, then $B$ is a finitely generated $A$-algebra.

The Artin–Tate lemma is a result in algebraic geometry and number theory that relates to the behavior of certain types of sheaves on algebraic varieties, particularly in the context of étale cohomology. It's often discussed in the context of the study of motives and in particular with K-theory or algebraic cycles.