Remark 1

It amounts to saying that $A$ satisfies the following condition: every integral $A$-algebra $B$ integral over $A$, contained in a finitely generated extension of the field of fractions $K$ of $A$, is a finite $A$-algebra. In fact, the field of fractions $L$ of $B$ is an algebraic extension of $K$, and therefore is of finite degree over $K$ (A, V, p. 112, cor. 1 of prop. 17). The $A$-algebra $B$ is contained in the integral closure of $A$ in $L$, and is therefore finite if the latter is finite.