Exercise 4, § 6

Let us take up again the notations and hypotheses of Exercise 2. We say that the filtration $\mathscr{F}$ is a filtration by quasi-powers if there exists an integer $k > 0$ such that

Prove that if the ring $A$ is noetherian, the following conditions are equivalent: (i) The filtration $\mathscr{F}$ is a filtration by quasi-powers. (ii) The ring $A$ is noetherian. (iii) The ideal $\mathscr{N} = \sum_{i \geqslant 1} v^{-i} \mathscr{F}_i$ of $A$ is of finite type. (iv) There exists an integer $k \geqslant 1$ such that $\mathscr{F}$ is the smallest filtration of $A$, for the relation of order « $\mathscr{J}_n \subset \mathscr{G}_n$ for all $n$ » whose first terms are $\mathscr{F}_1, ..., \mathscr{F}_k$. (v) The $A$-algebra $A$ is of finite type.