Lemma 5

Let A be a ring, M an A-module and $(P_n), (Q_n)$ two decreasing filtrations on M consisting of submodules. Suppose that one has $P_n \supset Q_n$ and $\operatorname{long}_A(P_n/Q_n) < +\infty$ for every $n \in \mathbf{Z}$ and that there exists an integer $n_1$ such that $Q_{n_1} = M$. In $\mathbf{Z}((T))$, one has the inequalities

It is a matter of proving that one has the inequalities

(15) $\operatorname{long}((P_{n+1} \cap Q_n)/Q_{n+1}) \leq \operatorname{long}(P_{n+1}/Q_{n+1})$, (16) $\operatorname{long}(P_{n+1}/Q_{n+1}) \leq \sum_{i \leq n} \operatorname{long}((P_{i+1} \cap Q_i)/Q_{i+1})$.

The first is obvious. On the other hand, one has $P_{n+1} \cap Q_i = P_{n+1}$ for $i \leq n_1$ and $P_{n+1} \cap Q_{n+1} = Q_{n+1}$; whence one deduces the inequality

But the A-module $(P_{n+1} \cap Q_i)/(P_{n+1} \cap Q_{i+1})$ is isomorphic to a submodule of $(P_{i+1} \cap Q_i)/Q_{i+1}$, and inequality (16) follows.