Lemma 6

Let A be a ring, M an A-module and $(F_n)$ a decreasing filtration on M consisting of submodules; suppose that there exists an integer $n_1$ such that $F_{n_1} = M$. Let f be an endomorphism of M, $M'$ its kernel and $M''$ its cokernel. We equip $M'$ with the filtration $(F'_n)$ induced by $(F_n)$ and $M''$ with the quotient filtration $(F''_n)$ of $(F_n)$. Suppose that $F_n/F_{n+1}$ is of finite length for every $n \in \mathbf{Z}$ and that there exists an integer $\delta$ such that $f(F_n) \subset F_{n+\delta}$. Let $\varphi$ be the graded endomorphism of degree $\delta$ of the graded module $\mathrm{gr}(M) = \bigoplus_{n \in \mathbf{Z}} F_n/F_{n+1}$ deduced from f. Between the following elements of $\mathbf{Z}((T))$

one has the inequalities

(17) $H_{M'} \leq P_{\mathrm{Ker}(\varphi)}$

(18) $(1 - T^\delta).H_M^{(1)} + T^\delta.P_{\mathrm{Ker}(\varphi)} \leq H_{M''}^{(1)} \leq (1 - T^\delta).H_M^{(1)} + T^\delta.P_{\mathrm{Ker}(\varphi)}^{(1)}$.

The sequence of submodules $G_n = f^{-1}(F_{n+\delta})$ of $M$ is a decreasing filtration, and one has $F_n \subset G_n$ for every integer $n$. By definition, one has $\mathrm{Ker}(\varphi_n) = (G_{n+1} \cap F_n)/F_{n+1}$, whence

(19) $P_{\mathrm{Ker}(\varphi)} = \sum_{n \in \mathbf{Z}} \mathrm{long}_A((G_{n+1} \cap F_n)/F_{n+1}).T^n$.

For every $n$, the A-module $(M' \cap F_n)/(M' \cap F_{n+1})$ identifies with a submodule of $(G_{n+1} \cap F_n)/F_{n+1}$, and inequality (17) follows at once from (19). By lemma 5, one has moreover

(20) $P_{\mathrm{Ker}(\varphi)} \leq \sum_{n \in \mathbf{Z}} \mathrm{long}_A(G_{n+1}/F_{n+1}).T^n \leq P_{\mathrm{Ker}(\varphi)}^{(1)}$.

For every $n \in \mathbf{Z}$, one has an exact sequence of A-modules

where $f_n$ is deduced from $f$ by passing to quotients. Consequently one has

$\mathrm{long}_A(M''/F''_{n+\delta+1}) = \mathrm{long}_A(M/F_{n+\delta+1}) - \mathrm{long}_A(M/F_{n+1}) + \mathrm{long}_A(G_{n+1}/F_{n+1})$.

Multiplying by $T^{n+\delta}$ and summing over $n$, one obtains

(21) $H_{M''}^{(1)} = (1 - T^\delta)H_M^{(1)} + T^\delta.\sum_{n \in \mathbf{Z}} \mathrm{long}_A(G_{n+1}/F_{n+1}).T^n,$

and inequality (18) follows at once from (20) and (21).