Lemma 7

Retain the notation of lemma 6.

a) One has the inequality $H_{M''}^{(1)} \geq \frac{1 - T^\delta}{1 - T} H_M$.

b) In order that one have equality, it is necessary and sufficient that $\varphi$ be injective.

c) If this is so, one has $M' \subset \bigcap_n F_n$, and the exact sequence of A-modules

where $v$ is the canonical mapping, is exact.

Assertions a) and b) follow from formula (18) of lemma 6, and from the definition $H_M^{(1)} = (1 - T)^{-1}.H_M$.

Suppose that $\varphi$ is injective. By III, § 2, no 8, th. 1, (i), one has

for all $n$, whence the first assertion of c). Moreover there is an exact sequence

where $f'$ is deduced from $f$ by passing to the quotient. If $\varphi$ is injective, one has as above ${f'}^{-1}(F_n) = F_{n-\delta}/M'$. It follows that the filtration on $M/M'$ deduced as inverse image by $f'$ of the filtration $F$ on $M$ is the filtration $n \mapsto F_{n-\delta}/M'$; the associated graded is $\mathrm{gr}(M)(-\delta)$ and there is an exact sequence of graded modules (III, § 2, no. 4, prop. 2)

where $\varphi'_n = \varphi_{n-\delta}$ for all $n$. This completes the proof of c).