Proposition 5

Suppose $H_0$ local and artinian, $H$ generated by $H_0 \cup H_1$, $H_1$ a finitely generated $H_0$-module, $M$ nonzero and a finitely generated $H$-module. Then $M_n$ is an $H_0$-module of finite length for each $n$, and there exists $Q(T) \in \mathbf{Z}[T, T^{-1}]$ such that $Q(1) > 0$ and that one has in the ring $\mathbf{Z}((T))$

where $d = \dim_H(M)$.

The ring $S^{-1}H$ is local and noetherian (No. 1, remark 1), the $S^{-1}H$-module $S^{-1}M$ is nonzero and finitely generated, and of dimension $d = \dim_H(M)$ (prop. 4). Moreover, $S^{-1}H_{\geq 1}$ is an ideal of definition of $S^{-1}H$ (§ 3, No. 2, lemma 2), and $S^{-1}M_{\geq n}$ is an $S^{-1}H_{\geq 1}$-good filtration on $S^{-1}M$ (lemma 3). Finally, one has $\operatorname{long}_{S^{-1}H} S^{-1}M_{\geq n}/S^{-1}M_{\geq n+1} = \operatorname{long}_{H_0}(M_n)$ for all $n$. It is enough therefore to apply Theorems 2 and 3 of § 4 (Nos. 3 and 4).