Lemma 3

Suppose that $H$ is generated by $H_0 \cup H_1$ and $M$ generated by $\bigoplus_{i \leq n_0} M_i$ for a suitable integer $n_0$. Then the filtration $(S^{-1}M_{\geq n})$ on $S^{-1}M$ is good for the ideal $S^{-1}H_{\geq 1}$ of $S^{-1}H$.

One has, for $n \geq n_0$, $M_{\geq n+1} = H_1 \cdot M_{\geq n}$, hence $S^{-1}M_{\geq n+1} = H_1 \cdot S^{-1}M_{\geq n} = S^{-1}H_{\geq 1} \cdot S^{-1}M_{\geq n}$.