Proposition 9

Let us denote by $p_1, ..., p_r$ those prime ideals of the graded ring $\mathrm{gr}(A) = \bigoplus_n (q^n/q^{n+1})$ which are associated with the graded module $\mathrm{gr}(M) = \bigoplus_n (q^n M/q^{n+1} M)$ and do not contain $\mathrm{gr}_1(A) = q/q^2$. Let $\delta$ be an integer $> 0$, $\xi$ an element of $\mathrm{gr}_\delta(A)$, and $\varphi : \mathrm{gr}(M) \to \mathrm{gr}(M)$ the homothety of ratio $\xi$ in $\mathrm{gr}(M)$. For $\varphi_n$ to be injective for every $n$ sufficiently large, it is necessary and sufficient that $\xi$ belong to none of the $p_i$. In fact, the prime ideals associated with the $\mathrm{gr}(A)$-module $\mathrm{Ker}\, \varphi$ are those of the prime ideals associated with $\mathrm{gr}(M)$ which contain $\xi$ (IV, § 1, No. 1, Def. 1). By Lemma 3, $(\mathrm{Ker}\, \varphi)_n$ is zero for $n$ sufficiently large, if and only if all these ideals contain $\mathrm{gr}_+(A)$ (or, what amounts to the same, $\mathrm{gr}_1(A)$), whence the proposition.