Remark 1

Let $\delta$ be an integer $> 0$. One sometimes says that an element x of A is superficial of order $\delta$ for M relative to q if $x \in q^\delta$, and if, for all $n$ sufficiently large, the mapping $q^n M/q^{n+1} M \to q^{n+\delta} M/q^{n+\delta+1} M$ induced by multiplication by x is injective. With this terminology, the superficial elements in the sense of Def. 2 are the superficial elements of order 1. 2) With the notation of Prop. 9, x is superficial of order $\delta$ if and only if its class $\xi$ in $\mathrm{gr}_\delta(A)$ does not belong to any of the $p_i$. 3) By III, § 1, No. 4, Prop. 8, there exists a homogeneous element of $\mathrm{gr}(A)$ of degree $> 0$ which does not belong to any of the $p_i$. Consequently, there exist an integer $\delta > 0$ and a superficial element of order $\delta$ for M. 4) Suppose A local with residue field k, and consider the canonical surjective mapping $\lambda : q \to q \otimes_A k$. It is the composite of the canonical mappings $q \to q/q^2$ and $\bar{\lambda} : q/q^2 \to q \otimes_A k$. By Nakayama's lemma, each of the vector subspaces $V_i = \bar{\lambda}(p_i \cap (q/q^2))$ of $q \otimes_A k$ is distinct from $q \otimes_A k$; if $\alpha \in q \otimes_A k$ does not belong to any of the $V_i$, then $\lambda^{-1}(\alpha)$ consists of superficial elements for M (Prop. 9). If k is infinite, the union of the $V_i$ is distinct from $q \otimes_A k$ and there therefore exist superficial elements for M.