map $\times\ell: [M]_{d+1} \rightarrow [M]_{d+2}$ is surjective.

Consider the sequence $$[M]_d \stackrel{\times\ell}{\longrightarrow} [M]_{d+1} \rightarrow [M/\ell M]_{d+1} \rightarrow 0.$$ Notice the first map is surjective if and only if $[M/\ell M]_{d+1} = 0.$ By assumption the map is surjective, so $[M/\ell M]_{d+1} = 0$. Hence $[M/\ell M]_{d+2}$ is zero unless there is a generator of $M$ with degree beyond $d$. However, the assumption is that no generators exist with degree beyond $d.$

From this we get a result analogous to [@MMN Proposition 2.1(b)] for non-level algebras.

\[pro:inj\] Let $A = R/I$ be an artinian standard graded $K$-algebra, and let $\ell$ be a general linear form. If the map $\times \ell: [A]_{d-1} \rightarrow [A]_{d}$ is injective, and $d$ is no greater than the smallest socle degree of $A$, then $\times \ell: [A]_{d-2} \rightarrow [A]_{d-1}$ is injective.

The $K$-dual of $A$, $M$, is a shift of the canonical module of $A$ and is generated in degrees that are a linear shift of the socle degrees of $A$. Consider now the map $\times \ell: [M]_i \rightarrow [M]_{i+1}$. Using Lemma \[lem:mod-surj\] we see that once $i$ is at least as large as the largest degree in which $M$ is generated, and the map is surjective, 