\[cor:mac-poly\]).

Compiling the tool-chain {#sec:toolchain}
========================

Let $R = K[x_1, \ldots, x_n]$ be the standard graded $n$-variate polynomial ring over the infinite field $K$, and let $A$ be an artinian standard graded $K$-algebra over $R$. Then the minimal free resolution of $A$ ends with the free module $\bigoplus_{i=1}^{m} R(-t_i)^{r_i},$ where $0 < t_1 < \cdots < t_m$ and $0 < r_i$ for all $i$. In this case, $A$ is called [*level*]{} if $m = 1$, the [*socle degrees*]{} of $A$ are $t_i - n$, for all $i$, and the [*socle type*]{} of $A$ is the sum $\sum_{i=1}^{m} t_i$.

We recall that once multiplication by a general linear form is surjective, then it remains surjective.

[[@MMN Proposition 2.1(a)]]{} \[pro:surj\] Let $A = R/I$ be an artinian standard graded $K$-algebra, and let $\ell$ be a general linear form. If the map $\times \ell: [A]_{d} \rightarrow [A]_{d+1}$ is surjective, then $\times \ell: [A]_{d+1} \rightarrow [A]_{d+2}$ is surjective.

This generalises to modules generated in degrees that are sufficiently small.

\[lem:mod-surj\] Let $M$ be an $R$-module generated in degrees bounded by $e$, and let $\ell$ be a general linear form. If the map $\times\ell: [M]_d \rightarrow [M]_{d+1}$ is surjective, and $d \geq e$, then the 