then the map is surjective thereafter. The result then follows by duality.

Further recall that a monomial algebra has the weak Lefschetz property exactly when the sum of the variables is a Lefschetz element.

[[@MMN Proposition 2.2]]{} \[pro:mono\] Let $A = R/I$ be an artinian standard graded $K$-algebra with $I$ generated by monomials. Then $A$ has the weak Lefschetz property if and only if $x_1 + \cdots + x_n$ is a Lefschetz element of $A$.

Hence, the weak Lefschetz property can be decided for monomial ideals, in a small number of cases, by simple invariants. The following lemma is a generalisation of [@LZ Proposition 3.7].

\[lem:wlp-p\] Let $A = R/I$ be an artinian standard graded $K$-algebra with $I$ generated by monomials. Suppose that $a$ is the least positive integer such that $x_i^a \in I$, for $1 \leq i \leq n$, and suppose that the Hilbert function of $R/I$ weakly increases to degree $s+1$. Then, for any positive prime $p$ such that $a \leq p^m \leq s+1$ for some positive integer $m$, $A$ fails to have the weak Lefschetz property in characteristic $p$.

By Proposition \[pro:mono\], we need only consider $\ell = x_1 + \cdots + x_n$. Suppose the characteristic of 