$K$ is $p$, then by the Frobenius endomorphism $\ell \cdot \ell^{p^m-1} = \ell^{p^m} = x_1^{p^m} + \cdots + x_n^{p^m}$. Moreover, as $a \leq p^m$, then $\ell^{p^m} = 0$ in $A$ while $\ell \neq 0$ in $A$. Hence $\times\ell^{p^m-1}: [A]_{1} \rightarrow [A]_{p^m}$ is not injective and thus $A$ does not have the weak Lefschetz property.

Further, for monomial ideals, if the weak Lefschetz property holds in characteristic zero, then it holds for almost every characteristic.

\[lem:wlp-ae\] Let $I$ be an artinian monomial ideal in $R$. If $R/I$ has the weak Lefschetz property when $\operatorname{char}{K} = 0$, then $R/I$ has the weak Lefschetz property for $\operatorname{char}{K}$ sufficiently large.

By Proposition \[pro:mono\], we need only consider $\ell = x_1 + \cdots + x_n$. As $R/I$ is artinian, then there are finitely many maps that need to be checked for the maximal rank property, and this in turn implies finitely many determinants that need to be computed. Further, because of the form of $\ell$, the matrices in question are all zero-one matrices. Thus, the determinants to be checked are integers. Simply let $p$ be the smallest prime larger than all prime divisors of the determinants, then the determinants are all non-zero modulo $p$ and 