  Let $R=k[x,y]$ be a polynomial ring over a field $k$ of prime characteristic
$p$ and let $E$ denote the injective hull of $k$ (which is isomorphic to
$H^2_{(x,y)}(R)$). We prove that $E$ is not an injective object in the category
of graded $F$-modules over $R$. This answers in the negative a question raised
by Lyubeznik-Singh-Walther.
