On the Category of Graded F Modules

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 H2 (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.