A remark on the category of graded 𝐹-modules

Abstract

Let R = k [ x , y ] R=k[x,y] be a polynomial ring over a field k k of prime characteristic p p and let E E denote the injective hull of k k (which is isomorphic to H ( x , y ) 2 ( R ) H^2_{(x,y)}(R) ). We prove that E E is not an injective object in the category of graded F F -modules over R R . This answers in the negative a question raised by Lyubeznik-Singh-Walther [J. Eur. Math. Soc. (JEMS) 18 (2016), pp. 2545- 2578].