A Remark on the Category of Graded F-Modules

TL;DR

This paper demonstrates that the injective hull of the residue field in a polynomial ring over a field of prime characteristic is not injective in the category of graded F-modules, answering a question posed by Lyubeznik-Singh-Walther.

Contribution

It provides a counterexample showing that the injective hull is not injective in the graded F-module category, clarifying the structure of these modules.

Findings

E is not injective in the category of graded F-modules over R.

Answers a previously open question by Lyubeznik-Singh-Walther.

Highlights differences between injective modules and graded F-modules in characteristic p.

Abstract

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.