Graded $F$-modules and Local Cohomology

TL;DR

This paper introduces graded F-modules over polynomial rings in positive characteristic, characterizes those with zero-dimensional support, and describes their structure in relation to local cohomology modules, extending understanding of their algebraic properties.

Contribution

It defines graded F-modules as a refinement of F-modules and proves their structure when support is zero-dimensional, linking to local cohomology modules and their graded structure.

Findings

Graded F-modules with zero-dimensional support are direct sums of shifted injective hulls.

Local cohomology modules of certain compositions are isomorphic to finite direct sums of shifted injective hulls.

The results hold in characteristic p > 0; the characteristic zero case remains open.

Abstract

Let $R=k[x_1,..., x_n]$ be a polynomial ring over a field $k$ of characteristic $p>0,$ let $\m=(x_1,..., x_n)$ be the maximal ideal generated by the variables, let $^*E$ be the naturally graded injective hull of $R/\m$ and let $^*E(n)$ be $^*E$ degree shifted downward by $n.$ We introduce the notion of graded $F$-modules (as a refinement of the notion of $F$-modules) and show that if a graded $F$-module $\M$ has zero-dimensional support, then $\M,$ as a graded $R$-module, is isomorphic to a direct sum of a (possibly infinite) number of copies of $^*E(n).$ As a consequence, we show that if the functors $T_1,...,T_s$ and $T$ are defined by $T_{j}=H^{i_j}_{I_j}(-)$ and $T=T_1\circ...\circ T_s,$ where $I_1,..., I_s$ are homogeneous ideals of $R,$ then as a naturally graded $R$-module, the local cohomology module $H^{i_0}_{\m}(T(R))$ is isomorphic to $^*E(n)^c,$ where $c$ is a finite number. If $\text{char}k=0,$ this question is open even for $s=1.$