Graded $F$-modules and Local Cohomology

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 ${}^{\ast }E$ be the naturally graded injective hull of $R/\m$ and let ${}^{\ast }E(n)$ be ${}^{\ast }E$ 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 ${}^{\ast }E(n).$ 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 ${}^{\ast }E(n{)}^c,$ where $c$ is a finite number. If $\text{char}k=0,$ this question is open even for $s=1.$

Revised result in section 3

Cited by in corpus (2)

• Local cohomology -- an invitation
• A Remark on the Category of Graded F-Modules