Fiber-full modules and a local freeness criterion for local cohomology modules

TL;DR

This paper introduces a local criterion for determining when local cohomology modules of graded modules over certain algebras are free over the base ring, characterizing fiber-full modules and analyzing their loci.

Contribution

It provides a new local freeness criterion for local cohomology modules and characterizes fiber-full modules, extending the understanding of their geometric properties.

Findings

Fiber-full modules satisfy the new local freeness criterion.

The fiber-full locus in the spectrum of a base ring is always open.

The fiber-full locus is dense when the base ring is generically reduced.

Abstract

Let $R$ be a finitely generated positively graded algebra over a Noetherian local ring $B$, and $\mathfrak{m} = [R]_+$ be the graded irrelevant ideal of $R$. We provide a local criterion characterizing the $B$-freeness of all the local cohomology modules $\text{H}_\mathfrak{m}^i(M)$ of a finitely generated graded $R$-module $M$. We show that fiber-full modules are exactly the ones that satisfy this criterion. When we change $B$ by an arbitrary Noetherian ring $A$, we study the fiber-full locus of a module in $\text{Spec}(A)$: we show that the fiber-full locus is always an open subset of $\text{Spec}(A)$ and that it is dense when $A$ is generically reduced.