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

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.