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

Let R be a finitely generated positively graded algebra over a Noetherian local ring B , and 𝔪= [R]_+ be the graded irrelevant ideal of R . We provide a local criterion characterizing the B -freeness of all the local cohomology modules H_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 Spec(A) : we show that the fiber-full locus is always an open subset of Spec(A) and that it is dense when A is generically reduced.