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.