A cohomological stability result for projective schemes over surfaces

Abstract

Let π : X → X0 be a projective morphism of schemes such that X0 is noetherian and essentially of finite type over a field K. Let i N0, let F be a coherent sheaf of -modules and let L be an ample invertible sheaf over X. We show that the set of associated points of the higher direct image sheaf ultimately becomes constant if n tends to −∞, provided X0 has dimensione 2. If , this stability result need not hold any more. To prove this, we show that the set of associated primes of the n-th graded component of the i-th local cohomology module of a finitely generated graded module M over a homogeneous noetherian ring which is essentially of finite type over a field becomes ultimately constant in codimension 2 if n tends to −∞.