Let π : X → X 0 be a projective morphism of schemes, such that X 0 is noetherian and essentially of finite type over a field K. Let i ∈ N 0 , let F be a coherent sheaf of O X -modules and let L be an ample invertible sheaf over X. Let Z 0 ⊆ X 0 be a closed set. We show that the depth of the higher direct image sheafThere are various examples which show that the mentioned asymptotic stability may fail if dim(X 0 ) ≥ 3. To prove our stability result, we show that for a finitely generated graded module M over a homogeneous noetherian ring R = n≥0 R n for which R 0 is essentially of finite type over a field and an ideal a 0 ⊆ R 0 the a 0 -depth of the n-th graded component H i R+ (M ) n of the i-th local cohomology module of M with respect to R + := k>0 R k ultimately becomes constant in codimension ≤ 2 if n tends to −∞.
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