Hilbert-samuel Coefficients and Postulation Numbers of Graded Components of Certain Local Cohomology Modules

Let R = ⊕ n≥0 Rn be a Noetherian homogeneous ring with onedimensional local base ring (R0,m0). Let q0 ⊆ R0 be an m0-primary ideal, let M be a finitely generated graded R-module and let i ∈ N0. Let Hi R+ (M) denote the i-th local cohomology module of M with respect to the irrelevant ideal R+ := ⊕ n>0 Rn of R. We show that the first Hilbert-Samuel coefficient e1 ( q0, Hi R+ (M)n ) of the n-th graded component of Hi R+ (M) with respect to q0 is antipolynomial of degree < i in n. In addition, we prove that the postulation numbers of the components Hi R+ (M)n with respect to q0 have a common upper bound.