On Depth of Rees Modules and Hilbert Functions

Let (A, m) be a Noetherian local ring and q be an m-primary ideal. Let 𝔽 be a good q-filtration and 𝕄 = {M j } j≥0 be a good 𝔽-filtration of a finitely generated A-module M. We associate 𝕄 two graded ℝ(𝔽)-module ℜ(𝕄), ℜ*(𝕄). We investigate their dimensions and depths and prove that if M is a finitely generated Cohen–Macaulay A-module, then ℜ(𝕄) is an almost Cohen–Macaulay ℜ(𝔽)-module if and only if gr 𝕄(M) is an almost Cohen–Macaulay G(𝔽)-module. We establish some connections between the Hilbert function of 𝕄 and the local cohomology module of ℜ(𝕄) and particularly prove that if M is a Cohen–Macaulay A-module of dimension d ≥ 2, then for any n ∈ Z, where is the Hilbert–Samuel polynomial of 𝕄.