Annihilator of local cohomology of homogeneous parts of a graded module.

Let S = ⊕ n ≥ 0 S n be a homogeneous graded ring, where (S 0 , 𝔪) is a Noetherian local ring. Let M = ⊕ n ≥ 0 M n be a finitely generated graded S -module. For n , i ∈ ℕ , set 𝔞 i (M n) : = Ann S 0 H 𝔪 i (M n). Denote by Var (𝔞 i (M n)) the set of all prime ideals of S 0 containing 𝔞 i (M n). For r ∈ ℕ , let (Var (𝔞 i (M n))) ≥ r be the set of all 𝔭 ∈ Var (𝔞 i (M n)) such that dim (S 0 / 𝔭) ≥ r. In this paper, we prove that the sets ⋃ j ≤ i Var (𝔞 j (M n)) and Var (𝔞 i (M n)) ≥ i − 1 do not depend on n for n ≫ 0. We show that the annihilators 𝔞 0 (M n) , 𝔞 d (M n) are eventually stable, where d = dim S 0 M n for n ≫ 0. As an application, we prove the asymptotic stability of some loci contained in the non-Cohen–Macaulay locus of M n . [ABSTRACT FROM AUTHOR]