On Hilbert-Samuel Coefficients of Graded Local Cohomology Modules

• Published in: Algebras and representation theory 2023-12, Vol.26 (6), p.2383-2397
• Main Authors: Freitas, T. H., Pérez, V. H. Jorge, Lima, P. H.
• Format: Article
• Language: English
• Subjects: Associative Rings and Algebras; Asymptotic properties; Coefficients; Commutative Rings and Algebras; Homology; Mathematics; Mathematics and Statistics; Modules; Non-associative Rings and Algebras
• ISSN: 1386-923X; 1572-9079
• DOI: 10.1007/s10468-022-10178-7

Let R = ⊕ n ∈ ℕ 0 R n be a homogeneous Noetherian ring with local base ring ( R 0 , m 0 ) . Let R + = ⊕ n ∈ ℕ R n denote the irrelevant ideal of R and let M = ⊕ n ∈ ℤ M n be a finitely generated graded R -module. In this paper, we extend the results of Brodmann et al. (Proc. Amer. Math. Soc. 131 , 2977–2985, 2003 ) and Brodmann and Rohrer (Proc. Amer. Math. Soc. 193 , 987–993, 2005 ) when dim ( R 0 ) = 2 . Actually, we show that the Hilbert-Samuel coefficient e 1 ( q 0 , H R + i ( M ) n ) has asymptotic behavior for all n ≪ 0 and also we establish in certain cases the asymptotic behavior of the Hilbert-Samuel coefficient e 2 ( q 0 , H R + i ( M ) n ) for all n ≪ 0, where H R + i ( M ) n is the n -th graded component of the local cohomology H R + i ( M ) and q 0 is an m 0 -primary ideal of R .