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2009•Communications in AlgebraRequires access

Upper and Lower Bounds for Finiteness of Graded Local Cohomology Modules

Reza Sazeedeh

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Abstract

Let R = ⨁ n ∈ ℕ0 R n be a noetherian homogeneous ring with local base ring (R 0, 𝔪0) and irrelevant ideal R +, let M be a finitely generated graded R-module. We shall define the upper finiteness dimension (lower finiteness dimension) of M with respect to R + and 𝔪0, denoted by u R +, 𝔪0 (M)(l R +, 𝔪0 (M)) and we prove that are artinian for all i ≤ u R +, 𝔪0 (M) (i ≥ l R +, 𝔪0 (M)). These results yield us to get some similar results for new finiteness dimensions induced by minimax and cofinite graded modules.

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What this paper is about

Let R = ⨁ n ∈ ℕ0 R n be a noetherian homogeneous ring with local base ring (R 0, 𝔪0) and irrelevant ideal R +, let M be a finitely generated graded R-module. We shall define the upper finiteness dimension (lower finiteness dimension) of M with respect to R + and 𝔪0, denoted by u R +, 𝔪0 (M)(l R +, 𝔪0 (M)) and we prove that are artinian for all i ≤ u R +, 𝔪0 (M) (i ≥ l R +, 𝔪0 (M)). These results yield us to get some similar results for new finiteness dimensions induced by minimax and cofinite graded modules.

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Available abstract

Let R = ⨁ n ∈ ℕ0 R n be a noetherian homogeneous ring with local base ring (R 0, 𝔪0) and irrelevant ideal R +, let M be a finitely generated graded R-module. We shall define the upper finiteness dimension (lower finiteness dimension) of M with respect to R + and 𝔪0, denoted by u R +, 𝔪0 (M)(l R +, 𝔪0 (M)) and we prove that are artinian for all i ≤ u R +, 𝔪0 (M) (i ≥ l R +, 𝔪0 (M)). These results yield us to get some similar results for new finiteness dimensions induced by minimax and cofinite graded modules.

Key concepts: Mathematics, Local cohomology, Dimension (graph theory), Local ring, Noetherian, Mathematics Subject Classification, Ideal (ethics), Pure mathematics

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