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2011•Canadian Mathematical BulletinOpen access

Artinianness of Certain Graded Local Cohomology Modules

Амир Мафи, Hero Saremi

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Abstract

Abstract We show that ifR= ⴲn∈ℕ0Rnis a Noetherian homogeneous ring with local base ring (R0, m0), irrelevant idealR+, andMa finitely generated gradedR-module, then is Artinian forj= 0, 1 wheret= inf﹛i∈ ℕ0: is not finitely generated﹜. Also, we prove that if cd(R+,M) = 2, then for eachi∈ ℕ0, ) is Artinian if and only if is Artinian, where cd(R+,M) is the cohomological dimension ofMwith respect toR+. This improves some results of R. Sazeedeh.

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Abstract We show that ifR= ⴲn∈ℕ0Rnis a Noetherian homogeneous ring with local base ring (R0, m0), irrelevant idealR+, andMa finitely generated gradedR-module, then is Artinian forj= 0, 1 wheret= inf﹛i∈ ℕ0: is not finitely generated﹜. Also, we prove that if cd(R+,M) = 2, then for eachi∈ ℕ0, ) is Artinian if and only if is Artinian, where cd(R+,M) is the cohomological dimension ofMwith respect toR+. This improves some results of R. Sazeedeh.

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Available abstract

Abstract We show that ifR= ⴲn∈ℕ0Rnis a Noetherian homogeneous ring with local base ring (R0, m0), irrelevant idealR+, andMa finitely generated gradedR-module, then is Artinian forj= 0, 1 wheret= inf﹛i∈ ℕ0: is not finitely generated﹜. Also, we prove that if cd(R+,M) = 2, then for eachi∈ ℕ0, ) is Artinian if and only if is Artinian, where cd(R+,M) is the cohomological dimension ofMwith respect toR+. This improves some results of R. Sazeedeh.

Key concepts: Mathematics, Local cohomology, Noetherian ring, Pure mathematics, Noetherian, Finitely-generated abelian group, Artinian ring, Local ring

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