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2012β€’Algebra ColloquiumRequires access

Noetherianness and Local Cohomology Modules

Kazem Khashyarmanesh

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Abstract

Let R be a commutative Noetherian ring, π”ž an ideal of R, and M an R-module. We show that, whenever [Formula: see text], M is Noetherian if and only if there exists a submodule N of M such that the R-modules M/π”ž N and [Formula: see text] are Noetherian. By using this result, we establish Noetherian properties for local cohomology modules [Formula: see text] in several cases. For instance, we obtain a new version of the Lichtenbaum-Hartshorne Vanishing Theorem.

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What this paper is about

Let R be a commutative Noetherian ring, π”ž an ideal of R, and M an R-module. We show that, whenever [Formula: see text], M is Noetherian if and only if there exists a submodule N of M such that the R-modules M/π”ž N and [Formula: see text] are Noetherian. By using this result, we establish Noetherian properties for local cohomology modules [Formula: see text] in several cases. For instance, we obtain a new version of the Lichtenbaum-Hartshorne Vanishing Theorem.

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Available abstract

Let R be a commutative Noetherian ring, π”ž an ideal of R, and M an R-module. We show that, whenever [Formula: see text], M is Noetherian if and only if there exists a submodule N of M such that the R-modules M/π”ž N and [Formula: see text] are Noetherian. By using this result, we establish Noetherian properties for local cohomology modules [Formula: see text] in several cases. For instance, we obtain a new version of the Lichtenbaum-Hartshorne Vanishing Theorem.

Key concepts: Local cohomology, Mathematics, Noetherian, Noetherian ring, Ideal (ethics), Commutative property, Pure mathematics, Local ring

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