Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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CHARACTERIZATION OF WEAKLY COFINITE LOCAL COHOMOLOGY MODULES

  • Received : 2022.01.13
  • Accepted : 2023.02.17
  • Published : 2023.05.31
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Abstract

Let R be a commutative Noetherian ring, 𝔞 an ideal of R, M an arbitrary R-module and X a finite R-module. We prove a characterization for Hi𝔞(M) and Hi𝔞(X, M) to be 𝔞-weakly cofinite for all i, whenever one of the following cases holds: (a) ara(𝔞) ≤ 1, (b) dim R/𝔞 ≤ 1 or (c) dim R ≤ 2. We also prove that, if M is a weakly Laskerian R-module, then Hi𝔞(X, M) is 𝔞-weakly cofinite for all i, whenever dim X ≤ 2 or dim M ≤ 2 (resp. (R, m) a local ring and dim X ≤ 3 or dim M ≤ 3). Let d = dim M < ∞, we prove an equivalent condition for top local cohomology module Hd𝔞(M) to be weakly Artinian.

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Acknowledgement

The authors like to thank the referee for his/her careful reading and many helpful suggestions on this paper.

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