Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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COLOCALIZATION OF GENERALIZED LOCAL HOMOLOGY MODULES

  • Received : 2021.07.20
  • Accepted : 2022.02.04
  • Published : 2022.07.31
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Abstract

Let R be a commutative Noetherian ring and I an ideal of R. In this paper, we study colocalization of generalized local homology modules. We intend to establish a dual case of local-global principle for the finiteness of generalized local cohomology modules. Let M be a finitely generated R-module and N a representable R-module. We introduce the notions of the representation dimension rI(M, N) and artinianness dimension aI(M, N) of M, N with respect to I by rI(M, N) = inf{i ∈ ℕ0 : HIi(M, N) is not representable} and aI(M, N) = inf{i ∈ ℕ0 : HIi(M, N) is not artinian} and we show that aI(M, N) = rI(M, N) = inf{rIR𝔭 (M𝔭,𝔭N) : 𝔭 ∈ Spec(R)} ≥ inf{aIR𝔭 (M𝔭,𝔭N) : 𝔭 ∈ Spec(R)}. Also, in the case where R is semi-local and N a semi discrete linearly compact R-module such that N/∩t>0ItN is artinian we prove that inf{i : HIi(M, N) is not minimax}=inf{rIR𝔭 (M𝔭,𝔭N) : 𝔭 ∈ Spec(R)＼Max(R)}.

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Acknowledgement

The author thanks the referee for his/her careful reading and many helpful suggestions on this paper.

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