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AN ABELIAN CATEGORY OF WEAKLY COFINITE MODULES

  • Received : 2023.02.24
  • Accepted : 2023.06.22
  • Published : 2024.01.31
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Abstract

Let I be an ideal of a commutative Noetherian semi-local ring R and M be an R-module. It is shown that if dim M ≤ 2 and SuppR M ⊆ V (I), then M is I-weakly cofinite if (and only if) the R-modules HomR(R/I, M) and Ext1R(R/I, M) are weakly Laskerian. As a consequence of this result, it is shown that the category of all I-weakly cofinite modules X with dim X ≤ 2, forms an Abelian subcategory of the category of all R-modules. Finally, it is shown that if dim R/I ≤ 2, then for each pair of finitely generated R-modules M and N and each pair of the integers i, j ≥ 0, the R-modules TorRi(N, HjI(M)) and ExtiR(N, HjI(M)) are I-weakly cofinite.

Keywords

Acknowledgement

The author is deeply grateful to the referee for a very careful reading of the manuscript and many valuable suggestions.

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