Abelian Category of Weakly Cofinite Modules and Local Cohomology
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Abelian Category of Weakly Cofinite Modules and Local Cohomology Eslam Hatami1 · Moharram Aghapournahr1 Received: 23 March 2020 / Revised: 13 August 2020 / Accepted: 11 September 2020 © Iranian Mathematical Society 2020
Abstract Let R be a commutative Noetherian ring, a an ideal of R, and M an R-module. We prove that the category of a-weakly cofinite modules is a Melkersson subcategory of R-modules whenever dim R ≤ 1 and is an Abelian subcategory whenever dim R ≤ 2. We also prove that if (R, m) is a local ring with dim R/a ≤ 2 and Supp R (M) ⊆ V(a), then M is a-weakly cofinite if (and only if) Hom R (R/a, M), Ext 1R (R/a, M) and Ext 2R (R/a, M) are weakly Laskerian. In addition, we prove that if (R, m) is a local ring with dim R/a ≤ 2 and n ∈ N0 , such that ExtiR (R/a, M) is weakly Laskerian for all i, then Hia (M) is a-weakly cofinite for all i if (and only if) Hom R (R/a, Hia (M)) is weakly Laskerian for all i. Keywords Local cohomology · Weakly Laskerian modules · Weakly cofinite modules · Melkersson subcategory Mathematics Subject Classification 13D45 · 14B15 · 13E05
1 Introduction Throughout this paper, R is a commutative Noetherian ring with non-zero identity and a an ideal of R. For an R-module M, the i th local cohomology module M with respect to ideal a is defined as: Hia (M) ∼ = lim ExtiR (R/an , M). − → n
Communicated by Mohammad-Taghi Dibaei.
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Moharram Aghapournahr [email protected] Eslam Hatami [email protected]
1
Department of Mathematics, Faculty of Science, Arak University, Arak 38156-8-8349, Iran
123
Bulletin of the Iranian Mathematical Society
We refer the reader to [11] for more details about the local cohomology. Hartshorne [18] defined a module M to be a-cofinite if Supp R (M) ⊆ V(a) and ExtiR (R/a, M) are finitely generated for all i ≥ 0. He asked: (i) For which rings R and ideals a are the modules Hia (M) a-cofinite for all i and all finitely generated modules M? (ii) Whether the category C (R, a)co f of a- cofinite modules forms an abelian subcategory of the category of all R-modules? That is, if f : M −→ N is an R-homomorphism of a-cofinite modules, are Ker f and Coker f a-cofinite? With respect to the question (i), there are several papers devoted to this question; for example, see [3,6–8,14,19,20,22,23]. With respect to the question (ii), Hartshorne with an example showed that this is not true in general. However, it is proved in [13, Theorem 2.2(ii)], [22, Theorem 2.6] and [23, Theorem 7.4] and with a referent proof in [24, Corollary 2.6] that the category C (R, a)co f of a-cofinite modules forms an Abelian subcategory of the category of all R-modules, respectively, in cases cd(a, R) ≤ 1, dim R/a ≤ 1 and dim R ≤ 2. Also, in [9, Theorem 3.5], Bahmanpour et al. as a generalization of [22, Theorem 2.3] have shown that if a is an ideal of a Noetherian local ring (R, m) with dim R/a = 2 and M is an R-module, such that Supp R (M) ⊆ V (a), then M is a-cofinite if (and only if) Hom R (R/a, M), Ext 1R (R/a, M), and Ext 2R (R/a, M) are finitely generated. Recall that a
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