• 대한수학회보
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• 제54권1호
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• pp.289-298
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• 2017

Let R denote a commutative Noetherian (not necessarily local) ring and let I be an ideal of R of dimension one. The main purpose of this note is to show that the category ${\mathfrak{M}}(R,\;I)_{com}$ of I-cominimax R-modules forms an Abelian subcategory of the category of all R-modules. This assertion is a generalization of the main result of Melkersson in [15]. As an immediate consequence of this result we get some conditions for cominimaxness of local cohomology modules for ideals of dimension one. Finally, it is shown that the category ${\mathcal{C}}^1_B(R)$ of all R-modules of dimension at most one with finite Bass numbers forms an Abelian subcategory of the category of all R-modules.

• 대한수학회논문집
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• 제33권1호
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• pp.53-71
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• 2018

In this paper, we provide two different descriptions for a relative derived category with respect to a subcategory ${\mathcal{X}}$ of an abelian category ${\mathcal{A}}$. First, we construct an exact model structure on certain exact category which has as its homotopy category the relative derived category of ${\mathcal{A}}$. We also show that a relative derived category is equivalent to homotopy category of certain complexes. Moreover, we investigate the existence of certain recollements in such categories.

• 대한수학회지
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• 제56권4호
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• pp.1031-1048
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• 2019

In this paper we study recollements of abelian categories and balanced pairs. The main results are: recollements induce new balanced pairs from the middle category; the resolution dimensions are bounded under certain conditions. As an application, the resolution dimensions with respect to cotilting objects of abelian categories involved in recollements are recovered.

• 대한수학회보
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• 제61권1호
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• pp.273-280
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• 2024

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 Supp_R M ⊆ V (I), then M is I-weakly cofinite if (and only if) the R-modules Hom_R(R/I, M) and Ext¹_R(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 Tor^R_i(N, H^j_I(M)) and Extⁱ_R(N, H^j_I(M)) are I-weakly cofinite.

• 대한수학회지
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• 제56권3호
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• pp.623-644
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• 2019

We investigate the homotopy category ${\mathcal{K}}_N(Inj{\mathfrak{A}})$ of N-complexes of injectives in a Grothendieck abelian category ${\mathfrak{A}}$ not necessarily locally noetherian, and prove that the inclusion ${\mathcal{K}}_N(Inj{\mathfrak{A}}){\rightarrow}{\mathcal{K}}({\mathfrak{A}})$ has a left adjoint and ${\mathcal{K}}_N(Inj{\mathfrak{A}})$ is well generated. We also show that the homotopy category ${\mathcal{K}}_N(PrjR)$ of N-complexes of projectives is compactly generated whenever R is right coherent.

• 호남수학학술지
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• 제45권2호
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• pp.269-284
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• 2023

In this paper we study generalizations of the concept of pure-direct-projectivity from module categories to Grothendieck categories. We examine for which categories or under what conditions pure-direct-projective objects are direct-projective, quasi-projective, pure-projective, projective and flat. We investigate classes all of whose objects are pure-direct-projective. We give applications of some of the results to comodule categories.

• 대한수학회지
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• 제43권1호
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• pp.99-109
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• 2006

Approximate fibrations form a useful class of maps. By definition fibrators provide instant detection of maps in this class, and PL fibrators do the same in the PL category. We show that every closed s-hopfian t-aspherical manifold N with sparsely Abelian, hopfian fundamental group and X(N) $\neq$ 0 is a codimension-(t + 1) PL fibrator.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제28권2호
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• pp.187-198
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• 2021

This work presents an exposition of both the internal structure of derived category of an abelian category D^*(𝓐) and its contribution in solving problems, particularly in algebraic geometry. Calculation of some morphisms will be presented between objects in D^*(𝓐) as elements in appropriate cohomology groups along with their compositions with the help of Yoneda construction under the assumption that the homological dimension of D^*(𝓐) is greater than or equal to 2. These computational settings will then be considered under sheaf cohomological context with a particular case from projective geometry.

• 대한수학회논문집
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• 제11권4호
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• pp.1077-1085
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• 1996

This paper provides examples which can not be approximate fibrations and shows that if $N^n$ is a closed aspherical manifold, $\pi_1(N)$ is hyperhophian, normally cohophian, and $\pi_1(N)$ has no nontrivial Abelian normal subgroup, then the product of $N^n$ and a sphre $S^m$ satisfies the property that all PL maps from an orientable manifold M to a polyhedron B for which each point preimage is homotopy equivalent to $N^n \times S^m$ necessarily are approximate fibrations.

• 대한수학회논문집
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• 제25권2호
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• pp.283-289
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• 2010

Approximate fibrations provide a useful class of maps. Fibrators give instant detection of maps in this class, and PL fibrators do the same in the PL category. We show that rational homology spheres with some additional conditions are codimension-k PL fibrators and PL manifolds with trivial homology groups in some range can be codimension-k (k > 2) PL fibrators.