Reflective subcategories

Rada, Juan; Saorı́n, Manuel; Valle, A. Del

doi:10.1017/s0017089500010120

Cited by 7 publications

(5 citation statements)

References 14 publications

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“…In the previous Corollary 3.7 we showed that the class F of locally torsion-free quasi-coherent sheaves is the right part of a torsion theory in Qcoh(X). One immediate consecuence of this is that each M ∈ Qcoh(X) admits an F -reflection and thus F is a reflective class in Qcoh(X) (see MacLane (1971) for notation and terminology and Saorín et al (2000) for a nice treatment and characterization of reflective subcategories). So, in particular, we deduce that F is enveloping.…”

Section: Locally Torsion-free Quasi-coherent Sheavesmentioning

confidence: 99%

Locally torsion-free quasi-coherent sheaves

Odabasi

2014

Journal of Pure and Applied Algebra

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Let X be an arbitrary scheme. The category Qcoh(X) of quasi-coherent sheaves on X is known that admits arbitrary direct products. However their structure seems to be rather mysterious. In the present paper we will describe the structure of the product object of a family of locally torsion-free objects in Qcoh(X), for X an integral scheme. Several applications are provided. For instance it is shown that the class of flat quasi-coherent sheaves on a Dedekind scheme X is closed under arbitrary direct products, and that the class of all locally torsion-free quasi-coherent sheaves induces a hereditary torsion theory on Qcoh(X). Finally torsion-free covers are shown to exist in Qcoh(X).

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Section: Locally Torsion-free Quasi-coherent Sheavesmentioning

confidence: 99%

Locally torsion-free quasi-coherent sheaves

Odabasi

2014

Journal of Pure and Applied Algebra

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“…The equivalence of (1) and (6) in Corollary 3.10 has been proven in [15,18] using different methods (see [ …”

Section: Global Dimension and Left Derived Functors Of Hommentioning

confidence: 99%

Global dimension and left derived functors of Hom

Mao

Ding

2007

SCI CHINA SER A

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It is well known that the right global dimension of a ring R is usually computed by the right derived functors of Hom and the left projective resolutions of right R-modules. In this paper, for a left coherent and right perfect ring R, we characterize the right global dimension of R, from another point of view, using the left derived functors of Hom and the right projective resolutions of right R-modules. It is shown that rD(R) n (n 2) if and only if the gl right Proj-dim MR n − 2 if and only if Extn−1(N, M) = 0 for all right R-modules N and M if and only if every (n − 2)th Projcosyzygy of a right R-module has a projective envelope with the unique mapping property. It is also proved that rD(R) n (n 1) if and only if every (n − 1)th Proj-cosyzygy of a right R-module has an epic projective envelope if and only if every nth Proj-cosyzygy of a right R-module is projective. As corollaries, the right hereditary rings and the rings R with rD(R) 2 are characterized.

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“…closed under extensions, direct summands and taking cones. On the other hand, it is an immediate consequence of the dual of [30,Theorem 3.1] that if C is cocomplete abelian, a subcategory B is coreflective if, and only if, it is precovering and closed under taking cokernels.…”

Section: Introductionmentioning

confidence: 99%

Reflective and coreflective subcategories

Cortés-Izurdiaga¹,

Crivei²,

Saorı́n³

2021

Preprint

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Given any additive category C with split idempotents, pseudokernels and pseudocokernels, we show that a subcategory B is coreflective if, and only if, it is precovering, closed under direct summands and each morphism in B has a pseudocokernel in C that belongs to B. We apply this result and its dual to, among others, preabelian and pretriangulated categories. As a consequence, we show that a subcategory of a preabelian category is coreflective if, and only it, it is precovering and closed under taking cokernels. On the other hand, if C is pretriangulated with split idempotents, then a subcategory B is coreflective and invariant under the suspension functor if, and only if, it is precovering and closed under taking direct summands and cones. These are extensions of well-known results for AB3 abelian and triangulated categories, respectively.By-side applications of these results allow us: a) To characterize the coreflective subcategories of a given AB3 abelian category which have a set of generators and are themselves abelian, abelian exact or module categories; b) to extend to module categories over arbitrary small preadditive categories a result of Gabriel and De la Peña stating that all fully exact subcategories are bireflective; c) to show that, in any Grothendieck category, the direct limit closure of its subcategory of finitely presented objects is a coreflective subcategory.

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Reflective subcategories

Cited by 7 publications

References 14 publications

Locally torsion-free quasi-coherent sheaves

Locally torsion-free quasi-coherent sheaves

Global dimension and left derived functors of Hom

Reflective and coreflective subcategories

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