Grothendieck groups arising from contravariantly finite subcategories (original) (raw)

Some remarks on homologically finite subcategories

Boletín de la Sociedad Matemática Mexicana, 2017

Let be an Artin algebra and C a full subcategory of-mod closed under direct summands and closed under extensions. It is known that if C is functorially finite, then it has almost split sequences. Here we review an example of a covariantly finite subcategory that has right almost split morphisms except for one isomorphism class, and we compute its almost split sequences.

2001

Ideals are used to define homological functors in additive categories. In abelian categories the ideals corresponding to the usual universal objects are principal, and the construction reduces, in a choice dependent way, to homology groups. The applications considered in this paper are: derived categories and functors. 2000 Mathematics Subject Classification: 18G50, 18A05.

Some Remarks on Categories of Modules Modulo Morphisms with Essential Kernel or Superfluous Image

Journal of the Korean Mathematical Society, 2013

For an ideal I of a preadditive category A, we study when the canonical functor C : A → A/I is local. We prove that there exists a largest full subcategory C of A for which the canonical functor C : C → C/I is local. Under this condition, the functor C turns out to be a weak equivalence between C and C/I. If A is additive (with splitting idempotents), then C is additive (with splitting idempotents). The category C is ample in several cases, such as the case when A = Mod-R and I is the ideal ∆ of all morphisms with essential kernel. In this case, the category C contains, for instance, the full subcategory F of Mod-R whose objects are all the continuous modules. The advantage in passing from the category F to the category F /I lies in the fact that, although the two categories F and F /I are weakly equivalent, every endomorphism has a kernel and a cokernel in F /∆, which is not true in F. In the final section, we extend our theory from the case of one ideal I to the case of n ideals I 1 ,. .. , In.

Abelian Subcategories Closed Under Extensions: K -Theory and Decompositions

Communications in Algebra, 2007

A full subcategory of modules over a commutative ring R is wide if it is abelian and closed under extensions. Hovey [Hov01] gave a classification of the wide subcategories of finitely presented modules over regular coherent rings in terms of certain specialisation closed subsets of Spec(R). We use this classification theorem to study K-theory and Krull-Schmidt type decompositions for wide subcategories. It is shown that the K-group, in the sense of Grothendieck, of a wide subcategory W of finitely presented modules over a regular coherent ring is isomorphic to that of the thick subcategory of perfect complexes whose homology groups belong to W. We also show that the wide subcategories of finitely generated modules over a noetherian regular ring can be decomposed uniquely into indecomposable ones. This result is then applied to obtain a decomposition for the K-groups of wide subcategories.

Remarks on equivalences of additive subcategories

We study category equivalences between some additive subcategories of module categories. As its application, we show that the group of aut- ofunctors of the category of reflexive modules over a normal domain is isomorphic to the divisor class group.

Torsion and torsion-free classes from objects of finite type in Grothendieck categories

Journal of Algebra

In an arbitrary Grothendieck category, we find necessary and sufficient conditions for the class of FPn-injective objects to be a torsion class. By doing so, we propose a notion of n-hereditary categories. We also define and study the class of FPn-flat objects in Grothendieck categories with a generating set of small projective objects, and provide several equivalent conditions for this class to be torsion-free. In the end, we present several applications and examples of n-hereditary categories in the contexts modules over a ring, chain complexes of modules and categories of additive functors from an additive category to the category of abelian groups. Concerning the latter setting, we find a characterization of when these functor categories are n-hereditary in terms of the domain additive category.

On ideals and homology in additive categories

International Journal of Mathematics and Mathematical Sciences, 2002

Atomical Grothendieck categories

International Journal of Mathematics and Mathematical Sciences, 2003

Motivated by the study of Gabriel dimension of a Grothendieck category, we introduce the concept of atomical Grothendieck category, which has only two localizing subcategories, and we give a classification of this type of Grothendieck categories.

Grothendieck groups arising from contravariantly finite subcategories (original) (raw)

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