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1.
Let 𝒳 ? 𝒜 be subcategories of a triangulated category 𝒯, and 𝒳 a functorially finite subcategory of 𝒜. If 𝒜 has the properties that any 𝒳-monomorphism of 𝒜 has a cone and any 𝒳-epimorphism has a cocone, then the subfactor category 𝒜/[𝒳] forms a pretriangulated category in the sense of [4]. Moreover, the above pretriangulated category 𝒜/[𝒳] with 𝒯(𝒳, 𝒳[1]) = 0 becomes a triangulated category if and only if (𝒜, 𝒜) forms an 𝒳-mutation pair and 𝒜 is closed under extensions. 相似文献
2.
A notion of mutation pairs of subcategories in an abelian category is defined in this article. For an extension closed subcategory 𝒵 and a rigid subcategory 𝒟 ? 𝒵, the subfactor category 𝒵/[𝒟] is also a triangulated category whenever (𝒵, 𝒵) forms a 𝒟-mutation pair. Moreover, if 𝒟 and 𝒵 satisfy certain conditions in modΛ, the category of finitely generated Λ-modules over an artin algebra Λ, the triangulated category 𝒵/[𝒟] has a Serre functor. 相似文献
3.
Zhi-Wei Li 《代数通讯》2013,41(9):3725-3753
Beligiannis and Marmaridis in 1994, constructed the one-sided triangulated structures on the stable categories of additive categories induced from some homologically finite subcategories. We extend their results to slightly more general settings. As an application of our results, we give some new examples of one-sided triangulated categories arising from abelian model categories. An interesting outcome is that we can describe the pretriangulated structures of the homotopy categories of abelian model categories via those of stable categories. 相似文献
4.
Hiroyuki Nakaoka 《代数通讯》2013,41(10):4302-4326
In this article, we investigate the condition for the hearts of twin cotorsion pairs to be equivalent, compatibly with the associated functors. This is related to the vanishing of components of pairs through the associated functors. 相似文献
5.
George Ciprian Modoi 《代数通讯》2013,41(3):995-1011
We say that a projective class in a triangulated category with coproducts is perfect if the corresponding ideal is closed under coproducts of maps. We study perfect projective classes and the associated phantom and cellular towers. Given a perfect generating projective class, we show that every object is isomorphic to the homotopy colimit of a cellular tower associated to that object. Using this result and the Neeman's Freyd-style representability theorem, we give a new proof of Brown Representability Theorem. 相似文献
6.
A notion of mutation of subcategories in a right triangulated category is defined in this article. When (𝒵, 𝒵) is a 𝒟-mutation pair in a right triangulated category 𝒞, the quotient category 𝒵/𝒟 carries naturally a right triangulated structure. Moreover, if the right triangulated category satisfies some reasonable conditions, then the right triangulated quotient category 𝒵/𝒟 becomes a triangulated category. When 𝒞 is triangulated, our result unifies the constructions of the quotient triangulated categories by Iyama-Yoshino and by Jørgensen, respectively. 相似文献
7.
设{D',D,D'';i^*,i*=i!,i^!,j!,j^*=j^!,j*)是一个recollement,本文证明了当D有AR-三角时,D',D''也有AR-三角,并且它们的AR-三角完全可由D中AR-三角诱导. 相似文献
8.
The relationship between the Ziegler spectrum of (the category of modules over) a ring and the Ziegler spectrum of its derived
category is investigated. Over von Neumann regular rings and hereditary rings the spectrum of the derived category is a disjoint
union of copies of the spectrum of the ring but in general there are further indecomposable pure-injective objects of the
derived category.
Presented by A. Verschoren
Mathematics Subject Classifications (2000) Primary: 18E30; secondary: 03C60. 相似文献
9.
利用Abdian范畴中子对象的交与并的性质,将模范畴中的蝴蝶结引理推广到Abelian范畴中,并据此证明了Abelian范畴中的Jordan-Holder定理. 相似文献
10.
11.
Cyclic posets are generalizations of cyclically ordered sets. In this article, we show that any cyclic poset gives rise to a Frobenius category over any discrete valuation ring R. The stable category of a Frobenius category is always triangulated and has a cluster structure in many cases. The continuous cluster categories of [14], the infinity-gon of [12], and the m-cluster category of type A ∞ (m ≥ 3) [13] are examples of this construction as well as some new examples such as the cluster category of ?2. An extension of this construction and further examples are given in [16]. 相似文献
12.
Let 𝒞 be a triangulated category. When ω is a functorially finite subcategory of 𝒞, Jøtrgensen showed that the stable category 𝒞/ω is a pretriangulated category. A pair (𝒳, 𝒴) of subcategories of 𝒞 with ω ? 𝒳 ∩ 𝒴 gives rise to a pair (𝒳/ω, 𝒴/ω) of subcategories of 𝒞/ω. In this article, we find conditions for (𝒳/ω, 𝒴/ω) to be a torsion pair in terms of properties of the pair (𝒳, 𝒴). In particular, we obtain necessary and sufficient conditions for (𝒳/ω, 𝒴/ω) to be a torsion pair in the stable category 𝒞/ω when τω = ω, where τ is the Auslander–Reiten translation. 相似文献
14.
Given a triangle functor F : A → B, the authors introduce the half image hIm F,which is an additive category closely related to F. If F is full or faithful, then hIm F admits a natural triangulated structure. However, in general, one can not expect that hIm F has a natural triangulated structure. The aim of this paper is to prove that hIm F admits a natural triangulated structure if and only if F satisfies the condition(SM). If this is the case, hIm F is triangle-equivalent to the Verdier quotient A/Ker F. 相似文献
15.
We show that each category enriched in Abelian groupoids is a linear track extension and hence is determined up to weak equivalence by a characteristic chomology class. We also discuss compatibility with coproducts. 相似文献
16.
本文在紧Riemann曲面上引入了拟距离函数和圆环域的概念,并给出了这种圆环域上的Hadamard定理. 相似文献
17.
We introduce a notion of an extended operation which should serve as a new tool for the study of categories like Mal’tsev, unital, strongly unital and subtractive categories.
However, in the present paper we are only concerned with subtractive categories, and accordingly, most of the time we will
deal with extended subtractions, which are particular instances of extended operations. We show that these extended subtractions provide new conceptual characterizations
of subtractive categories and moreover, they give an enlarged “algebraic tool” for working in a subtractive category—we demonstrate
this by using them to describe the construction of associated abelian objects in regular subtractive categories with finite colimits. Also, the definition and some basic properties of abelian objects
in a general subtractive category is given for the first time in the present paper.
The second author acknowledges the support of Claude Leon Foundation, INTAS (06-1000017-8609) and Georgian National Science
Foundation (GNSF/ST06/3-004). 相似文献
18.
19.
Phùng Hô` 《Compositio Mathematica》2002,132(1):27-48
We show that, with some technical conditions, an Abelian monoidal category admits a monoidal embedding into the category of bimodules over a ring. The case of semisimple rigid monoidal categories is studied in more detail. 相似文献
20.
Amnon Neeman 《K-Theory》2000,20(3):243-298
This article is a sequel to K-theory for triangulated categories 3 1/2 (A). In this article, we complete the proof of the main theorem of the previous article. 相似文献