共查询到20条相似文献,搜索用时 46 毫秒
1.
Panyue Zhou 《代数通讯》2018,46(1):82-89
Gentle and Todorov proved that in an abelian category with enough projective objects, the extension subcategory of two covariantly finite subcategories is covariantly finite. We prove a right triangulated version of Gentle-Todorov’s theorem by introducing the notion of right homotopy cartesian square. 相似文献
2.
Consider a monad on an idempotent complete triangulated category with the property that its Eilenberg–Moore category of modules inherits a triangulation. We show that any other triangulated adjunction realizing this monad is ‘essentially monadic’, i.e. becomes monadic after performing the two evident necessary operations of taking the Verdier quotient by the kernel of the right adjoint and idempotent completion. In this sense, the monad itself is ‘intrinsically monadic’. It follows that for any highly structured ring spectrum, its category of homotopy (aka naïve) modules is triangulated if and only if it is equivalent to its category of highly structured (aka strict) modules. 相似文献
3.
The Orlov spectrum is a new invariant of a triangulated category. It was introduced by D. Orlov, building on work of A. Bondal-M. Van den Bergh and R. Rouquier. The supremum of the Orlov spectrum of a triangulated category is called the ultimate dimension. In this work, we study Orlov spectra of triangulated categories arising in mirror symmetry. We introduce the notion of gaps and outline their geometric significance. We provide the first large class of examples where the ultimate dimension is finite: categories of singularities associated to isolated hypersurface singularities. Similarly, given any nonzero object in the bounded derived category of coherent sheaves on a smooth Calabi-Yau hypersurface, we produce a generator, by closing the object under a certain monodromy action, and uniformly bound this generator’s generation time. In addition, we provide new upper bounds on the generation times of exceptional collections and connect generation time to braid group actions to provide a lower bound on the ultimate dimension of the derived Fukaya category of a symplectic surface of genus greater than one. 相似文献
4.
We propose a natural definition of a category of matrix factorizations for nonaffine Landau–Ginzburg models. For any LG-model
we construct a fully faithful functor from the category of matrix factorizations defined in this way to the triangulated category
of singularities of the corresponding fiber. We also show that this functor is an equivalence if the total space of the LG-model
is smooth. 相似文献
5.
A pretriangulated category is an additive category with left and right triangulations such that these two triangulations are compatible. In this paper, we first show that the idempotent completion of a left triangulated category admits a unique structure of left triangulated category and dually this is true for a right triangulated category. We then prove that the idempotent completion of a pretriangulated category has a natural structure of pretriangulated category. As an application, we show that a torsion pair in a pretriangulated category extends uniquely to a torsion pair in the idempotent completion. 相似文献
6.
Hiroyuki Nakaoka 《Applied Categorical Structures》2011,19(6):879-899
In the paper of Keller and Reiten, it was shown that the quotient of a triangulated category (with some conditions) by a cluster
tilting subcategory becomes an abelian category. After that, Koenig and Zhu showed in detail, how the abelian structure is
given on this quotient category, in a more abstract setting. On the other hand, as is well known since 1980s, the heart of
any t-structure is abelian. We unify these two constructions by using the notion of a cotorsion pair. To any cotorsion pair in
a triangulated category, we can naturally associate an abelian category, which gives back each of the above two abelian categories,
when the cotorsion pair comes from a cluster tilting subcategory, or a t-structure, respectively. 相似文献
7.
Xiao-Wu Chen 《Archiv der Mathematik》2009,93(1):29-35
Gentle and Todorov proved that in an abelian category with enough projective objects, the extension subcategory of two covariantly
finite subcategories is covariantly finite. We give an example to show that Gentle–Todorov’s theorem may fail in an arbitrary
abelian category; however we prove a triangulated version of Gentle–Todorov’s theorem which holds for arbitrary triangulated
categories; we apply Gentle–Todorov’s theorem to obtain short proofs of a classical result by Ringel and a recent result by
Krause and Solberg.
This project is partially supported by China Postdoctoral Science Foundation (No.s 20070420125 and 200801230). The author
also gratefully acknowledges the support of K. C. Wong Education Foundation, Hong Kong. 相似文献
8.
We exhibit examples of triangulated categories which are neither the stable category of a Frobenius category nor a full triangulated
subcategory of the homotopy category of a stable model category. Even more drastically, our examples do not admit any non-trivial
exact functors to or from these algebraic respectively topological triangulated categories.
Mathematics Subject Classification (1991) 18E30, 55P42 相似文献
9.
We study a triangulated category of graded matrix factorizations for a polynomial of type ADE. We show that it is equivalent to the derived category of finitely generated modules over the path algebra of the corresponding Dynkin quiver. Also, we discuss a special stability condition for the triangulated category in the sense of T. Bridgeland, which is naturally defined by the grading. 相似文献
10.
Paul Balmer 《Advances in Mathematics》2011,(5):4352
We prove that the category of modules over a separable ring object in a tensor triangulated category admits a unique structure of triangulated category which is compatible with the original one. This applies in particular to étale algebras. More generally, we do this for exact separable monads. 相似文献
11.
Yu-Han Liu 《代数通讯》2013,41(8):3013-3031
We compute Balmer's prime spectrum for the derived category of quiver representations for a finite ordered quiver with the vertex-wise tensor product and show that it does not recover the quiver. We then associate an algebra to every k-linear triangulated tensor category and show that the path algebra can be recovered in this way. 相似文献
12.
Changjian Fu 《代数通讯》2013,41(7):2410-2418
We show that a tilting module over the endomorphism algebra of a cluster-tilting object in a 2-Calabi–Yau triangulated category lifts to a cluster-tilting object in this 2-Calabi–Yau triangulated category. This generalizes a recent work of Smith for cluster categories. 相似文献
13.
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
14.
Xiao Yan Yang 《数学学报(英文版)》2013,29(11):2137-2154
Let T be a triangulated category and ξ a proper class of triangles.Some basics properties and diagram lemmas are proved directly from the definition of ξ. 相似文献
15.
Ya-nan LIN & Lin XIN Department of Mathematics Xiamen University Xiamen China Department of Mathematics Fujian Normal University Puzhou China 《中国科学A辑(英文版)》2007,50(1):13-26
Motivated by the concept of a torsion pair in a pre-triangulated category induced by Beligiannis and Reiten, the notion of a left (right) torsion pair in the left (right) triangulated category is introduced and investigated. We provide new connections between different aspects of torsion pairs in one-sided triangulated categories, pre-triangulated categories, stable categories and derived categories. 相似文献
16.
In the preceding part (I) of this paper, we showed that for any torsion pair (i.e., t-structure without the shift-closedness) in a triangulated category, there is an associated abelian category, which we call
the heart. Two extremal cases of torsion pairs are t-structures and cluster tilting subcategories. If the torsion pair comes from a t-structure, then its heart is nothing other than the heart of this t-structure. In this case, as is well known, by composing certain adjoint functors, we obtain a homological functor from the
triangulated category to the heart. If the torsion pair comes from a cluster tilting subcategory, then its heart coincides
with the quotient category of the triangulated category by this subcategory. In this case, the quotient functor becomes homological.
In this paper, we unify these two constructions, to obtain a homological functor from the triangulated category, to the heart
of any torsion pair. 相似文献
17.
Xiao-Wu Chen 《Abhandlungen aus dem Mathematischen Seminar der Universit?t Hamburg》2010,80(2):207-212
Let
\mathbbX\mathbb{X} be a separated Noetherian scheme of finite Krull dimension which has enough locally free sheaves of finite rank and let
U í \mathbbXU\subseteq \mathbb{X} be an open subscheme. We prove that the singularity category of U is triangle equivalent to the Verdier quotient triangulated category of the singularity category of
\mathbbX\mathbb{X} with respect to the thick triangulated subcategory generated by sheaves supported in the complement of U. The result unifies two results of Orlov. We also prove a noncommutative version of this result. 相似文献
18.
We prove that, in a triangulated category with combinatorial models, every localizing subcategory is coreflective and every colocalizing subcategory is reflective if a certain large-cardinal axiom (Vopěnka?s principle) is assumed true. It follows that, under the same assumptions, orthogonality sets up a bijective correspondence between localizing subcategories and colocalizing subcategories. The existence of such a bijection was left as an open problem by Hovey, Palmieri and Strickland in their axiomatic study of stable homotopy categories and also by Neeman in the context of well-generated triangulated categories. 相似文献
19.
Gonçalo Tabuada 《Selecta Mathematica, New Series》2016,22(2):735-764
This article is the sequel to (Marcolli and Tabuada in Sel Math 20(1):315–358, 2014). We start by developing a theory of noncommutative (=NC) mixed motives with coefficients in any commutative ring. In particular, we construct a symmetric monoidal triangulated category of NC mixed motives, over a base field k, and a full subcategory of NC mixed Artin motives. Making use of Hochschild homology, we then apply Ayoub’s weak Tannakian formalism to these motivic categories. In the case of NC mixed motives, we obtain a motivic Hopf dg algebra, which we describe explicitly in terms of Hochschild homology and complexes of exact cubes. In the case of NC mixed Artin motives, we compute the associated Hopf dg algebra using solely the classical category of mixed Artin–Tate motives. Finally, we establish a short exact sequence relating the Hopf algebra of continuous functions on the absolute Galois group with the motivic Hopf dg algebras of the base field k and of its algebraic closure. Along the way, we describe the behavior of Ayoub’s weak Tannakian formalism with respect to orbit categories and relate the category of NC mixed motives with Voevodsky’s category of mixed motives. 相似文献
20.