Mutation pairs and quotient categories of Abelian categories

The notion of 𝒟-mutation pairs of subcategories in an abelian category is defined in this article. When (𝒵,𝒵) is a 𝒟-mutation pair in an abelian category 𝒜, the quotient category 𝒵∕𝒟 carries naturally a triangulated structure. Moreover, our result generalize the construction of the quotient triangulated category by Happel [10 Happel, D. (1988). Triangulated Categories in the Representation of Finite Dimensional Algebras. London Mathematical Society, LMN, Vol. 119. Cambridge: Cambridge University Press.[Crossref] , [Google Scholar], Theorem 2.6]. Finally, we find a one-to-one correspondence between cotorsion pairs in 𝒜 and cotorsion pairs in the quotient category 𝒵∕𝒟, and study homological finiteness of subcategories in a mutation pair.     

Right n-angulated categories arising from covariantly finite subcategories

We define right n-angulated categories, which are analogous to right triangulated categories. Let 𝒞 be an additive category and 𝒳 a covariantly finite subcategory of 𝒞. We show that under certain conditions, the quotient 𝒞∕[𝒳] is a right n-angulated category. This has immediate applications to n-angulated quotient categories.     

Mutation Pairs in Abelian Categories

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.     

Subfactor Categories of Triangulated Categories

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 Beligiannis , A. , Reiten , I. ( 2007 ). Homological and Homotopical Aspects of Torsion Theories . Memoirs of the AMS 883 : 426 – 454 . [Google Scholar]]. 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.     

Torsion Pairs in Stable Categories

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.     

Endomorphism category of an abelian category

Let 𝒞 be an additive category. Denote by End(𝒞) the endomorphism category of 𝒞, i.e., the objects in End(𝒞) are pairs (C,c) with C∈𝒞,c∈End_𝒞(C), and a morphism f:(C,c)→(D,d) is a morphism f∈Hom_𝒞(C,D) satisfying fc?=?df. This paper is devoted to an approach of the general theory of the endomorphism category of an arbitrary additive category. It is proved that the endomorphism category of an abelian category is again abelian with an induced structure without nontrivial projective or injective objects. Furthermore, the endomorphism category of any nontrivial abelian category is nonsemisimple and of infinite representation type. As an application, we show that two unital rings are Morita equivalent if and only if the endomorphism categories of their module categories are equivalent.     

Locally finite triangulated categories

A k-linear triangulated category is called locally finite provided for any indecomposable object Y in . It has Auslander–Reiten triangles. In this paper, we show that if a (connected) triangulated category has Auslander–Reiten triangles and contains loops, then its Auslander–Reiten quiver is of the form :

Full-size image (<1K)

By using this, we prove that the Auslander–Reiten quiver of any locally finite triangulated category is of the form , where Δ is a Dynkin diagram and G is an automorphism group of . For most automorphism groups G, the triangulated categories with as their Auslander–Reiten quivers are constructed. In particular, a triangulated category with as its Auslander–Reiten quiver is constructed.     

Calculus of Functorial Morphisms

We prove that, if F, G: 𝒞 → 𝒟 are two right exact functors between two Grothendieck categories such that they commute with coproducts and U is a generator of 𝒞, then there is a bijection between Nat(F, G) and the centralizer of Hom_𝒟(F(U), G(U)) considered as an Hom_𝒞(U, U)-Hom_𝒞(U, U)-bimodule. We also prove a dual of this result and give applications to Frobenius functors between Grothendieck categories.     

Models for homotopy categories of injectives and Gorenstein injectives

A natural generalization of locally noetherian and locally coherent categories leads us to define locally type FP_∞ categories. They include not just all categories of modules over a ring, but also the category of sheaves over any concentrated scheme. In this setting we generalize and study the absolutely clean objects recently introduced in [5 Bravo, D., Gillespie, J., Hovey, M. The stable module category of a general ring (arXiv:1405.5768). [Google Scholar]]. We show that 𝒟(𝒜𝒞), the derived category of absolutely clean objects, is always compactly generated and that it is embedded in K(Inj), the chain homotopy category of injectives, as a full subcategory containing the DG-injectives. Assuming the ground category 𝒢 has a set of generators satisfying a certain vanishing property, we also show that there is a recollement relating 𝒟(𝒜𝒞) to the (also compactly generated) derived category 𝒟(𝒢). Finally, we generalize the Gorenstein AC-injectives of [5 Bravo, D., Gillespie, J., Hovey, M. The stable module category of a general ring (arXiv:1405.5768). [Google Scholar]], showing that they are the fibrant objects of a cofibrantly generated model structure on 𝒢.     

On one-sided torsion pair

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.     

Balanced pairs induce recollements

Let 𝒜 be an abelian category with arbitrary (set-indexed) coproducts and exact products. Let (𝒫,?) be a complete balanced pair. Then as in the classical case, we prove that there exists a recollement with the middle term K(𝒜), the homotopy category of 𝒜. In particular, this implies that the relative derived category exists. Two applications are given.     

The derived category of an algebra with radical square zero

Koszul duality and covering theory are combined to realize the bounded derived category 𝒟 of an algebra with radical square zero as a certain orbit category of the bounded derived category of finitely presented representations of an associated infinite quiver. As a consequence, the possible shapes of the connected components of the Auslander–Reiten quiver of 𝒟 are described.     

Additivity of Lie multiplicative maps on triangular algebras

Let 𝒜 and ? be unital algebras over a commutative ring ?, and ? be a (𝒜,??)-bimodule, which is faithful as a left 𝒜-module and also as a right ?-module. Let 𝒰?=?Tri(𝒜,??,??) be the triangular algebra and 𝒱 any algebra over ?. Assume that Φ?:?𝒰?→?𝒱 is a Lie multiplicative isomorphism, that is, Φ satisfies Φ(ST???TS)?=?Φ(S)Φ(T)???Φ(T)Φ(S) for all S, T?∈?𝒰. Then Φ(S?+?T)?=?Φ(S)?+?Φ(T)?+?Z _S,T for all S, T?∈?𝒰, where Z _S,T is an element in the centre 𝒵(𝒱) of 𝒱 depending on S and T.     

Relative homological dimensions and Tate cohomology of complexes with respect to cotorsion pairs

Let (𝒳,𝒴) be a complete and hereditary cotorsion pair in a bicomplete abelian category 𝒜. We introduce a Gorenstein category 𝒢(𝒳) and 𝒢(𝒳)-resolution dimension of complexes with respect to (𝒳,𝒴). For complexes with finite 𝒢(𝒳)-resolution dimension, Tate 𝒳-resolutions are constructed. Furthermore, we study relative Tate cohomology of complexes, which is useful for detecting the finiteness of the relative homological dimensions of complexes.     

Exceptional Pairs in Hereditary Categories

We study the category 𝒞(X, Y) generated by an exceptional pair (X, Y) in a hereditary category ?. If r = dim _k Hom(X, Y) ≥ 1 we show that there are exactly 3 possible types for 𝒞(X, Y), all derived equivalent to the category of finite dimensional modules mod(H _r ) over the r-Kronecker algebra H _r . In general 𝒞(X, Y) will not be equivalent to a module category. More specifically, if ? is the category of coherent sheaves over a weighted projective line 𝕏, then 𝒞(X, Y) is equivalent to the category of coherent sheaves on the projective line ?¹ or to mod(H _r ) and, if 𝕏 is wild, then every r ≥ 1 can occur in this way.     

三角范畴和Abel范畴的Torsion理论

本文主要研究了三角范畴在Abel化过程中torsion理论的保持问题.利用三角范畴的coherent函子范畴是Abel范畴,证明了T的coherent函子范畴A(T)是A(D)的thick子范畴;若(X,Y)是D的torsion理论,且D=X*Y的扩张是可裂的,那么(A(X),A(Y))是A(D)的torsion理论.     

On categories associated to a Quasi-Hopf algebra

A quasi-Hopf algebra H can be seen as a commutative algebra A in the center 𝒵(H-Mod) of H-Mod. We show that the category of A-modules in 𝒵(H-Mod) is equivalent (as a monoidal category) to H-Mod. This can be regarded as a generalization of the structure theorem of Hopf bimodules of a Hopf algebra to the quasi-Hopf setting.     

Cluster-Tilted Algebras of Type D n

Let H be a hereditary algebra of Dynkin type D _n over a field k and 𝒞 _H be the cluster category of H. Assume that n ≥ 5 and that T and T′ are tilting objects in 𝒞 _H . We prove that the cluster-tilted algebra Γ = End_{𝒞 H} (T)^op is isomorphic to Γ′ = End_{𝒞 H} (T′)^op if and only if T = τ ⁱ T′ or T = στ ^j T′ for some integers i and j, where τ is the Auslander–Reiten translation and σ is the automorphism of 𝒞 _H defined in Section 4.     

Notes on proper class of triangles

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.     

Equivalences Induced by Adjoint Functors

Abstract

Let 𝒜 and ? be two Grothendieck categories, R : 𝒜 → ?, L : ? → 𝒜 a pair of adjoint functors, S ∈ ? a generator, and U = L(S). U defines a hereditary torsion class in 𝒜, which is carried by L, under suitable hypotheses, into a hereditary torsion class in ?. We investigate necessary and sufficient conditions which assure that the functors R and L induce equivalences between the quotient categories of 𝒜 and ? modulo these torsion classes. Applications to generalized module categories, rings with local units and group graded rings are also given here.