Finite Subcategories Research Articles

Right n-angulated categories arising from covariantly finite subcategories

ABSTRACTWe 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.

Artin Algebras of Finite Type and Finite Categories of Δ-Good Modules

We give an alternative proof to the fact that, if the square of the infinite radical of the module category of an Artin algebra is equal to zero, then the algebra is of finite type by making use of the theory of postprojective and preinjective partitions. Further, we use this new approach in order to get a characterization of finite subcategories of Δ-good modules of a quasi-hereditary algebra in terms of depth of morphisms similar to a recently obtained characterization of Artin algebras of finite type.

Auslander-Reiten Sequences or Triangles Related to Rigid Subcategories

Let 𝒞 be a triangulated category which has Auslander-Reiten triangles, and ℛ a functorially finite rigid subcategory of 𝒞. It is well known that there exist Auslander-Reiten sequences in mod ℛ. In this paper, we give explicitly the relations between the Auslander-Reiten translations, sequences in mod ℛ and the Auslander-Reiten functors, triangles in 𝒞, respectively. Furthermore, if 𝒯 is a cluster-tilting subcategory of 𝒞 and mod 𝒯 is a Frobenius category, we also get the Auslander-Reiten functor and the translation functor of mod 𝒯 corresponding to the ones in 𝒞. As a consequence, we get that if the quotient of a d-Calabi-Yau triangulated category modulo a cluster tilting subcategory is Frobenius, then its stable category is (2d-1)-Calabi-Yau. This result was first proved by Keller and Reiten in the case d=2, and then by Dugas in the general case, using different methods.

On Kan-injectivity of Locales and Spaces

In the category T o p 0 of T 0-spaces and continuous maps, embeddings are just those morphisms with respect to which the Sierpinski space is Kan-injective, and the Kan-injective hull of the Sierpinski space is the category of continuous lattices and maps preserving directed suprema and arbitrary infima. In the category L o c of locales and localic maps, we give an analogous characterization of flat embeddings; more generally, we characterize n-flat embeddings, for each cardinal n, as those morphisms with respect to which a certain finite subcategory is Kan-injective. As a consequence, we obtain similar characterizations of the n-flat embeddings in the category T o p 0, and we show that several well-known subcategories of L o c and T o p 0 are Kan-injective hulls of finite subcategories. Moreover, we show that there is a subcategory of spatial locales whose Kan-injective hull is the entire category L o c.

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]. 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.

The Left and Right Triangulated Structures of Stable Categories

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.

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.

Relative singularity categories

We study the properties of the relative derived category DCb(A) of an abelian category A relative to a full and additive subcategory C. In particular, when A=A-mod for a finite-dimensional algebra A over a field and C is a contravariantly finite subcategory of A-mod which is admissible and closed under direct summands, the C-singularity category DC-sg(A)=DCb(A)/Kb(C) is studied. We give a sufficient condition when this category is triangulated equivalent to the stable category of the Gorenstein category G(C) of C.

On Artin Algebras Arising from Morita Contexts

We study Morita rings \(\Lambda _{(\phi ,\psi )}=\left (\begin {array}{cc}A &_{A}N_{B} _{B}M_{A} & B \end {array}\right )\) in the context of Artin algebras from various perspectives. First we study covariantly finite, contravariantly finite, and functorially finite subcategories of the module category of a Morita ring when the bimodule homomorphisms \(\phi \) and \(\psi \) are zero. Further we give bounds for the global dimension of a Morita ring \(\Lambda _{(0,0)}\), as an Artin algebra, in terms of the global dimensions of A and B in the case when both \(\phi \) and \(\psi \) are zero. We illustrate our bounds with some examples. Finally we investigate when a Morita ring is a Gorenstein Artin algebra and then we determine all the Gorenstein-projective modules over the Morita ring \(\Lambda _{\phi ,\psi }\) in case \(A=N=M=B\) and A an Artin algebra.

Gorenstein derived equivalences and their invariants

The main objective of this paper is to study the relative derived categories from various points of view. Let A be an abelian category and C be a contravariantly finite subcategory of A. One can define C-relative derived category of A, denoted by DC⁎(A). The interesting case for us is when A has enough projective objects and C=GP-A is the class of Gorenstein projective objects, where DC⁎(A) is known as the Gorenstein derived category of A. We explicitly study the relative derived categories, specially over artin algebras, present a relative version of Rickardʼs theorem, specially for Gorenstein derived categories, provide some invariants under Gorenstein derived equivalences and finally study the relationships between relative and (absolute) derived categories.

Order-preserving reflectors and injectivity

We investigate a Galois connection in poset enriched categories between subcategories and classes of morphisms, given by means of the concept of right-Kan injectivity, and, specially, we study its relationship with a certain kind of subcategories, the KZ-reflective subcategories. A number of well-known properties concerning orthogonality and full reflectivity can be seen as a particular case of the ones of right-Kan injectivity and KZ-reflectivity. On the other hand, many examples of injectivity in poset enriched categories encountered in the literature are closely related to the above connection. We give several examples and show that some known subcategories of the category of T 0 -topological spaces are right-Kan injective hulls of a finite subcategory.

Monomorphism categories, cotilting theory, and Gorenstein-projective modules

The monomorphism category S n ( X ) is introduced, where X is a full subcategory of the module category A-mod of an Artin algebra A. The key result is a reciprocity of the monomorphism operator S n and the left perpendicular operator ⊥: for a cotilting A-module T, there is a canonical construction of a cotilting module m ( T ) over the upper triangular matrix algebra T n ( A ) , such that S n ( T ⊥ ) = m ⊥ ( T ) . As applications, S n ( X ) is a resolving contravariantly finite subcategory in T n ( A ) -mod with S n ( X ) ˆ = T n ( A ) -mod if and only if X is a resolving contravariantly finite subcategory in A-mod with X ˆ = A -mod. For a Gorenstein algebra A, the category T n ( A ) - G proj of Gorenstein-projective T n ( A ) -modules can be explicitly determined as S n ( A ⊥ ) . Also, self-injective algebras A can be characterized by the property T n ( A ) - G proj = S n ( A ) . Finally, we obtain a characterization of those categories S n ( A ) which have finite representation type in terms of Auslanderʼs representation dimension.

On Auslander–Reiten translates in functorially finite subcategories and applications

We consider functorially finite subcategories in module categories over Artin algebras. One main result provides a method, in the setup of bounded derived categories, to compute approximations and the end terms of relative Auslander–Reiten sequences

Extensions of covariantly finite subcategories

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.

Contravariantly finite subcategories closed under predecessors

Let A be an Artin algebra and mod A be the category of finitely generated right A-modules. We prove that an additive full subcategory C of mod A closed under predecessors is contravariantly finite if and only if its right Ext-orthogonal is covariantly finite, or if and only if the Ext-injectives in C define a cotilting module (over the support algebra of C ) or, equivalently, if and only if C is the support of the representable functors given by the Ext-injectives.

Pure Projective Approximations

AbstractA notion of good behavior is introduced for a definable subcategory of left R-modules. It is proved that every finitely presented left R-module has a pure projective left -approximation if and only if the associated torsion class of finite type in the functor category (mod-R, Ab) is coherent, i.e., the torsion subobject of every finitely presented object is finitely presented. This yields a bijective correspondence between such well-behaved definable subcategories and preenveloping subcategories of the category Add(R-mod) of pure projective left R-modules. An example is given of a preenveloping subcategory ⊆ Add(R-mod) that does not arise from a covariantly finite subcategory of finitely presented left R-modules. As a general example of this good behavior, it is shown that if R is a ring over which every left cotorsion R-module is pure injective, then the smallest definable subcategory (R-proj) containing every finitely generated projective module is well-behaved.

Liftings of distributive lattices by locally matricial algebras with respect to the Idc functor

We study representations of distributive \( {\left\langle {0,1} \right\rangle } \)-lattices, considered as join-semilattices, by semilattices of finitely generated two-sided ideals of locally matricial algebras over a field k, aiming to find a functorial solution of the problem. We find simple examples of a finite subcategory of the category Ld of distributive \( {\left\langle {0,1} \right\rangle } \)-lattices and of a subcategory of Ld corresponding to a partially ordered class which cannot be lifted with respect to the Idc functor. On the other hand, we prove that there is such a lifting of every diagram in Ld or of a subcategory Ld1 of Ld whose objects are all distributive \( {\left\langle {0,1} \right\rangle } \)-lattices and whose morphisms are \( {\left\langle { \vee , \wedge ,0,1} \right\rangle } \)-embeddings.

Tilting categories with applications to stratifying systems

In the study of standardly stratified algebras and stratifying systems, we find an object which is either a tilting module or one whose properties strongly remind us of a tilting module. This tilting module appeared already in Dlab and Ringel's work on quasi-hereditary algebras (see [V. Dlab, C.M. Ringel, The module theoretical approach to quasi-hereditary algebras, in: Repr. Theory and Related Topics, in: London Math. Soc. Lecture Note Ser. 168 (1992) 200–224] and [C.M. Ringel, The category of modules with good filtrations over a quasi-hereditary algebra has almost split sequences, Math. Z. 208 (1991) 209–223]). Also, this tilting module appears on the work on standardly stratified algebras of I. Ágoston, D. Happel, E. Lukács, and L. Unger, and in the paper of M.I. Platzeck and I. Reiten (see [I. Ágoston, D. Happel, E. Lukács, L. Unger, Standardly stratified algebras and tilting, J. Algebra 226 (2000) 144–160] and [M.I. Platzeck, I. Reiten, Modules of finite projective dimension for standardly stratified algebras, Comm. Algebra 29 (3) (2001) 973–986]). Inspired by them, we introduce the notion of tilting category in order to give a unified approach of these situations for stratifying systems. To do so, we use the ideas of M. Auslander, O. Buchweitz and I. Reiten related to approximation theory (see [M. Auslander, R.O. Buchweitz, The homological theory of maximal Cohen–Macaulay approximations. Mem. Soc. Math. Fr. (N.S.) 38 (1989) 5–37; M. Auslander, I. Reiten, Applications of contravariantly finite subcategories, Adv. Math. 86 (1991) 111–152]).

Coherent Functors and Covariantly Finite Subcategories

Given a locally presentable additive category A, we study a class of covariantly finite subcategories which we call definable. A definable subcategory arises from a set of coherent functors F i on A by taking all objects X in A such that F i X=0 for all i. We give various characterizations of definable subcategories, demonstrating that all covariantly finite subcategories which arise in practice are of this form. This is based on a filtration of the category of all coherent functors on A.

Subcategories of the derived category and cotilting complexes

We show that there is a one-to-one correspondence between basic cotilting complexes and certain contravariantly finite subcategories of the bounded derived category of an artin algebra. This is a triangulated version of a result by Auslander and Reiten. We use this to find an existence criterion for complements to exceptional complexes. Introduction. Homologically finite subcategories were introduced by Auslander and Smalo (3), and they have proved to be important in the study of artin algebras. Homologically finite subcategories of the category of finitely generated modules have been studied by several authors. In (1), Auslander and Reiten showed that there is a correspondence between cer- tain contravariantly finite subcategories and basic cotilting modules. In this paper we consider some subcategories of the bounded derived category of an artin algebra that are associated with cotilting complexes. In the first sec- tion we give definitions and basic results that we use in the second section, where we show that there is a correspondence between cotilting complexes and certain contravariantly finite subcategories of the derived category. The third section is devoted to examples. In the fourth section we use the cor- respondence to prove an existence criterion for complements of exceptional complexes. 1. Subcategories of the derived category. Let Λ be an artin algebra. Let mod Λ be the category of finitely generated left Λ-modules, and let D = D b (mod Λ) be the bounded derived category. This is a triangulated category. We denote the shift functor by (1), and its inverse by (−1). We let I(Λ) be the full subcategory of mod Λ formed by the injective objects, and, similarly, P(Λ) stands for the projectives. Then D b (mod Λ) is equivalent to K +,b (I(Λ)). We consider this an identification, and let K b (I(Λ)) denote the coperfect complexes. By a subcategory, we will always mean a full additive