Finite Subcategories Research Articles

The radical of functorially finite subcategories

Let Λ be an artin algebra and C be a functorially finite subcategory of mod Λ which contains Λ or DΛ. We use the concept of the infinite radical of C and show that C has an additive generator if and only if radC∞ vanishes. In this case we describe the morphisms in powers of the radical of C in terms of its irreducible morphisms. Moreover, under a mild assumption, we prove that C is of finite representation type if and only if any family of monomorphisms (epimorphisms) between indecomposable objects in C is noetherian (conoetherian). Also, by using injective envelopes, projective covers, left C-approximations and right C-approximations of simple Λ-modules, we give other criteria to describe whether C is of finite representation type. In addition, we give a nilpotency index of the radical of C which is independent from the maximal length of indecomposable Λ-modules in C.

The completion of d-abelian categories

Let A be a finite-dimensional algebra, and M be a d-cluster tilting subcategory of mod A. From the viewpoint of higher homological algebra, a natural question to ask is when M induces a d-cluster tilting subcategory in Mod A. In this paper, we investigate this question in a more general form. We consider M as an essentially small d-abelian category, known to be equivalent to a d-cluster tilting subcategory of an abelian category A. The completion of M, denoted by Ind(M), is defined as the universal completion of M with respect to filtered colimits. We explore Ind(M) and demonstrate its equivalence to the full subcategory Ld(M) of ModM, comprising left d-exact functors. Notably, Ind(M) as a subcategory of ModMEff(M) falls short of being a d-cluster tilting subcategory since it satisfies all properties of a d-cluster tilting subcategory except d-rigidity. For a d-cluster tilting subcategory M of mod A, M→ consists of all filtered colimits of objects from M, is a generating-cogenerating, functorially finite subcategory of Mod A. The question of whether M→ is a d-rigid subcategory remains unanswered. However, if it is indeed d-rigid, it qualifies as a d-cluster tilting subcategory. In the case d=2, employing cotorsion theory, we establish that M→ is a 2-cluster tilting subcategory if and only if M is of finite type. Thus, the question regarding whether M→ is a d-cluster tilting subcategory of Mod A appears to be equivalent to Iyama's question about the finiteness of M. Furthermore, for general d, we address the problem and present several equivalent conditions for Iyama's question.

On algebras of Ωn-finite and Ω∞-infinite representation type

Co-Gorenstein algebras were introduced by Beligiannis in [A. Beligiannis, The homological theory of contravariantly finite subcategories: Auslander–Buchweitz contexts, Gorenstein categories and co-stabilization, Comm. Algebra 28(10) (2000) 4547–4596]. In [S. Kvamme and R. Marczinzik, Co-Gorenstein algebras, Appl. Categorical Struct. 27(3) (2019) 277–287], the authors propose the following conjecture (co-GC): if [Formula: see text] is extension closed for all [Formula: see text], then [Formula: see text] is right co-Gorenstein, and they prove that the generalized Nakayama conjecture implies the co-GC, also that the co-GC implies the Nakayama conjecture. In this paper, we characterize the subcategory [Formula: see text] for algebras of [Formula: see text]-finite representation type. As a consequence, we characterize when a truncated path algebra is a co-Gorenstein algebra in terms of its associated quiver. We also study the behavior of Artin algebras of [Formula: see text]-infinite representation type. Finally, an example of a non-Gorenstein algebra of [Formula: see text]-infinite representation type and an example of a finite dimensional algebra with infinite [Formula: see text]-dimension are presented.

A new characterization of silting subcategories in the stable category of a Frobenius extriangulated category

We give a new characterization of silting subcategories in the stable category of a Frobenius extriangulated category, generalizing the result of Di, Liu, Wang and Wei (J. Algebra 525 (2019) 42–63) about the Auslander-Reiten type correspondence for silting subcategories over triangulated categories. More specifically, for any Frobenius extriangulated category , we establish a bijective correspondence between silting subcategories of the stable category and certain covariantly finite subcategories of . As a consequence, a characterization of silting subcategories in the stable category of a Frobenius exact category is given. This result is applied to homotopy categories over abelian categories with enough projectives, derived categories over Grothendieck categories with enough projectives, and to the stable category of Gorenstein projective modules over a ring R.

Pre-(n + 2)-angulated categories

In this article, we introduce the notion of a pre-(n+2)-angulated category as a higher dimensional analogue of a pre-triangulated category defined by Beligiannis-Reiten. We first show that the idempotent completion of a pre-(n+2)-angulated category admits a unique pre-(n+2)-angulated structure. Let (C,E,s) be an n-exangulated category and X be a strongly functorially finite subcategory of C. We then show that the quotient category C/X is a pre-(n+2)-angulated category. These results allow to construct several examples of pre-(n+2)-angulated categories. Moreover, we also give a necessary and sufficient condition for the quotient C/X to be an (n+2)-angulated category.

Axiomatizing subcategories of Abelian categories

We investigate how to characterize subcategories of abelian categories in terms of intrinsic axioms. In particular, we find axioms which characterize generating cogenerating functorially finite subcategories, precluster tilting subcategories, and cluster tilting subcategories of abelian categories. As a consequence we prove that any d -abelian category is equivalent to a d -cluster tilting subcategory of an abelian category, without any assumption on the categories being projectively generated.

Triangular matrix categories II: Recollements and functorially finite subcategories

In this paper we continue the study of triangular matrix categories $\mathbf {{\varLambda }}=\left [\begin {smallmatrix} \mathcal {T} & 0 \\ M & \mathcal {U} \end {smallmatrix}\right ]$ initiated in León-Galeana et al. (2022). First, given a additive category $\mathcal {C}$ and an ideal $\mathcal {I}_{{\mathscr{B}}}$ in $\mathcal {C}$ , we prove a well known result that there is a canonical recollement We show that given a recollement between functor categories we can induce a new recollement between triangular matrix categories, this is a generalization of a result given by Chen and Zheng in (J. Algebra, 321 (9), 2474–2485 2009, [Theorem 4.4]). In the case of dualizing K-varieties we can restrict the recollement we obtained to the categories of finitely presented functors. Given a dualizing variety $\mathcal {C}$ , we describe the maps category of $\text {mod}(\mathcal {C})$ as modules over a triangular matrix category and we study its Auslander-Reiten sequences and contravariantly finite subcategories, in particular we generalize several results from Martínez-Villa and Ortíz-Morales (Inter. J Algebra, 5 (11), 529–561 2011). Finally, we prove a generalization of a result due to Smalø (2011, [Theorem 2.1]), which give us a way of construct functorially finite subcategories in the category $\text {Mod}\Big (\left [\begin {smallmatrix} \mathcal {T} & 0 \\ M & \mathcal {U} \end {smallmatrix}\right ]\Big )$ from those of $\text {Mod}(\mathcal {T})$ and $\text {Mod}(\mathcal {U})$ .

The recollements induced by contravariantly finite subcategories

The recollements induced by contravariantly finite subcategories

Notes on Relative Extension Functors

The objective of this paper is to study the relative homological properties of contravariantly and covariantly finite subcategories. Some sufficient conditions for are obtained. We also give the conditions under which the stable categories are one‐side triangulated categories.

Triangular matrix coalgebras: Representation theory and recollement

For any triangular matrix coalgebra [Formula: see text], in this paper, we first examine some connections between coalgebra properties of [Formula: see text] and its constituent coalgebras C, D, which contain semiperfectness, computability and row/column-finiteness of their left Cartan matices. Then we devote to considering the coresolution dimensions of recollement of comodule categories by investigating covariantly finite subcategories.

Quotient categories of extriangulated categories

Let be a subcategory of an extriangulated category and let be a strongly functorially finite subcategory of Then the subfactor category admits a pre-triangulated structure in the sense of Beligiannis-Reiten where has the following properties: For any -triangle in where is -monic, the third term C is also in For any -triangle in where is -epic, the first term A is also in Moreover, if the above pre-triangulated category with becomes a triangulated category, then forms an -mutation pair and is closed under extensions.

QUOTIENT CATEGORIES OF n-ABELIAN CATEGORIES

AbstractThe notion of mutation pairs of subcategories in an n-abelian category is defined in this paper. Let ${\cal D} \subseteq {\cal Z}$ be subcategories of an n-abelian category ${\cal A}$. Then the quotient category ${\cal Z}/{\cal D}$ carries naturally an (n + 2) -angulated structure whenever $ ({\cal Z},{\cal Z}) $ forms a ${\cal D} \subseteq {\cal Z}$-mutation pair and ${\cal Z}$ is extension-closed. Moreover, we introduce strongly functorially finite subcategories of n-abelian categories and show that the corresponding quotient categories are one-sided (n + 2)-angulated categories. Finally, we study homological finiteness of subcategories in a mutation pair.

On the representation type of subcategories of derived categories

Let [Formula: see text] be a finite-dimensional [Formula: see text]-algebra. In this paper, we mainly study the representation type of subcategories of the bounded derived category [Formula: see text]. First, we define the representation type and some homological invariants including cohomological length, width, range for subcategories. In this framework, we provide a characterization for derived discrete algebras. Moreover, for a finite-dimensional algebra [Formula: see text], we establish the first Brauer–Thrall type theorem of certain contravariantly finite subcategories [Formula: see text] of [Formula: see text], that is, [Formula: see text] is of finite type if and only if its cohomological range is finite.

An Auslander–Buchweitz approximation approach to (pre)silting subcategories in triangulated categories

We apply the Auslander–Buchweitz approximation triangles to study (pre)silting subcategories in a triangulated catego- ry T. An Auslander–Reiten type correspondence between the class of silting subcategories of T and that of certain covariantly finite subcategories of T is established. We introduce a relation on presilting subcategories of T, which can be used to obtain another silting subcategory from a given one. We also give a Bazzoni's characterization for two presilting subcategories satisfying such a relation. The results can be used to improve the Auslander–Reiten type correspondence and Bazzoni's characterization for small silting complexes established by Wei.

Recollements for Dualizing k-Varieties and Auslander’s Formulas

Given the pair of a dualizing $k$-variety and its functorially finite subcategory, we show that there exists a recollement consisting of their functor categories of finitely presented objects. We provide several applications for Auslander's formulas: The first one realizes a module category as a Serre quotient of a suitable functor category. The second one shows a close connection between Auslander-Bridger sequences and recollements. The third one gives a new proof of the higher defect formula which includes the higher Auslander-Reiten duality as a special case.

English

Extriangulated categories were introduced by Nakaoka and Palu by extracting the similarities between exact categories and triangulated categories. A notion of homotopy cartesian square in an extriangulated category is defined in this article. We prove that in an extriangulated category with enough projective objects, the extension subcategory of two covariantly finite subcategories is covariantly finite. As an application, we give a simultaneous generalization of a result of X. W. Chen (2009) and of a result of R. Gentle, G. Todorov (1996).

Cosilting complexes and AIR-cotilting modules

We introduce and study the new concepts of cosilting complexes, cosilting modules and AIR-cotilting modules. We prove that the three concepts AIR-cotilting modules, cosilting modules and quasi-cotilting modules coincide with each other, in contrast with the dual fact that AIR-tilting modules, silting modules and quasi-tilting modules are different. Further, we show that there are bijections between the following four classes (1) equivalence classes of AIR-cotilting (resp., cosilting, quasi-cotilting) modules, (2) equivalence classes of 2-term cosilting complexes, (3) torsion-free cover classes and (4) torsion-free special precover classes. We also extend a classical result of Auslander and Reiten on the correspondence between certain contravariantly finite subcategories and cotilting modules to the case of cosilting complexes.

Some remarks on homologically finite subcategories

Let \(\Lambda \) be an Artin algebra and \(\mathcal {C}\) a full subcategory of \(\Lambda \)-mod closed under direct summands and closed under extensions. It is known that if \(\mathcal {C}\) 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.

A right triangulated version of Gentle-Todorov’s theorem

ABSTRACTGentle 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.

Pro-Species of Algebras I: Basic Properties

In this paper, we generalise part of the theory of hereditary algebras to the context of pro-species of algebras. Here, a pro-species is a generalisation of Gabriel’s concept of species gluing algebras via projective bimodules along a quiver to obtain a new algebra. This provides a categorical perspective on a recent paper by Geis et al. (2016). In particular, we construct a corresponding preprojective algebra, and establish a theory of a separated pro-species yielding a stable equivalence between certain functorially finite subcategories.