For a length abelian category, we show that all torsion-free classes can be classified by using only the information on bricks, including non functorially-finite ones. The idea is to consider the set of simple objects in a torsion-free class, which has the following property: it is a set of bricks where every non-zero map between them is an injection. We call such a set a monobrick. In this paper, we provide a uniform method to study torsion-free classes and wide subcategories via monobricks. We show that monobricks are in bijection with left Schur subcategories, which contains all subcategories closed under extensions, kernels and images, thus unifies torsion-free classes and wide subcategories. Then we show that torsion-free classes bijectively correspond to cofinally closed monobricks. Using monobricks, we deduce several known results on torsion(-free) classes and wide subcategories (e.g. finiteness result and bijections) in length abelian categories, without using $τ$-tilting theory. For Nakayama algebras, left Schur subcategories are the same as subcategories closed under extensions, kernels and images, and we show that its number is related to the large Schröder number.
For a finite dimensional algebra $A$, we establish correspondences between torsion classes and wide subcategories in $mod(A)$. In case $A$ is representation finite, we obtain an explicit bijection between these two classes of subcategories. Moreover, we translate our results to the language of ring epimorphisms and universal localisations. It turns out that universal localisations over representation finite algebras are classified by torsion classes and support $τ$-tilting modules.
In this paper, we compute the posets of wide subcategories and ICE-closed subcategories from the lattice of torsion classes in an abelian length category in a purely lattice-theoretical way, by using the kappa map in a completely semidistributive lattice. As for the poset of wide subcategories, we give two more simple constructions via a bijection between wide subcategories and torsion classes with canonical join representations. More precisely, for a completely semidistributive lattice, we give two poset structures on the set of elements with canonical join representations: the kappa order (d…
In this paper, we study the relationship between wide subcategories and torsion classes of an abelian length category $\mathcal{A}$ from the point of view of lattice theory. Motivated by $τ$-tilting reduction of Jasso, we mainly focus on intervals $[\mathcal{U},\mathcal{T}]$ in the lattice $\operatorname{\mathsf{tors}} \mathcal{A}$ of torsion classes in $\mathcal{A}$ such that $\mathcal{W}:=\mathcal{U}^\perp \cap \mathcal{T}$ is a wide subcategory of $\mathcal{A}$; we call these intervals wide intervals. We prove that a wide interval $[\mathcal{U},\mathcal{T}]$ is isomorphic to the lattice $\o…
The main theme of this paper is to study $τ$-tilting subcategories in an abelian category $\mathscr{A}$ with enough projective objects. We introduce the notion of $τ$-cotorsion torsion triples and show a bijection between the collection of $τ$-cotorsion torsion triples in $\mathscr{A}$ and the collection of $τ$-tilting subcategories of $\mathscr{A}$, generalizing the bijection by Bauer, Botnan, Oppermann and Steen between the collection of cotorsion torsion triples and the collection of tilting subcategories of $\mathscr{A}$. General definitions and results are exemplified using persistent mod…
In the category of finitely generated modules over an artinian ring, we classify all the abelian exact subcategories closed under predecessors or, equivalently, all the split torsion pairs with torsion-free class closed under quotients.
In this paper, we study ICE-closed (= Image-Cokernel-Extension-closed) subcategories of an abelian length category using torsion classes. To each interval $[\mathcal{U},\mathcal{T}]$ in the lattice of torsion classes, we associate a subcategory $\mathcal{T} \cap \mathcal{U}^\perp$ called the heart. We show that every ICE-closed subcategory can be realized as a heart of some interval of torsion classes, and give a lattice-theoretic characterization of intervals whose hearts are ICE-closed. In particular, we prove that ICE-closed subcategories are precisely torsion classes in some wide subcatego…
We give a combinatorial classification of cluster tilting subcategories and torsion pairs in Igusa--Todorov cluster categories of Dynkin type $A_{ \infty }$.
We study IE-closed subcategories of a module category, subcategories which are closed under taking Images and Extensions. We investigate the relation between IE-closed subcategories and torsion pairs, and characterize $τ$-tilting finite algebras using IE-closed subcategories. For the hereditary case, we show that IE-closed subcategories can be classified by twin rigid modules, pairs of rigid modules satisfying some homological conditions. Moreover, we introduce mutation of twin rigid modules analogously to tilting modules, which gives a way to calculate all twin rigid modules for the represent…
We introduce image-cokernel-extension-closed (ICE-closed) subcategories of module categories. This class unifies both torsion classes and wide subcategories. We show that ICE-closed subcategories over the path algebra of Dynkin type are in bijection with basic rigid modules, that ICE-closed subcategories are precisely torsion classes in some wide subcategories, and that the number does not depend on the orientation of the quiver. We give an explicit formula of this number for each Dynkin type, and in particular, it is equal to the large Schröder number for type A case.
Let $A$ be an artinian algebra, and let $\mathcal{C}$ be a subcategory of mod$A$ that is closed under extensions. When $\mathcal{C}$ is closed under kernels of epimorphisms (or closed under cokernels of monomorphisms), we describe the smallest class of modules that filters $\mathcal{C}$. As a consequence, we obtain sufficient conditions for the finitistic dimension of an algebra over a field to be finite. We also apply our results to the torsion pairs. In particular, when a torsion pair is induced by a tilting module, we show that the smallest classes of modules that filter the torsion and tor…
If (A,B) and (A',B') are co-t-structures of a triangulated category, then (A',B') is called intermediate if A \subseteq A' \subseteq ΣA. Our main results show that intermediate co-t-structures are in bijection with two-term silting subcategories, and also with support tau-tilting subcategories under some assumptions. We also show that support tau-tilting subcategories are in bijection with certain finitely generated torsion classes. These generalise results by Adachi, Iyama, and Reiten.
We give a full classification of representation types of the subcategories of representations of an $m \times n$ rectangular grid with monomorphisms (dually, epimorphisms) in one or both directions, which appear naturally in the context of clustering as two-parameter persistent homology in degree zero. We show that these subcategories are equivalent to the category of all representations of a smaller grid, modulo a finite number of indecomposables. This equivalence is constructed from a certain cotorsion torsion triple, which is obtained from a tilting subcategory generated by said indecomposa…
$r$ICE-closed subcategories induced by the morphism category of projective modules
Nanjing University of Information Science & Technology（南京信息工程大学）；University of Isfahan（伊斯法罕大学）；Institute for Research in Fundamental Sciences (IPM)（伊朗基础科学研究所）；Zhejiang Normal University（浙江师范大学）
Let $Λ$ be a finite-dimensional algebra over a field $K$. We describe how Buan and Marsh's $τ$-exceptional sequences can be used to give a "brick labeling" of a certain poset of wide subcategories of finitely-generated $Λ$-modules. When $Λ$ is representation-directed, we prove that there exists a total order on the set of bricks which makes this into an EL-labeling. Motivated by the connection between classical exceptional sequences and noncrossing partitions, we then turn our attention towards the study of (well-separated) completely semidistributive lattices. Such lattices come equipped with…
For finite semidistributive lattices the map $κ$ gives a bijection between the sets of completely join-irreducible elements and completely meet-irreducible elements. Here we study the $κ$-map in the context of torsion classes. It is well-known that the lattice of torsion classes for an artin algebra is semidistributive, but in general it is far from finite. We show the $κ$-map is well-defined on the set of completely join-irreducible elements, even when the lattice of torsion classes is infinite. We then extend $κ$ to a map on torsion classes which have canonical join representations given by …
For a path algebra $A$ over a quiver $Q$, there are bijections between the support-tilting modules of $A$, torsion classes in $\mathrm{mod}(A)$ and wide subcategories in $\mathrm{mod}(A)$; these are part of the Ingalls-Thomas bijections. As a blueprint for further study, we show how these bijections manifest themselves for higher Auslander algebras of linearly oriented type $A$. In particular, we introduce a higher analogue of torsion classes in $d$-representation-finite algebras.
This is a survey on the categorification of the poset of generalized non-crossing partitions, using the representation theory of a hereditary artin algebra H, looking at the set P of exceptional subcategories in mod H. This categorification is due to Ingalls and Thomas, and a subsequent paper by Igusa and Schiffler. Starting point is a refinement of the classical tilting theory for mod H, replacing torsion pairs by torsion triples, thus putting it into the realm of the stability theory of King. The torsion pairs in mod H correspond nicely to the perpendicular pairs of exceptional subcategories…
We consider the closure space on the set of strings of a gentle algebra of finite representation type. Palu, Pilaud, and Plamondon proved that the collection of all biclosed sets of strings forms a lattice, and moreover, that this lattice is congruence-uniform. Many interesting examples of finite congruence-uniform lattices may be represented as the lattice of torsion classes of an associative algebra. We introduce a generalization, the lattice of torsion shadows, and we prove that the lattice of biclosed sets of strings is isomorphic to a lattice of torsion shadows. Finite congruence-uniform …
Higher homological algebra was introduced by Iyama. It is also known as $n$-homological algebra where $n \geq 2$ is a fixed integer, and it deals with $n$-cluster tilting subcategories of abelian categories. All short exact sequences in such a subcategory are split, but it has nice exact sequences with $n+2$ objects. This was recently formalised by Jasso in the theory of $n$-abelian categories. There is also a derived version of $n$-homological algebra, formalised by Geiss, Keller, and Oppermann in the theory of $( n+2 )$-angulated categories (the reason for the shift from $n$ to $n+2$ is that…
The exchange graph of a 2-acyclic quiver is the graph of mutation-equivalent quivers whose edges correspond to mutations. When the quiver admits a nondegenerate Jacobi-finite potential, the exchange graph admits a natural acyclic orientation called the oriented exchange graph, as shown by Brüstle and Yang. The oriented exchange graph is isomorphic to the Hasse diagram of the poset of functorially finite torsion classes of a certain finite dimensional algebra. We prove that lattices of torsion classes are semidistributive lattices, and we use this result to conclude that oriented exchange graph…
