We show that for a given Nakayama algebra Θ, there exist countably many cyclic Nakayama algebras Λ_i, where i ∈ℕ, such that the syzygy filtered algebra of Λ_i is isomorphic to Θ and we describe those algebras Λ_i. We show, among these algebras, there exists a unique algebra Λ where the defects, representing the number of indecomposable injective but not projective modules, remain invariant for both Θ and Λ. As an application, we achieve the classification of cyclic Nakayama algebras that are minimal Auslander-Gorenstein and dominant Auslander-regular algebras of global dimension three. Specifically, by using the Auslander-Iyama correspondence, we obtain cluster-tilting objects for certain Nakayama algebras. Additionally, we introduce cosyzygy filtered algebras and show that it is dual of syzygy filtered algebra.

For finite semidistributive lattices the map kappa gives a bijection between the sets of completely join-irreducible elements and completely meet-irreducible elements. Here we study the kappa-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 kappa-map is well-defined on the set of completely join-irreducible elements, even when the lattice of torsion classes is infinite. We then extend kappa to a map on torsion classes which have canonical join representations given by the special torsion classes associated to the minimal extending modules introduced by the first and third authors and A. Carroll in 2019. For hereditary algebras, we show that the extended kappa-map on torsion classes is essentially the same as Ringel's epsilon-map on wide subcategories. Also in the hereditary case, we relate the square of kappa to the Auslander-Reiten translation. Published by Elsevier B.V.

We study the existence and uniqueness of minimal right determiners in various categories. Particularly in a $${{\,\mathrm{Hom}\,}}$$-finite hereditary abelian category with enough projectives, we prove that the Auslander–Reiten–Smalø–Ringel formula of the minimal right determiner still holds. As an application, we give a formula of minimal right determiners in the category of finitely presented representations of strongly locally finite quivers.

Let Λ be a finite-dimensional associative algebra. The torsion classes of modΛ form a lattice under containment, denoted by torsΛ. In this paper, we characterize the cover relations in torsΛ by certain indecomposable modules. We consider three applications: First, we show that the completely join-irreducible torsion classes (torsion classes which cover precisely one element) are in bijection with bricks. Second, we characterize faces of the canonical join complex of torsΛ in terms of representation theory. Finally, we show that, in general, the algebra Λ is not characterized by its lattice torsΛ. In particular, we study the torsion theory of a quotient of the preprojective algebra of type A n . We show that its torsion class lattice is isomorphic to the weak order on A n .

We study which algebras have tilting modules that are both generated and cogenerated by projective–injective modules. Crawley–Boevey and Sauter have shown that Auslander algebras have such tilting modules; and for algebras of global dimension 2, Auslander algebras are classified by the existence of such tilting modules. In this paper, we show that the existence of such a tilting module is equivalent to the algebra having dominant dimension at least 2, independent of its global dimension. In general such a tilting module is not necessarily cotilting. Here, we show that the algebras which have a tilting–cotilting module generated–cogenerated by projective–injective modules are precisely 1-minimal Auslander–Gorenstein algebras. When considering such a tilting module, without the assumption that it is cotilting, we study the global dimension of its endomorphism algebra, and discuss a connection with the Finitistic Dimension Conjecture. Furthermore, as special cases, we show that triangular matrix algebras obtained from Auslander algebras and certain injective modules, have such a tilting module. We also give a description of which Nakayama algebras have such a tilting module.

Let be the path algebra of a tree-type quiver Q, and lambda be a nonzero element in a field . We construct irreducible morphisms in the Auslander-Reiten quiver of the transjective component of the bounded derived category of that satisfy what we call the lambda-relations. When lambda = 1, the relations are known as mesh relations. When , they are known as commutativity relations. We give a new description of the preprojective algebra of and using our technique of constructing irreducible maps together with the results given by Baer-Geigle-Lenzing, Crawley-Boevey, Ringel, and others, we show that for any tree-type quiver, our description is equivalent to several other definitions of preprojective algebras, previously introduced in various contexts.