Tilting modules over Auslander algebras of Nakayama algebras with radical cube zero

Let [Formula: see text] be the Auslander algebra of a finite-dimensional basic connected Nakayama algebra [Formula: see text] with radical cube zero and [Formula: see text] simple modules. Then the cardinality [Formula: see text] of the set consisting of isomorphism classes of basic tilting [Formula: see text]-modules is [Formula: see text]

Classifying tilting modules over the Auslander algebras of radical square zero Nakayama algebras

Let [Formula: see text] be a radical square zero Nakayama algebra with [Formula: see text] simple modules and let [Formula: see text] be the Auslander algebra of [Formula: see text]. Then every indecomposable direct summand of a tilting [Formula: see text]-module is either simple or projective. Moreover, if [Formula: see text] is self-injective, then the number of tilting [Formula: see text]-modules is [Formula: see text]; otherwise, the number of tilting [Formula: see text]-modules is [Formula: see text].

From subcategories to the entire module categories

In this paper we show that how the representation theory of subcategories (of the category of modules over an Artin algebra) can be connected to the representation theory of all modules over some algebra. The subcategories dealing with are some certain subcategories of the morphism categories (including submodule categories studied recently by Ringel and Schmidmeier) and of the Gorenstein projective modules over (relative) stable Auslander algebras. These two kinds of subcategories, as will be seen, are closely related to each other. To make such a connection, we will define a functor from each type of the subcategories to the category of modules over some Artin algebra. It is shown that to compute the almost split sequences in the subcategories it is enough to do the computation with help of the corresponding functors in the category of modules over some Artin algebra which is known and easier to work. Then as an application the most part of Auslander–Reiten quiver of the subcategories is obtained only by the Auslander–Reiten quiver of an appropriate algebra and next adding the remaining vertices and arrows in an obvious way. As a special case, when Λ is a Gorenstein Artin algebra of finite representation type, then the subcategories of Gorenstein projective modules over the {2\times 2} lower triangular matrix algebra over Λ and the stable Auslander algebra of Λ can be estimated by the category of modules over the stable Cohen–Macaulay Auslander algebra of Λ.

Morphism Categories of Gorenstein-projective Modules

Let Λ be an algebra of finite Cohen-Macaulay type and Γ its Cohen-Macaulay Auslander algebra. We are going to characterize the morphism category Mor(Λ-Gproj) of Gorenstein-projective Λ-modules in terms of the module category Γ-mod by a categorical equivalence. Based on this, we obtain that some factor category of the epimorphism category Epi(Λ-Gproj) is a Frobenius category, and also, we clarify the relations among Mor(Λ-Gproj), Mor(T2Λ-Gproj) and Mor(Δ-Gproj), where T2(Λ) and Δ are respectively the lower triangular matrix algebra and the Morita ring closely related to Λ.

STRONGLY QUASI-HEREDITARY ALGEBRAS AND REJECTIVE SUBCATEGORIES

Ringel’s right-strongly quasi-hereditary algebras are a distinguished class of quasi-hereditary algebras of Cline–Parshall–Scott. We give characterizations of these algebras in terms of heredity chains and right rejective subcategories. We prove that any artin algebra of global dimension at most two is right-strongly quasi-hereditary. Moreover we show that the Auslander algebra of a representation-finite algebra $A$ is strongly quasi-hereditary if and only if $A$ is a Nakayama algebra.

A new characterization of Auslander algebras

Let [Formula: see text] be a finite dimensional Auslander algebra. For a [Formula: see text]-module [Formula: see text], we prove that the projective dimension of [Formula: see text] is at most one if and only if the projective dimension of its socle soc[Formula: see text][Formula: see text] is at most one. As an application, we give a new characterization of Auslander algebras [Formula: see text] and prove that a finite dimensional algebra [Formula: see text] is an Auslander algebra provided its global dimension gl.d[Formula: see text][Formula: see text] and an injective [Formula: see text]-module is projective if and only if the projective dimension of its socle is at most one.