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].

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 NOTE ON RESOLUTION QUIVERS

Recently, Ringel introduced the resolution quiver for a connected Nakayama algebra. It is known that each connected component of the resolution quiver has a unique cycle. We prove that all cycles in the resolution quiver are of the same size. We introduce the notion of weight for a cycle in the resolution quiver. It turns out that all cycles have the same weight.