Examples of Auslander-Reiten components in the bounded derived Category

TL;DR

This paper investigates the structure of Auslander-Reiten components in the bounded derived category of finite-dimensional algebras, providing classification results, explicit computations for Nakayama algebras, and conditions for certain components to appear.

Contribution

It introduces a necessary condition for Euclidean tree class components, classifies some irreducible maps, and generalizes conditions for $ ext{Z}[A_{ ext{infinity}}]$-components.

Findings

Finitely many Euclidean tree class components up to shift.

Explicit Auslander-Reiten quivers for Nakayama algebras.

Generalized condition for $ ext{Z}[A_{ ext{infinity}}]$-components.

Abstract

We deduce a necessary condition for Auslander-Reiten components of the bounded derived category of a finite dimensional algebra to have Euclidean tree class by classifying certain types of irreducible maps in the category of complexes. This result shows that there are only finitely many Auslander-Reiten components with Euclidean tree class up to shift. Also the Auslander-Reiten quiver of certain classes of Nakayama are computed directly and it is shown that they are piecewise hereditary. Finally we state a condition for $\Z[A_{\infty}]$-components to appear in the Auslander-Reiten quiver generalizing a result in \cite{W}.