The bounded derived categories of an algebra with radical squared zero

TL;DR

This paper explores the structure of bounded derived categories of modules over an algebra with radical squared zero, revealing their Galois coverings, indecomposable objects, and Auslander-Reiten components.

Contribution

It establishes a Galois covering of the derived category by representations of a gradable quiver and classifies algebras based on their Auslander-Reiten components.

Findings

Derived category admits a Galois covering from a gradable quiver representation

Complete description of indecomposable objects in the derived category

Classification of algebras with finitely many Auslander-Reiten components

Abstract

Let $\La$ be an elementary locally bounded linear category over a field with radical squared zero. We shall show that the bounded derived category $D^b(\ModbLa)$ of finitely supported left $\La$-modules admits a Galois covering which is the bounded derived category of almost finitely co-presented representations of a gradable quiver. Restricting to the bounded derived category $D^b({\rm mod}^b\hspace{-2pt}\La)$ of finite dimensional left $\La$-modules, we shall be able to describe its indecomposable objects, obtain a complete description of the shapes of its Auslander-Reiten components, and classify those $\La$ such that $D^b({\rm mod}^b\hspace{-2.3pt}\La)$ has only finitely many Auslander-Reiten components.