Representation theory of graded algebras given by locally finite quivers

TL;DR

This paper develops a new framework for understanding the representation theory of graded algebras from locally finite quivers, establishing almost split sequences and triangles in various categories of graded modules.

Contribution

It introduces a graded Nakayama functor and derives new Auslander-Reiten formulas, enabling the construction of almost split sequences and triangles in graded module categories.

Findings

Existence of almost split sequences in categories of finitely presented and copresented graded modules.

Characterization of indecomposable complexes with almost split triangles in bounded derived categories.

Conditions under which the bounded derived category has almost split triangles based on simple modules' projective and injective dimensions.

Abstract

This paper aims to study graded modules over a graded algebra $\La$ given by a locally finite quiver with homogeneous relations. By constructing a graded Nakayama functor, we discover a novel approach to establish Auslander-Reiten formulas, from which we derive almost split sequences in the category of all graded $\La$-modules. In case $\La$ is locally left (respectively, right) bounded, the category of finitely presented graded modules and that of finitely copresented graded modules both have almost split sequences on the left (respectively, right). We shall also obtain existence theorems for almost split triangles in various derived categories of graded $\La$-modules. In case $\La$ is locally bounded, an indecomposable complex in the bounded derived category of finite dimensional graded modules is the starting (respectively, ending) term of an almost split triangle if and only if it has a finite graded projective reso\-lution (respectively, injective coresolution); and consequently, this bounded derived category has almost split triangles on the right (respectively, left) if and only if every graded simple module is of finite graded projective (respectively, injective) dimension. Finally, we specialize to the existence of almost split sequences and almost split triangles for graded representations of any locally finite quiver.