Realizing stable categories as derived categories

TL;DR

This paper establishes new triangle-equivalences connecting the stable categories of graded modules over self-injective algebras with derived categories of finite global dimension, revealing deep structural relationships.

Contribution

It constructs explicit triangle-equivalences linking stable categories of graded modules over self-injective algebras to derived categories of finite global dimension, including orbit categories.

Findings

Established a triangle-equivalence between stable category of $ ext{Z}$-graded modules and derived category of an algebra $ ext{ extGamma}$.

Proved a triangle-equivalence between stable category of $ ext{ extZ}/ extell extZ$-graded modules and a derived-orbit category of $ ext{ extGamma}$.

Connected representation theory of graded self-injective algebras with that of finite global dimension algebras.

Abstract

In this paper, we discuss a relationship between representation theory of graded self-injective algebras and that of algebras of finite global dimension. For a positively graded self-injective algebra $A$ such that $A_0$ has finite global dimension, we construct two types of triangle-equivalences. First we show that there exists a triangle-equivalence between the stable category of $\mathbb{Z}$-graded $A$-modules and the derived category of a certain algebra $\Gamma$ of finite global dimension. Secondly we show that if $A$ has Gorenstein parameter $\ell$, then there exists a triangle-equivalence between the stable category of $\mathbb{Z}/\ell\mathbb{Z}$-graded $A$-modules and a derived-orbit category of $\Gamma$, which is a triangulated hull of the orbit category of the derived category.