Modules determined by their composition factors in higher homological algebra

TL;DR

This paper extends classical module theory by defining Grothendieck groups for $d$-abelian categories and demonstrating that indecomposable objects are determined by their composition factors, generalizing Auslander-Reiten formulas.

Contribution

It introduces a Grothendieck group framework for $d$-abelian categories and generalizes the Auslander-Reiten translation formula to higher homological algebra.

Findings

Grothendieck groups of $ ext{mod} ext{-}\Phi$ and $d$-cluster tilting subcategories are isomorphic.

Indecomposable objects in $d$-cluster tilting subcategories are determined by their composition factors.

Generalization of Auslander-Reiten formula to higher dimensions.

Abstract

ABSTRACT. Let $\Phi$ be a finite dimensional $K$-algebra and let $\mathscr{C} = \textrm{mod}\: \Phi$ be the abelian category of finitely generated right $\Phi$-modules. In their 1985 paper ``Modules determined by their composition factors'', Auslander and Reiten showed that under certain conditions modules in $\textrm{mod}\: \Phi$ are determined by their composition factors, and show an important formula related to the Auslander-Reiten translation. Let $\mathscr{T}$ be a $d$-cluster tilting subcategory of $\mathscr{C}$, which by definition is also $d$-abelian. In this paper we will define the Grothendieck group for a $d$-abelian category, and show that the Grothendieck groups of $\mathscr{C}$ and $\mathscr{T}$ are isomorphic. We show also that under certain conditions, the indecomposable objects of $\mathscr{T}$ are determined up to isomorphism by their composition factors in $\mathscr{C}$. Finally, we generalise the formula from Auslander and Reiten involving the higher dimensional Auslander-Reiten translation.