From triangulated categories to abelian categories--cluster tilting in a general framework

TL;DR

This paper generalizes the concept of cluster tilting by demonstrating that quotients of triangulated categories modulo tilting subcategories naturally form abelian categories, specifically module categories of Gorenstein algebras of low dimension.

Contribution

It introduces a broad framework connecting triangulated and abelian categories through tilting subcategories, expanding the understanding of their relationship.

Findings

Quotients of triangulated categories modulo tilting subcategories are abelian.

These abelian categories are equivalent to module categories of Gorenstein algebras.

The Gorenstein algebras involved have dimension at most one.

Abstract

We put cluster tilting in ageneral framework by showing that any quotient of a triangulated category modulo a tilting subcategory (that is, a maximal one-orthogonal subcategory) carries an abelian structure. These abelian quotients turn out to be module categories of Goreisten algebras of dimension at most one.