Proper abelian subcategories of triangulated categories and their tilting theory

TL;DR

This paper introduces a new approach in triangulated categories by replacing hearts of t-structures with proper abelian subcategories, enabling a generalized tilting theory applicable even when hearts are absent.

Contribution

It proposes the concept of proper abelian subcategories as a replacement for hearts in triangulated categories and develops a tilting theory for these subcategories.

Findings

Proper abelian subcategories can have vanishing negative self-extensions.

A generalized tilting theory is established for these subcategories.

Application to negative cluster categories demonstrates the approach.

Abstract

In the theory of triangulated categories, we propose to replace hearts of $t$-structures by proper abelian subcategories, which may be plentiful even when hearts are not. For instance, this happens in negative cluster categories. In support of our proposal, we show that proper abelian subcategories with a few vanishing negative self extensions permit a tilting theory which is a direct generalisation of Happel-Reiten-Smal{\o} tilting of hearts.