On abelian subcategories of triangulated categories

TL;DR

This paper investigates the structure of abelian subcategories within triangulated categories, focusing on their relation to invariants of selfinjective algebras and the conditions under which these subcategories arise as hearts of t-structures.

Contribution

It explores the connection between invariants of selfinjective algebras and abelian subcategories in their stable module categories, providing new insights into their structural relationships.

Findings

Full abelian subcategories may not be hearts of t-structures.

Relations between algebra invariants and abelian subcategories are established.

Conditions for abelian subcategories to arise as hearts are characterized.

Abstract

The stable module category of a selfinjective algebra is triangulated, but need not have any nontrivial $t$-structures, and in particular, full abelian subcategories need not arise as hearts of a $t$-structure. The purpose of this paper is to investigate full abelian subcategories of triangulated categories whose exact structures are related, and more precisely, to explore relations between invariants of finite-dimensional selfinjective algebras and full abelian subcategories of their stable module categories.