Almost split sequences in tri-exact categories

TL;DR

This paper investigates the existence of almost split sequences within tri-exact categories, extending known results from abelian and triangulated categories to broader contexts, with applications to derived categories of modules.

Contribution

It unifies and generalizes theorems on almost split sequences and triangles across various categorical frameworks, providing new existence results in derived categories.

Findings

Extended existence theorems for almost split sequences

Unified framework for abelian, exact, and triangulated categories

New results on derived categories of modules

Abstract

We shall study the existence of almost split sequences in tri-exact categories, that is, extension-closed subcategories of triangulated categories. Our results unify and extend the existence theorems for almost split sequences in abelian categories and exact categories (that is, extension-closed subcategories of abelian categories), and those for almost split triangles in triangulated categories. As applications, we shall obtain some new results on the existence of almost split sequences in the derived categories of all modules over an algebra with a unity or a locally finite dimensional algebra given by a quiver with relations.