Stratifications of finite directed categories and generalized APR tilting modules

TL;DR

This paper studies representations of finite directed categories, explores their stratification properties, and demonstrates the existence of generalized APR tilting modules for certain triangular matrix algebras, unifying various algebraic structures.

Contribution

It introduces the concept of stratifications for finite directed categories and proves the existence of generalized APR tilting modules in this context.

Findings

Finite directed categories unify multiple algebraic structures.

Stratification properties of these categories are characterized.

Existence of generalized APR tilting modules for certain algebras is established.

Abstract

A finite directed category is a $k$-linear category with finitely many objects and an underlying poset structure, where $k$ is an algebraically closed field. This concept unifies structures such as $k$-linerizations of posets and finite EI categories, quotient algebras of finite-dimensional hereditary algebras, triangular matrix algebras, etc. In this paper we study representations of finite directed categories, discuss their stratification properties, and show the existence of generalized APR tilting modules for triangular matrix algebras under some assumptions.