Stratifications of Finite Directed Categories and Generalized APR Tilting Modules

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 article, we study representations of finite directed categories and discuss their stratification properties. In particular, we show the existence of generalized Auslander-Platzeck-Reiten tilting modules for triangular matrix algebras under some assumptions.

Key Words:

• Directed categories
• Quasi hereditary
• Stratification
• Tilting modules

ACKNOWLEDGMENT

The author would like to thank the referee for carefully reading the preprint, pointing out some existed results on this topic which are unknown to the author, and correcting typos.

Notes

This result is true for any locally finite k-linear category with finitely many objects, even if it is not directed.

In many cases people assume that the preorder is actually a partial order or even a linear order. In this paper we have to deal with preorders since the endomorphism algebras of objects in finite directed categories might not be local. However, we remind the reader that some results are only true for partial orders. For example, quasi-hereditary algebras with respect to preorders may not have finite global dimensions.

Communicated by D. Zacharia.