The Ingalls-Thomas Bijections

TL;DR

This paper extends the known bijections between support-tilting modules and thick subcategories from path algebras of quivers to all hereditary artin algebras, providing new insights and generalizations.

Contribution

It introduces additional bijections and proves their validity for arbitrary hereditary artin algebras, broadening the scope of previous results.

Findings

Bijections hold for all hereditary artin algebras

Extended the correspondence beyond path algebras

Proofs are relevant for quiver path algebras as well

Abstract

Given a finite acyclic quiver Q with path algebra kQ, Ingalls and Thomas have exhibited a bijection between the set of Morita equivalence classes of support-tilting modules and the set of thick subcategories of mod kQ and they have collected a large number of further bijections with these sets. We add some additional bijections and show that all these bijections hold for arbitrary hereditary artin algebras. The proofs presented here seem to be of interest also in the special case of the path algebra of a quiver.