Maximal self-orthogonal modules and a new generalization of tilting modules

TL;DR

This paper introduces a new class of modules called projectively Wakamatsu tilting modules, generalizing tilting modules, and explores their properties, relations, and implications for representation theory and homological conjectures.

Contribution

It defines projectively Wakamatsu tilting modules, proves their equivalence with other classes under certain conditions, and studies their structure and connections to homological conjectures.

Findings

Equivalence of various self-orthogonal modules under finiteness conditions

Every self-orthogonal module over a representation-finite Iwanaga-Gorenstein algebra has finite projective dimension

Introduces Bongartz completion and extends the poset of tilting modules

Abstract

We introduce a generalization of tilting modules of finite projective dimension, projectively Wakamatsu tilting modules, which are self-orthogonal and Ext-progenerators in their Ext-perpendicular categories. Under a certain finiteness condition, we prove that the following modules coincide: projectively Wakamatsu tilting, Wakamatsu tilting, maximal self-orthogonal, and self-orthogonal modules with the same rank as the algebra. This provides another proof of the weak Gorensteinness of representation-finite algebras. To prove this, we introduce Bongartz completion of self-orthogonal modules and characterize its existence. Moreover, we study a binary relation on Wakamatsu tilting modules which extends the poset of tilting modules, and use it to prove that every self-orthogonal module over a representation-finite Iwanaga-Gorenstein algebra has finite projective dimension. Finally, we discuss several conjectures related to self-orthogonal modules and their connections to famous homological conjectures.