Large tilting modules and representation type

TL;DR

This paper investigates the properties of large tilting modules over rings, linking their behavior to the representation type of hereditary artin algebras and extending results to wild cases.

Contribution

It establishes how the endomorphism ring of certain infinite dimensional tilting modules determines the representation type of hereditary artin algebras.

Findings

The behavior of tilting modules L over their endomorphism rings characterizes the algebra's representation type.

Results extend to wild hereditary algebras, generalizing previous tame cases.

Connections between tilting modules and module categories are clarified.

Abstract

We study finiteness conditions on large tilting modules over arbitrary rings. We then turn to a hereditary artin algebra R and apply our results to the (infinite dimensional) tilting module L that generates all modules without preprojective direct summands. We show that the behaviour of L over its endomorphism ring determines the representation type of R. A similar result holds true for the (infinite dimensional) tilting module W that generates the divisible modules. Finally, we extend to the wild case some results on Baer modules and torsion-free modules proven in [AHT] for tame hereditary algebras.