Tilting Modules over Tame Hereditary Algebras

TL;DR

This paper classifies all infinite dimensional tilting modules over tame hereditary algebras, revealing their structure in terms of universal localizations, Prüfer modules, and specific tilting modules like Lukas.

Contribution

It provides a complete classification of infinite dimensional tilting modules over tame hereditary algebras, extending previous results and including the structure of cotilting modules.

Findings

Infinite dimensional tilting modules are classified via universal localizations and Prüfer modules.

Over the Kronecker algebra, large tilting modules are mostly of a specific form, except for Lukas tilting module.

The classification includes finite dimensional summands and generalizes to arbitrary tame hereditary algebras.

Abstract

We give a complete classification of the infinite dimensional tilting modules over a tame hereditary algebra R. We start our investigations by considering tilting modules of the form T=R_U\oplus R_U /R where U is a union of tubes, and R_U denotes the universal localization of R at U in the sense of Schofield and Crawley-Boevey. Here R_U/R is a direct sum of the Pr\"ufer modules corresponding to the tubes in U. Over the Kronecker algebra, large tilting modules are of this form in all but one case, the exception being the Lukas tilting module L whose tilting class Gen L consists of all modules without indecomposable preprojective summands. Over an arbitrary tame hereditary algebra, T can have finite dimensional summands, but the infinite dimensional part of T is still built up from universal localizations, Pr\"ufer modules and (localizations of) the Lukas tilting module. We also recover the classification of the infinite dimensional cotilting R-modules due to Buan and Krause.