Quantum tilting modules over local rings

TL;DR

This paper establishes the existence and parametrization of tilting modules over quantum groups defined on local Noetherian domains, introducing a model category framework that captures torsion phenomena and tilting module structures.

Contribution

It introduces a model category approach to quantum tilting modules over local rings, providing new constructions and parametrizations of indecomposable tilting modules.

Findings

Tilting modules exist over quantum groups on local Noetherian domains.

Indecomposable tilting modules are parametrized by highest weight.

Torsion phenomena are analyzed, leading to torsion-free tilting modules.

Abstract

We show that tilting modules for quantum groups over local Noetherian domains exist and that the indecomposable tilting modules are parametrized by their highest weight. For this we introduce a model category ${\mathcal X}={\mathcal X}_{\mathscr A}(R)$ associated with a Noetherian ${\mathbb Z}[v,v^{-1}]$-domain ${\mathscr A}$ and a root system $R$. We show that if ${\mathscr A}$ is of quantum characteristic $0$, the model category contains all $U_{\mathscr A}$-modules that admit a Weyl filtration. If ${\mathscr A}$ is in addition local, we study torsion phenomena in the model category. This leads to a construction of torsion free objects in ${\mathcal X}$. We show that these correspond to tilting modules for the quantum group associated with ${\mathscr A}$ and $R$.