Quantum tilting modules over local rings

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 $R$.