Quantum tilting modules over local rings

We show that tilting modules for quantum groups over local Noetherian domains of characteristic 0 exist and the indecomposable are parametrized by their highest weight. For this, we introduce a model category X = A ( R ) ${\mathcal {X}}={\mathcal {X}}_{\mathcal A}(R)$ associated with Z [ v , − 1 ] ${\mathbb {Z}}[v,v^{-1}]$ -domain A}$ root system $R$ . if is $\hskip.001pt 0$ contains all U $U_{\mathcal -modules admit Weyl filtration. If in addition local, study torsion phenomena category. This leads to construction free, or “maximal” objects {X}}$ these correspond group