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_{...