Cotilting Sheaves over Weighted Noncommutative Regular Projective Curves

We consider the category Qcoh$mathbb{X}$ of quasicoherent sheaves where $mathbb{X}$ is a weighted noncommutative regular projective curve over a field $k$. This category is a hereditary, locally noetherian Grothendieck category. We classify all indecomposable pure-injective sheaves and all cotilting sheaves of slope $infty$. In the cases of nonnegative orbifold Euler characteristic this leads to a classification of pure-injective indecomposable sheaves and a description of all large cotilting sheaves in Qcoh$mathbb{X}$.