Cotilting sheaves over weighted noncommutative regular projective curves

Abstract

We consider the category $\mathrm{Qcoh}\mathbb{X}$ 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 $\mathrm{Qcoh}\mathbb{X}$.

28 pages, 1 figure. v2: Some typos and formulations fixed, abstract slightly changed, extended number 6.12, additional references, more streamlined recapitulation of the geometric setting