Large tilting sheaves over weighted noncommutative regular projective curves

Abstract

Let $\mathbb{X}$ be a weighted noncommutative regular projective curve over a field $k$. The category $\mathrm{Qcoh}\mathbb{X}$ of quasicoherent sheaves is a hereditary, locally noetherian Grothendieck category. We classify all tilting sheaves which have a non-coherent torsion subsheaf. In case of nonnegative orbifold Euler characteristic we classify all large (that is, non-coherent) tilting sheaves and the corresponding resolving classes. In particular we show that in the elliptic and in the tubular cases every large tilting sheaf has a well-defined slope.

52 pages, 1 figure. v2: revised Cor. 6.6 and 7.15; a few minor fixes. v3: revised proof of Prop. 5.7; a few minor fixes and improvements