Large tilting sheaves over weighted noncommutative regular projective curves

TL;DR

This paper classifies large tilting sheaves over weighted noncommutative regular projective curves, especially in elliptic and tubular cases, revealing their slope structures and resolving classes.

Contribution

It provides a complete classification of large tilting sheaves with non-coherent torsion subsheaves on these curves, extending understanding of their structure.

Findings

Classified all tilting sheaves with non-coherent torsion subsheaves.

Established slope existence for large tilting sheaves in elliptic and tubular cases.

Described resolving classes associated with these tilting sheaves.

Abstract

Let $\mathbb{X}$ be a weighted noncommutative regular projective curve over a field $k$. The category $\operatorname{Qcoh}\mathbb{X}$ 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.