Cotilting sheaves over weighted noncommutative regular projective curves

TL;DR

This paper classifies indecomposable pure-injective and cotilting sheaves over weighted noncommutative regular projective curves, providing a comprehensive understanding of their structure especially for nonnegative orbifold Euler characteristic cases.

Contribution

It offers a complete classification of pure-injective and cotilting sheaves in the category of quasicoherent sheaves over these curves, extending the understanding of their structure.

Findings

Classified all indecomposable pure-injective sheaves.

Described all cotilting sheaves of slope infinity.

Provided classifications for cases with nonnegative orbifold Euler characteristic.

Abstract

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