A recollement approach to Geigle-Lenzing weighted projective varieties

TL;DR

This paper introduces a new recollement-based method for expanding abelian categories, specifically applied to Geigle-Lenzing weighted projective varieties, leading to new geometric constructions and insights into their algebraic structures.

Contribution

It develops a novel approach using recollements to expand abelian categories and applies this to construct and analyze new weighted projective varieties.

Findings

Established a criterion for cotilting objects in expanded categories

Connected the construction to Geigle-Lenzing weighted projective lines

Analyzed endomorphism algebras of tilting bundles in new varieties

Abstract

We introduce a new method for expanding an abelian category and study it using recollements. In particular, we give a criterion for the existence of cotilting objects. We show, using techniques from noncommutative algebraic geometry, that our construction encompasses the category of coherent sheaves on Geigle-Lenzing weighted projective lines. We apply our construction to some concrete examples and obtain new weighted projective varieties and analyse the endomorphism algebras of their tilting bundles.