# Introduction to coherent sheaves on weighted projective lines

## TL;DR

This paper explores the structure of coherent sheaves on weighted projective lines, presenting two approaches—axiomatic and expansion-based—to describe their abelian categories, with detailed analysis and tilting theory insights.

## Contribution

It introduces a new axiomatic framework and an expansion approach for understanding the abelian categories of coherent sheaves on weighted projective lines, enriching the theoretical foundation.

## Key findings

- Descriptions of abelian categories for weighted projective lines
- Development of an axiomatic approach to expansions
- Discussion of tilting theory in this context

## Abstract

These notes provide a description of the abelian categories that arise as categories of coherent sheaves on weighted projective lines. Two different approaches are presented: one is based on a list of axioms and the other yields a description in terms of expansions of abelian categories. A weighted projective line is obtained from a projective line by inserting finitely many weights. So we describe the category of coherent sheaves on a projective line in some detail, and the insertion of weights amounts to adding simple objects. We call this process `expansion' and treat it axiomatically. Thus most of these notes are devoted to studying abelian categories, including a brief discussion of tilting theory. We provide many details and have tried to keep the exposition as self-contained as possible.