Pr\"ufer sheaves and generic sheaves over the weighted projective lines and elliptic curves

TL;DR

This paper introduces Pr"ufer and adic sheaves over weighted projective lines and elliptic curves, demonstrating their role in classifying coherent sheaves and constructing generic sheaves.

Contribution

It defines Pr"ufer and adic sheaves in this context and establishes their relationship with generic sheaves, offering new methods for their construction.

Findings

Pr"ufer and adic sheaves classify coherent sheaves.

Methods to construct generic sheaves using these sheaves.

Relationship between Pr"ufer sheaves and generic sheaves.

Abstract

In the present paper, we introduce the concepts of Pr\"{u}fer sheaves and adic sheaves over a weighted projective line of genus one or an elliptic curve, show that Pr\"{u}fer sheaves and adic sheaves can classify the category of coherent sheaves. Moreover, we describe the relationship between Pr\"{u}fer sheaves and generic sheaves, and provide two methods to construct generic sheaves by using coherent sheaves and Pr\"{u}fer sheaves.