Locally torsion-free quasi-coherent sheaves

TL;DR

This paper investigates the structure of products of locally torsion-free quasi-coherent sheaves on integral schemes, revealing new properties of flat sheaves, torsion theories, and torsion-free covers in the category.

Contribution

It characterizes the product structure of locally torsion-free sheaves and establishes new properties of flat sheaves and torsion theories in the category of quasi-coherent sheaves.

Findings

The product of locally torsion-free sheaves has a specific described structure.

The class of flat quasi-coherent sheaves on Dedekind schemes is closed under arbitrary products.

Locally torsion-free quasi-coherent sheaves induce a hereditary torsion theory.

Abstract

Let $X$ be an arbitrary scheme. The category $\mathfrak{Qcoh}(X)$ of quasi--coherent sheaves on $X$ is known that admits arbitrary direct products. However their structure seems to be rather mysterious. In the present paper we will describe the structure of the product object of a family of locally torsion-free objects in $\mathfrak{Qcoh}(X)$, for $X$ an integral scheme. Several applications are provided. For instance it is shown that the class of flat quasi--coherent sheaves on a Dedekind scheme $X$ is closed under arbitrary direct products, and that the class of all locally torsion-free quasi--coherent sheaves induces a hereditary torsion theory on $\mathfrak{Qcoh}(X)$. Finally torsion-free covers are shown to exist in $\mathfrak{Qcoh}(X)$.