Locally torsion-free quasi-coherent sheaves

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 \(\mathfrak{Qcoh}(X)\).