Locally torsion-free quasi-coherent sheaves

Let X   be an arbitrary scheme. It is known that the category Qcoh(X)Qcoh(X) of quasi-coherent sheaves admits arbitrary products. However its 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 Qcoh(X)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 Qcoh(X)Qcoh(X). Finally torsion-free covers are shown to exist in Qcoh(X)Qcoh(X).