  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 $\mathfrak{Qcoh}(X)$.
