  We prove the torsion freeness of the decomposable Orlik--Solomon algebra of a
simple matroid on ground set $[n]$. In the class of hypersolvable \&
non-supersolvable complex hyperplane arrangements, the torsion freeness, in a
certain degree, of this combinatorially defined object, associated to the
intersection lattice of the arrangement, impacts on the first non-vanishing
higher homotopy group of the complement of the arrangement.
