We study the transformation of torsion-free coherent analytic sheaves under proper modifications. More precisely, we study direct images of inverse image sheaves, and torsion-free preimages of direct image sheaves. Under some conditions, it is shown that torsion-free coherent sheaves can be realized as the direct image of locally free sheaves under modifications. Thus, it is possible to study coherent sheaves modulo torsion by reducing the problem to study vector bundles on manifolds. We apply this to reduced ideal sheaves and to the Grauert-Riemenschneider canonical sheaf of holomorphic n-forms.