Flatness and Fibred Powers over Smooth Varieties

We prove the equivalence between flatness of a complex algebraic map to a smooth variety of dimension n and torsion freeness of its nth fibered power, under the assumption that the source space is of pure dimension. This generalizes the corresponding result for finite maps due to Auslander and proves a conjecture of Vasconcelos for a large class of objects. We develop several constructions in local analytic geometry and commutative algebra useful for our proof.