Flatness and fibred powers over smooth varieties

We prove the equivalence between flatness of a complex algebraic map to a s mooth variety of dimension n and torsion freeness of its nth fibered power, under the assumption that the source space is of pure dimension. This gene ralizes the corresponding result for finite maps due to Auslander and prove s a conjecture of Vasconcelos for a large class of objects. We develop seve ral constructions in local analytic geometry and commutative algebra useful for our proof. (C) 2000 Academic Press.