Flatness testing and torsion freeness of analytic tensor powers

Let fξ:Xξ→Yη be a morphism of germs of complex analytic spaces, where Xξ is reduced of pure dimension and Yη is smooth of dimension n. We give several sufficient conditions for the following characterization of flatness to hold: fξ is flat if and only if the nth analytic tensor power OX,ξ⊗ˆOY,η⋯⊗ˆOY,ηOX,ξ︸ntimes is a torsion-free OY,η-module.