Flatness testing and torsion freeness of analytic tensor powers

Abstract

Let f ξ : X ξ → Y η be a morphism of germs of complex analytic spaces, where X ξ is reduced of pure dimension and Y η is smooth of dimension n. We give several sufficient conditions for the following characterization of flatness to hold: f ξ is flat if and only if the nth analytic tensor power O X , ξ ⊗ ˆ O Y , η ⋯ ⊗ ˆ O Y , η O X , ξ ︸ n times is a torsion-free O Y , η -module.