Flatness and fibred powers (Q1271526)

An analytic map with reduced and locally irreducible target is proved to be flat if and only if none of its fibered powers have vertical components. The proof uses \textit{H. Hironaka}'s criterion for flatness [Proc. Nordic Summer Sch., Symp. Math., Oslo 1976, 199-265 (1977; Zbl 0424.32004)]. The result is applied to demonstrate an analogous result in commutative algebra: flatness is equivalent to torsion freeness of all tensor powers.