Hironaka decomposition

In mathematics, a Hironaka decomposition is a representation of an algebra over a field as a finitely generated free module over a polynomial subalgebra or regular local ring. They are named after Heisuke Hironaka, who used this in his unpublished master's thesis at Kyoto (Nagata 1962, p.217).

Hironaka's criterion (Nagata, theorem 25.16), sometimes called miracle flatness, states that a local ring R that is a finite module over a regular Noetherian local ring S is Cohen-Macaulay if and only if it is a free module over S. There is a similar result for rings that are graded over a field rather than local.