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 a regular local ring. Such decompositions are named after Heisuke Hironaka, who used this in his unpublished master's thesis at Kyoto University . 
Hironaka's criterion , sometimes called miracle flatness, states that a local ring "R" that is a finitely generated 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.
Explicit decomposition of an invariant algebra.
Let formula_1 be a finite-dimensional vector space over an algebraically closed field of characteristic zero, formula_2, carrying a representation of a group formula_3, and consider the polynomial algebra on formula_4, formula_5. The algebra formula_6 carries a grading with formula_7, which is inherited by the invariant subalgebra 
A famous result of invariant theory, which provided the answer to Hilbert's fourteenth problem, is that if formula_9 is a linearly reductive group and formula_1 is a rational representation of formula_9, then formula_12 is finitely-generated. Another important result, due to Noether, is that any finitely-generated graded algebra formula_13 with formula_14 admits a (not necessarily unique) homogeneous system of parameters (HSOP). A HSOP (also termed "primary invariants") is a set of homogeneous polynomials, formula_15, which satisfy two properties:
Importantly, this implies that the algebra can then be expressed as a finitely-generated module over the subalgebra generated by the HSOP, formula_20. In particular, one may write 
where the formula_22 are called "secondary invariants". 
Now if formula_23 is Cohen–Macaulay, which is the case if formula_9 is linearly reductive, then it is a free (and as already stated, finitely-generated) module over any HSOP. Thus, one in fact has a Hironaka decomposition 
In particular, each element in formula_23 can be written uniquely as 􏰐formula_27, where formula_28, and the product of any two secondaries is uniquely given by formula_29, where formula_30. This specifies the multiplication in formula_23 unambiguously.