Hironaka decomposition

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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 Such decompositions are named after Heisuke Hironaka, who used this in his unpublished master's thesis at Kyoto University ( Nagata 1962 , p.217).

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Hironaka's criterion( Nagata 1962 , theorem 25.16), 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 be a finite-dimensional vector space over an algebraically closed field of characteristic zero, , carrying a representation of a group , and consider the polynomial algebra on , . The algebra carries a grading with , 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 is a linearly reductive group and is a rational representation of , then is finitely-generated. Another important result, due to Noether, is that any finitely-generated graded algebra with admits a (not necessarily unique) homogeneous system of parameters (HSOP). A HSOP (also termed primary invariants) is a set of homogeneous polynomials, , which satisfy two properties:

  1. The are algebraically independent.
  2. The zero set of the , , coincides with the nullcone (link) of .

Importantly, this implies that the algebra can then be expressed as a finitely-generated module over the subalgebra generated by the HSOP, . In particular, one may write

,

where the are called secondary invariants.

Now if is Cohen–Macaulay, which is the case if 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 can be written uniquely as 􏰐, where , and the product of any two secondaries is uniquely given by , where . This specifies the multiplication in unambiguously.

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