Hironaka decomposition explained

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 Such decompositions are named after Heisuke Hironaka, who used this in his unpublished master's thesis at Kyoto University .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

V be a finite-dimensional vector space over an algebraically closed field of characteristic zero, K , carrying a representation of a group G , and consider the polynomial algebra on V , K[V] . The algebra K[V] carries a grading with (K[V]) _{0} = K , which is inherited by the invariant subalgebra K[V] ^{G} = \{ f \in K[V] \mid g \circ f = f, \forall g \in G \} .

A famous result of invariant theory, which provided the answer to Hilbert's fourteenth problem, is that if

G is a linearly reductive group and V is a rational representation of G , then K[V] is finitely-generated. Another important result, due to Noether, is that any finitely-generated graded algebra R with R _{0} = K admits a (not necessarily unique) homogeneous system of parameters (HSOP). A HSOP (also termed primary invariants) is a set of homogeneous polynomials, \{ \theta _{i} \} , which satisfy two properties:

1. The

\{ \theta _{i} \} are algebraically independent.

1. The zero set of the

\{ \theta _{i} \} , \{v \in V | \theta _{i} = 0 \} , coincides with the nullcone (link) of R .

Importantly, this implies that the algebra can then be expressed as a finitely-generated module over the subalgebra generated by the HSOP,

K[\theta _{1,} ... , \theta _{l]} . In particular, one may write K[V] ^{G} = \sum _{k} η _{k} K[\theta _{1,} ... , \theta _{l]} , where the η _{k} are called secondary invariants.

Now if

K[V] ^{G} is Cohen–Macaulay, which is the case if G 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 K[V] ^{G} = o plus _{k} η _{k} K[\theta _{1,} ... , \theta _{l]} . In particular, each element in K[V] ^{G} can be written uniquely as �\sum\nolimits _{j} η _{j} f _{j} , where f _{j} \in K[\theta _{1,} ... , \theta _{l]} , and the product of any two secondaries is uniquely given by η _{k} η _{m} = \sum\nolimits _{j} η _{j} f ^{j} _{km} , where f ^{j} _{km} \in K[\theta _{1,} ... , \theta _{l]} . This specifies the multiplication in K[V] ^{G} unambiguously.

See also

• Rees decomposition
• Stanley decomposition