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 Such decompositions are named after Heisuke Hironaka, who used this in his unpublished master's thesis at Kyoto University.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 $$ V $$ 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.
 * 2) 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, $$ K[\theta_1, \dots, \theta_l] $$. In particular, one may write
 * $$ K[V]^G = \sum_{k} \eta_k K[\theta_1, \dots, \theta_l] $$,

where the $$ \eta_k $$ 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
 * $$ K[V]^G = \bigoplus_{k} \eta_k K[\theta_1, \dots, \theta_l] $$.

In particular, each element in $$ K[V]^G $$ can be written uniquely as 􏰐$$ \sum\nolimits_j \eta_j f_j $$, where $$ f_j \in K[\theta_1, \dots, \theta_l] $$, and the product of any two secondaries is uniquely given by $$ \eta_k \eta_m = \sum\nolimits_j \eta_j f^j_{km}$$, where $$f^j_{km} \in K[\theta_1, \dots, \theta_l]$$. This specifies the multiplication in $$ K[V]^G $$ unambiguously.