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 Such decompositions are named after Heisuke Hironaka, who used this in his unpublished master's thesis at Kyoto University Template:Harv.

Hironaka's criterion Template:Harv, 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

$K[V{]}^G=\{f\in K[V]\mid g\circ f=f,\mathrm{\forall }g\in G\}$.

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

$K[V{]}^G=\underset{k}{⨁}{\eta }_{k}K[{\theta }_1,\dots ,{\theta }_l]$.

In particular, each element in $K[V{]}^G$ can be written uniquely as 􏰐$\sum _j{\eta }_j{f}_j$, where $f_j\in K[{\theta }_1,\dots ,{\theta }_l]$, and the product of any two secondaries is uniquely given by ${\eta }_k{\eta }_m=\sum _j{\eta }_j{f}_{km}^j$, where $f_{km}^j\in K[{\theta }_1,\dots ,{\theta }_l]$. This specifies the multiplication in $K[V{]}^G$ unambiguously.

See also

• Rees decomposition
• Stanley decomposition

References

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