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 Such decompositions are named after Heisuke Hironaka, who used this in his unpublished master's thesis at Kyoto University ( Nagata 1962 , p.217).

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

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

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

1. The ${\displaystyle \{\theta _{i}\}}$ are algebraically independent.
2. The zero set of the ${\displaystyle \{\theta _{i}\}}$, ${\displaystyle \{v\in V|\theta _{i}=0\}}$, coincides with the nullcone (link) of ${\displaystyle R}$.

Importantly, this implies that the algebra can then be expressed as a finitely-generated module over the subalgebra generated by the HSOP, ${\displaystyle K[\theta _{1},\dots ,\theta _{l}]}$. In particular, one may write

${\displaystyle K[V]^{G}=\sum _{k}\eta _{k}K[\theta _{1},\dots ,\theta _{l}]}$,

where the ${\displaystyle \eta _{k}}$ are called secondary invariants.

Now if ${\displaystyle K[V]^{G}}$ is Cohen–Macaulay, which is the case if ${\displaystyle G}$ 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

${\displaystyle K[V]^{G}=\bigoplus _{k}\eta _{k}K[\theta _{1},\dots ,\theta _{l}]}$.

In particular, each element in ${\displaystyle K[V]^{G}}$ can be written uniquely as 􏰐${\displaystyle \sum \nolimits _{j}\eta _{j}f_{j}}$, where ${\displaystyle f_{j}\in K[\theta _{1},\dots ,\theta _{l}]}$, and the product of any two secondaries is uniquely given by ${\displaystyle \eta _{k}\eta _{m}=\sum \nolimits _{j}\eta _{j}f_{km}^{j}}$, where ${\displaystyle f_{km}^{j}\in K[\theta _{1},\dots ,\theta _{l}]}$. This specifies the multiplication in ${\displaystyle K[V]^{G}}$ unambiguously.

See also

• Rees decomposition
• Stanley decomposition

References

• Nagata, Masayoshi (1962), Local rings, Interscience Tracts in Pure and Applied Mathematics, vol. 13, New York-London: Interscience Publishers a division of John Wiley & Sons, ISBN   0-88275-228-6, MR   0155856
• Sturmfels, Bernd; White, Neil (1991), "Computing combinatorial decompositions of rings", Combinatorica, 11 (3): 275–293, doi:10.1007/BF01205079, MR   1122013