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 Such decompositions are named after Heisuke Hironaka, who used this in his unpublished master's thesis at Kyoto University .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 /math>. The algebra K carries a grading with (K _0 = K , which is inherited by the invariant subalgebra
: K 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 /math> 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, $\$, which satisfy two properties:

# The $\$ are algebraically independent.
# The zero set of the $\$, $\$, 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 G = \sum_ \eta_k K theta_1, \dots, \theta_l,
where the $\eta_k$ are called ''secondary invariants''.

Now if K 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 G = \bigoplus_ \eta_k K theta_1, \dots, \theta_l.
In particular, each element in K 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 $\eta_k \eta_m = \sum\nolimits_j \eta_j f^j_$, where f^j_ \in K theta_1, \dots, \theta_l/math>. This specifies the multiplication in K G unambiguously.

See also

*Rees decomposition
* Stanley decomposition

References

*
*{{citation, mr=1122013
, last1=Sturmfels, first1= Bernd, authorlink1=Bernd Sturmfels, last2= White, first2= Neil, title=Computing combinatorial decompositions of rings
, journal=Combinatorica , volume=11 , year=1991, issue= 3, pages= 275–293, doi=10.1007/BF01205079

Commutative algebra