Hilbert-Kunz theory/Invariant theory/Section

Let ${}G$ be a finite group acting linearly on a polynomial ring ${}B=K[x_{1},\ldots ,x_{n}]$ with invariant ring ${}A=B^{G}$. This is a positively graded ${}K$-algebra with irrelevant ideal ${}A_{+}$ consisting of all invariant polynomials of positive degree. The extended ideal ${}{\mathfrak {h}}=A_{+}B$ is called the Hilbert ideal. The residue class ring

${}B_{G}=B/A_{+}B\,$

is called the ring of coinvariants.

We are interested in the Hilbert-Kunz multiplicity of ${}A$ and of its localization at the irrelevant ideal. A result of Watanabe and Yoshida implies the following observation. It uses the fact that for regular rings the Hilbert-Kunz multiplicity of an ideal is just the colength.

Lemma

Let ${}G$ be a finite group acting linearly on a polynomial ring ${}B=K[x_{1},\ldots ,x_{n}]$ with invariant ring ${}A=B^{G}$ and let ${}{\mathfrak {h}}=A_{+}B$ be the Hilbert ideal in ${}B$. Let ${}R$ be the localization of ${}A$ at the irrelevant ideal. If ${}K$ has positive characteristic, then

${}e_{HK}(R)={\frac {\operatorname {dim} _{K}\,(B/{\mathfrak {h}})}{\#\left(G\right)}}\,.$

In particular, this is a rational number, and this quotient depends only on the invariant ring, not on its representation.

With this observation we can give a Hilbert-Kunz proof of the following theorem of invariant theory (which was proved in positive characteristic by Larry Smith).

Corollary

Let ${}G$ be a finite group acting linearly on a polynomial ring ${}B=K[x_{1},\ldots ,x_{n}]$ with invariant ring ${}A=B^{G}$ and let ${}{\mathfrak {h}}=A_{+}B$ be the Hilbert ideal. If ${}K$ has positive characteristic, then the invariant ring is a polynomial ring if and only if we have

${}\operatorname {dim} _{K}\,B/{\mathfrak {h}}={\#\left(G\right)}\,.$