# Hilbert-Kunz theory/Invariant theory/Section

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

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

is called the ring of coinvariants.

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

${\displaystyle {}e_{HK}(R)={\frac {\operatorname {dim} _{K}\,(B/{\mathfrak {h}})}{\#\left(G\right)}}\,.}$
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 ${\displaystyle {}G}$ be a finite group acting linearly on a polynomial ring ${\displaystyle {}B=K[x_{1},\ldots ,x_{n}]}$ with invariant ring ${\displaystyle {}A=B^{G}}$ and let ${\displaystyle {}{\mathfrak {h}}=A_{+}B}$ be the Hilbert ideal. If ${\displaystyle {}K}$ has positive characteristic, then the invariant ring is a polynomial ring if and only if we have

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