Hilbert-Kunz/Invariant ring/Coinvariants/Fact

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.

Write a proof