# Hilbert-Kunz theory/Invariant theory/Section

Let be a finite group acting linearly on a polynomial ring
with invariant ring
.
This is a positively graded -algebra with irrelevant ideal consisting of all invariant polynomials of positive degree. The extended ideal
is called the *Hilbert ideal*. The residue class ring

is called the *ring of coinvariants*.

We are interested in the Hilbert-Kunz multiplicity of 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.

Let be a finite group acting linearly on a polynomial ring with invariant ring and let be the Hilbert ideal in . Let be the localization of at the irrelevant ideal. If has positive characteristic, then

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).

Let be a finite group acting linearly on a polynomial ring with invariant ring and let be the Hilbert ideal. If has positive characteristic, then the invariant ring is a polynomial ring if and only if we have