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

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).


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