Hilbert-Kunz/Invariantenring/Koinvarianten/Polynomring/en/Fakt

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

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

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