Lemma 15.110.2. Let $R$ be a ring. Let $G$ be a finite group acting on $R$. Let $I \subset R$ be an ideal such that $\sigma (I) \subset I$ for all $\sigma \in G$. Then $R^ G/I^ G \subset (R/I)^ G$ is an integral extension of rings which induces a homeomorphism on spectra and purely inseparable extensions of residue fields.

**Proof.**
Since $I^ G = R^ G \cap I$ it is clear that the map is injective. Lemma 15.110.1 shows that Algebra, Lemma 10.46.11 applies.
$\square$

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