Lemma 15.110.4. Let $R$ be a ring. Let $G$ be a finite group acting on $R$. Let $R^ G \to A$ be a ring map. The map

\[ A \to (A \otimes _{R^ G} R)^ G \]

is an isomorphism if $R^ G \to A$ is flat. In general the map is integral, induces a homeomorphism on spectra, and induces purely inseparable residue field extensions.

**Proof.**
The first statement follows from Lemma 15.110.3 and Algebra, Lemma 10.46.11. To see the second consider the exact sequence $0 \to R^ G \to R \to \bigoplus _{\sigma \in G} R$ where the second map sends $x$ to $(\sigma (x) - x)$. Tensoring with $A$ the sequence remains exact if $R^ G \to A$ is flat.
$\square$

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