The Stacks project

Lemma 15.110.4. Let $R$ be a ring. Let $G$ be a finite group of order $n$ acting on $R$. Let $J \subset R^ G$ be an ideal. Then $R^ G/J \to (R/JR)^ G$ is ring map such that

  1. for $b \in (R/JR)^ G$ there is a monic polynomial $P \in R^ G/J[T]$ whose image in $(R/JR)^ G[T]$ is $(T - b)^ n$,

  2. for $a \in \mathop{\mathrm{Ker}}(R^ G/J \to (R/JR)^ G)$ we have $(T - a)^ n = T^ n$ in $R^ G/J[T]$.

In particular, $R^ G/J \to (R/JR)^ G$ is an integral ring map which induces homeomorphisms on spectra and purely inseparable extensions of residue fields.

Proof. Part (1) follow from Lemma 15.110.1 with $I = JR$. If $a$ is as in part (2), then $a$ is the image of $x \in R^ G \cap JR$. Hence $(T - x)^ n = \prod _{\sigma \in G} (T - \sigma (x))$ is congruent to $T^ n$ modulo $J$ by Lemma 15.110.3. This proves part (2). To see the final statement we may apply Algebra, Lemma 10.46.11. $\square$


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