Examples of étale extensions of Green functors (Q6572986)

In classical algebra, a commutative \(k\)-algebra \(R\) is called formally étale if its underlying \(k\)-module is flat and the module of Kähler differentials \(\Omega^1_{R\vert k}\) vanishes. An equivariant analogue of this notion, in the setting of \(G\)-Tambara functors for a finite ring \(G\), was defined by \textit{M. A. Hill} [J. Algebra 489, 115--137 (2017; Zbl 1376.18006)], where the equivariant replacement for \(\Omega^1_{R \vert k}\) now classifies so-called \textit{genuine derivations} that take into account the norm maps of the Tambara functor \(R\).\N\NIn this article, the authors provide various new examples of equivariant étale extensions:\N\begin{itemize}\N\item For every \(C_2\)-étale extension \(K \hookrightarrow L\) of fields, the induced map \(\underline{K}^c \to \underline{L}^{\mathrm{fix}}\) of \(C_2\)-Tambara functors is \(C_n\)-étale.\N\item For every \(C_n\)-Kummer extension \(K \hookrightarrow L\) of fields, the map \(\underline{K}^c \to \underline{L}^{\mathrm{fix}}\) is \(C_2\)-étale.\N\item For every étale extension \(K \hookrightarrow L\) the induced map \(\underline{K}^c \to \underline{L}^{c}\) of \(G\)-Tambara functors is \(G\)-étale for every finite group \(G\).\N\end{itemize}\NThe method of proof is computational in nature: the authors employ Hill's concrete formula for the module of Kähler differentials and explicitly compute that it returns zero. The proof shows that the extensions are in fact already étale as \textit{Green functors}, in which one ignores the norm maps.\N\NThe authors suggest that the extension \(\underline{K}^c \to \underline{L}^{\mathrm{fix}}\) might be \(G\)-étale for an arbitrary finite \(G\)-Galois extension \(K \to L\), but that such a claim would require a more conceptual proof.