Examples of \'etale extensions of Green functors

Abstract

We provide new examples of \'etale extensions of Green functors by transferring classical examples of \'etale extensions to the equivariant setting. Our examples are Tambara functors, and we prove Green \'etaleness for them, which implies Tambara \'etaleness. We show that every $C_2$-Galois extensions of fields gives rise to an \'etale extension of $C_2$-Green functors. Here we associate the constant Tambara functor to the base field and the fix-Tambara functor to the extension. We also prove that all $C_n$-Kummer extensions give rise to \'etale extensions for arbitrary finite $n$. \'Etale extensions of fields induce \'etale extension of $G$-Green functors for any finite group $G$ by passing to the corresponding constant $G$-Tambara functors.