Examples of étale extensions of Green functors
type: Item
instance of: scholarly article
title: Examples of étale extensions of Green functors
author: Ayelet Lindenstrauss
author: Birgit Richter
author: Foling Zou
published in: Proceedings of the American Mathematical Society. Series B
publication date: 2024-07-16T00:00:00Z
review text: 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.
reviewed by: Bastiaan Cnossen
Mathematics Subject Classification ID: 55P91
Mathematics Subject Classification ID: 13B40
zbMATH DE Number: 7881499
zbMATH Keywords: Mackey functors
zbMATH Keywords: Tambara functors
zbMATH Keywords: Green functors
zbMATH Keywords: étale extensions
MaRDI profile type: Publication
cites work: Q4591375
cites work: On the André-Quillen homology of Tambara functors
cites work: An equivariant tensor product on Mackey functors
cites work: Free Incomplete Tambara Functors are Almost Never Flat
cites work: Q2770573
cites work: Q3822866
cites work: Q4264049
DOI: 10.1090/BPROC/189
description: scientific article; zbMATH DE number 7881499