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Separable commutative algebras in equivariant homotopy theory

Niko Naumann, Luca Pol, Maxime Ramzi

Abstract

Given a finite group $G$ and a commutative ring $G$-spectrum $R$, we study the separable commutative algebras in the category of compact $R$-modules. We isolate three conditions on the geometric fixed points of $R$ which ensure that every separable commutative algebra is standard, i.e. arises from a finite $G$-set. In particular we show that all separable commutative algebras in the categories of compact objects in $G$-spectra and in derived $G$-Mackey functors are standard provided that $G$ is a $p$-group. In these categories we also show that for a general finite group $G$, not all separable commutative algebras are standard. We finally discuss how the classification of separable commutative algebras in compact $G$-spectra varies if we require the existence of multiplicative norms. We show that if $G$ is solvable, then any separable commutative algebra therein that is normed is automatically standard. However, if $G$ is not solvable, we provide examples of separable commutative algebras that are normed but not standard.

Separable commutative algebras in equivariant homotopy theory

Abstract

Given a finite group and a commutative ring -spectrum , we study the separable commutative algebras in the category of compact -modules. We isolate three conditions on the geometric fixed points of which ensure that every separable commutative algebra is standard, i.e. arises from a finite -set. In particular we show that all separable commutative algebras in the categories of compact objects in -spectra and in derived -Mackey functors are standard provided that is a -group. In these categories we also show that for a general finite group , not all separable commutative algebras are standard. We finally discuss how the classification of separable commutative algebras in compact -spectra varies if we require the existence of multiplicative norms. We show that if is solvable, then any separable commutative algebra therein that is normed is automatically standard. However, if is not solvable, we provide examples of separable commutative algebras that are normed but not standard.

Paper Structure

This paper contains 15 sections, 37 theorems, 120 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Homotopy $G$-fixed points and separable algebras
  4. Stable module categories and the Tate construction
  5. The indecomposable condition
  6. The retraction condition
  7. Separably closed commutative algebras
  8. Pullback decomposition of equivariant spectra
  9. The classification result
  10. Not all separable algebras are standard
  11. Further considerations
  12. Normed separable algebras
  13. Other examples
  14. Free $G$-spectra
  15. Borel $G$-spectra

Key Result

Theorem 1

Suppose that $G$ and $R$ satisfy conditions (1), (2) and (3) for all $K\subseteq G$. Then is an equivalence, and so any separable commutative algebra is standard. If moreover (4) holds for all $K \subseteq G$, then we have an equivalence

Theorems & Definitions (103)

  • Theorem : \ref{['main-theorem']}, \ref{['prop-fully-faulful']}
  • Theorem : \ref{['cor-galois']}, \ref{['cor-galois-cpq']}
  • Theorem 2.1: cf. Ramzi2023
  • Proposition 2.3
  • Definition 3.1
  • Example 3.2
  • Remark 3.3
  • Lemma 3.4
  • Remark 3.5
  • Remark 3.6
  • ...and 93 more