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Separable commutative algebras and Galois theory in stable homotopy theories

Niko Naumann, Luca Pol

TL;DR

The paper establishes a precise bridge between Mathew's finite covers and Balmer's separable commutative algebras in stable homotopy theories, proving that finite covers coincide with separable algebras whose underlying module is dualizable and whose degree function is finite and locally constant. Using axiomatic Galois theory and descent across limits, it provides comprehensive classifications of separable algebras across a wide array of contexts, including modules over connective and even-periodic $\mathbb{E}_\infty$-rings, complete modules over adic $\mathbb{E}_\infty$-rings, Lubin–Tate theories, topological $K$-theories, spectral Deligne–Mumford stacks, qcqs schemes, and stable module categories of finite groups. In favorable settings (e.g., chromatic/localized categories, stable module categories of $p$-rank-one groups), separable algebras are governed by finite étale data over base rings or residue fields, frequently reducing to classical étale theory via the Balmer spectrum. The work culminates in explicit descriptions of Galois groups and torsors, showing, for instance, that for rank-one $p$-groups the Galois group is a Weyl group, while providing counterexamples in higher-rank cases. Overall, the results unite Balmer’s tt-geometry with Mathew’s Galois-type descent to advance a robust holistic theory of étale-like phenomena in stable homotopy theory. $

Abstract

We relate two different proposals to extend the étale topology into homotopy theory, namely via the notion of finite cover introduced by Mathew and via the notion of separable commutative algebra introduced by Balmer. We show that finite covers are precisely those separable commutative algebras with underlying dualizable module, which have a locally constant and finite degree function. We then use Galois theory to classify separable commutative algebras in numerous categories of interest. Examples include the category of modules over a connective $\mathbb{E}_\infty$-ring $R$ which is either connective or even periodic with $π_0(R)$ regular Noetherian in which $2$ acts invertibly, the stable module category of a finite group of $p$-rank one and the derived category of a qcqs scheme.

Separable commutative algebras and Galois theory in stable homotopy theories

TL;DR

The paper establishes a precise bridge between Mathew's finite covers and Balmer's separable commutative algebras in stable homotopy theories, proving that finite covers coincide with separable algebras whose underlying module is dualizable and whose degree function is finite and locally constant. Using axiomatic Galois theory and descent across limits, it provides comprehensive classifications of separable algebras across a wide array of contexts, including modules over connective and even-periodic -rings, complete modules over adic -rings, Lubin–Tate theories, topological -theories, spectral Deligne–Mumford stacks, qcqs schemes, and stable module categories of finite groups. In favorable settings (e.g., chromatic/localized categories, stable module categories of -rank-one groups), separable algebras are governed by finite étale data over base rings or residue fields, frequently reducing to classical étale theory via the Balmer spectrum. The work culminates in explicit descriptions of Galois groups and torsors, showing, for instance, that for rank-one -groups the Galois group is a Weyl group, while providing counterexamples in higher-rank cases. Overall, the results unite Balmer’s tt-geometry with Mathew’s Galois-type descent to advance a robust holistic theory of étale-like phenomena in stable homotopy theory. $

Abstract

We relate two different proposals to extend the étale topology into homotopy theory, namely via the notion of finite cover introduced by Mathew and via the notion of separable commutative algebra introduced by Balmer. We show that finite covers are precisely those separable commutative algebras with underlying dualizable module, which have a locally constant and finite degree function. We then use Galois theory to classify separable commutative algebras in numerous categories of interest. Examples include the category of modules over a connective -ring which is either connective or even periodic with regular Noetherian in which acts invertibly, the stable module category of a finite group of -rank one and the derived category of a qcqs scheme.
Paper Structure (19 sections, 78 theorems, 174 equations)

This paper contains 19 sections, 78 theorems, 174 equations.

Table of Contents

  1. Introduction
  2. Finite covers and separable commutative algebras
  3. Preliminaries
  4. Connected stable homotopy theories
  5. Separable commutative algebras
  6. Degree functions of separable algebras
  7. Descent for separable commutative algebras
  8. Axiomatic Galois theory
  9. Finite covers and separability
  10. Weak finite covers and separable algebras in limit categories
  11. Applications: classification of separable commutative algebras
  12. Modules over an E-ring
  13. Complete modules over an adic E-ring
  14. Chromatic homotopy theory
  15. Chromatic localizations
  16. ...and 4 more sections

Key Result

Theorem A

Let ${\mathscr{C}}$ be a stable homotopy theory with unit object $\mathbbm{1}\in{\mathscr{C}}$. Suppose that $\pi_0(\mathbbm{1})$ decomposes as a finite product of indecomposable rings. Then we have an equality between the finite covers of ${\mathscr{C}}$ and those separable commutative algebras of ${\mathscr{C}}$ which have a dualizable underlying module, and whose degree function is finite and

Theorems & Definitions (200)

  • Theorem A: see Corollary \ref{['cor-sep-lcf=cov']}.
  • Theorem B
  • Definition 2.2
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • Lemma 2.7
  • proof
  • ...and 190 more