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Algebraic models for 1-dimensional categories of rational G-spectra

J. P. C. Greenlees

TL;DR

The paper develops algebraic models for rational $G$-spectra restricted to 1-dimensional blocks, building abelian categories $A(G|V)$ from equivariant sheaf data and a standard model with nub and vertex data. It proves a sequence of Quillen-equivalence steps, via pullback constructions and isotropic localizations, that identify $G$-spectra over $V$ with DG-modules in the abelian model, including detailed treatment of full subgroups, stalks, and two structure sheaves. The approach leverages a two-part framework—Part I (algebra) and Part II (topology)—and culminates in a cellular skeleton theorem that embeds the abelian model as the computable heart of rational $G$-spectra in dimension 1. The results cover blocks of 1-dimensional groups and several examples (toral blocks, semifree cases, dihedral/quaternion blocks), providing a concrete path toward broader 1-dimensional and higher-dimensional algebraic models with strong formality properties and isotropic pullbacks. These insights establish a scalable roadmap for extending abelian models to more general $1$-dimensional cases and connect group-theoretic structure with computable algebraic categories.

Abstract

In this paper we give algebraic models for rational G-spectra for a compact Lie group G when the geometric isotropy is restricted to lie in a 1-dimensional block of conjugacy classes. This includes all blocks of all groups of dimension 1, semifree spectra, and 1-dimensional blocks for many other groups G. The results were known previously for G=SO(2) or O(2) due to work of Barnes, Shipley and the author.

Algebraic models for 1-dimensional categories of rational G-spectra

TL;DR

The paper develops algebraic models for rational -spectra restricted to 1-dimensional blocks, building abelian categories from equivariant sheaf data and a standard model with nub and vertex data. It proves a sequence of Quillen-equivalence steps, via pullback constructions and isotropic localizations, that identify -spectra over with DG-modules in the abelian model, including detailed treatment of full subgroups, stalks, and two structure sheaves. The approach leverages a two-part framework—Part I (algebra) and Part II (topology)—and culminates in a cellular skeleton theorem that embeds the abelian model as the computable heart of rational -spectra in dimension 1. The results cover blocks of 1-dimensional groups and several examples (toral blocks, semifree cases, dihedral/quaternion blocks), providing a concrete path toward broader 1-dimensional and higher-dimensional algebraic models with strong formality properties and isotropic pullbacks. These insights establish a scalable roadmap for extending abelian models to more general -dimensional cases and connect group-theoretic structure with computable algebraic categories.

Abstract

In this paper we give algebraic models for rational G-spectra for a compact Lie group G when the geometric isotropy is restricted to lie in a 1-dimensional block of conjugacy classes. This includes all blocks of all groups of dimension 1, semifree spectra, and 1-dimensional blocks for many other groups G. The results were known previously for G=SO(2) or O(2) due to work of Barnes, Shipley and the author.
Paper Structure (33 sections, 23 theorems, 68 equations)

This paper contains 33 sections, 23 theorems, 68 equations.

Table of Contents

  1. Introduction
  2. Associated work in preparation
  3. Contents
  4. Algebra
  5. Full subgroups
  6. Partition by subgroups of the component group
  7. Reduction to full subgroups
  8. Ingredients for the abelian models
  9. Decomposing $\mathcal{}$V$$
  10. Auxiliary data
  11. Equivariant sheaf data in the geometric context
  12. Sheaves and component structures
  13. Localization
  14. The abelian models
  15. The standard model
  16. ...and 18 more sections

Key Result

Lemma 2.1

t2wqalg For a toral group $G$ as above, the space $\mathfrak{X}_G=\mathrm{Sub}(G)/G$ of conjugacy classes of subgroups of the toral group $G$ is partitioned into pieces, $\mathcal{}$V$^G_{ \overline$H$}$ one for each conjugacy class of subgroups $\overline$H$$ of $W$. If $\overline$H$\subseteq W$, t is clopen in the Hausdorff metric topology and closed under passage to cotoral subgroups. Furthermo

Theorems & Definitions (42)

  • Example 1.1
  • Lemma 2.1
  • Remark 2.2
  • Lemma 2.3
  • Remark 2.4
  • Lemma 2.5
  • Remark 2.6
  • Lemma 3.1
  • Remark 3.2
  • Remark 3.3
  • ...and 32 more