[Paper Review] Rational $G$-spectra for profinite $G$

Danny Sugrue|arXiv (Cornell University)|Oct 28, 2019

Homotopy and Cohomology in Algebraic Topology12 references4 citations

TL;DR

This paper establishes an algebraic model for rational $G$-spectra when $G$ is a profinite group, using chain complexes of Weyl-$G$-sheaves of $\mathbb{Q}$-modules over the space $SG$ of closed subgroups of $G$. The key contribution is showing that the injective dimension of this category—hence of rational $G$-Mackey functors—is determined by the Cantor-Bendixson rank of $SG$, with the injective dimension equal to $\operatorname{CB\text{-}rank}(SG) - 1$ when $SG$ is scattered and finite, and infinite otherwise.

ABSTRACT

In this thesis we will investigate rational G-spectra for a profinite group G. We will provide an algebraic model for this model category whose injective dimension can be calculated in terms of the Cantor-Bendixson rank of the space of closed subgroups of G, denoted SG. The algebraic model we consider is chain complexes of Weyl-G-sheaves of rational vector spaces over the spaces. The key step in proving that this is an algebraic model for G-spectra is in proving that the category of rational G-Mackey functors is equivalent to Weyl-G-sheaves. In addition to the fact that this sheaf description utilises the topology of G and the closed subgroups of G in a more explicit way than Mackey functors do, we can also calculate the injective dimension. In the final part of the thesis we will see that the injective dimension of the category of Weyl-G-sheaves can be calculated in terms of the Cantor-Bendixson rank of SG, hence giving the injective dimension of the category of Mackey functors via the earlier equivalence.

Motivation & Objective

• To develop an algebraic model for the homotopy category of rational $G$-spectra when $G$ is a profinite group.
• To characterize rational $G$-Mackey functors in terms of sheaf-theoretic structures over the space $SG$ of closed subgroups of $G$.
• To compute the injective dimension of the category of rational $G$-Mackey functors using topological invariants of $SG$, specifically its Cantor-Bendixson rank.
• To establish an equivalence between rational $G$-Mackey functors and Weyl-$G$-sheaves of $\mathbb{Q}$-modules, enabling homological algebra techniques.

Proposed method

• Construct an equivalence between the category of rational $G$-Mackey functors and the category of Weyl-$G$-sheaves of $\mathbb{Q}$-modules over $SG$.
• Utilize the $G$-equivariant Godement resolution to compute injective resolutions of $G$-sheaves of $\mathbb{Q}$-modules.
• Define the Cantor-Bendixson rank of the profinite $G$-space $SG$ as a topological invariant to bound the injective dimension of sheaf categories.
• Prove that the injective dimension of Weyl-$G$-sheaves over $SG$ is $\operatorname{CB\text{-}rank}(SG) - 1$ when $SG$ is scattered and of finite rank.
• Apply the equivalence between Mackey functors and Weyl-$G$-sheaves to transfer injective dimension results to the category of rational $G$-Mackey functors.
• Use the structure of the Burnside ring and idempotents in the Burnside ring to analyze the decomposition of Mackey functors.

Experimental results

Research questions

• RQ1 How can rational $G$-spectra for a profinite group $G$ be modeled algebraically using sheaf-theoretic structures?
• RQ2 What is the injective dimension of the category of rational $G$-Mackey functors, and how does it relate to the topology of $SG$?
• RQ3 Can the category of rational $G$-Mackey functors be equivalently described as a category of sheaves over $SG$?
• RQ4 How does the Cantor-Bendixson rank of $SG$ control the homological properties of rational $G$-spectra?
• RQ5 What is the injective dimension of $G$-equivariant sheaves of $\mathbb{Q}$-modules over $SG$ for a profinite group $G$?

Key findings

• The category of rational $G$-Mackey functors is equivalent to the category of Weyl-$G$-sheaves of $\mathbb{Q}$-modules over $SG$, providing a sheaf-theoretic description of Mackey functors.
• The injective dimension of the category of Weyl-$G$-sheaves of $\mathbb{Q}$-modules over $SG$ is equal to $\operatorname{CB\text{-}rank}(SG) - 1$ when $SG$ is scattered and of finite Cantor-Bendixson rank.
• If $SG$ has infinite Cantor-Bendixson rank, then the injective dimension of Weyl-$G$-sheaves over $SG$ is infinite.
• For $G = \mathbb{Z}_p$, the Cantor-Bendixson rank of $SG$ is 2, so the injective dimension of the algebraic model for rational $G$-spectra is 1.
• For a finite group $G$, $SG$ is discrete and finite, so $\operatorname{CB\text{-}rank}(SG) = 1$, and the injective dimension of the algebraic model is 0.
• For $G = \prod_{i=1}^n \mathbb{Z}_{p_i}$, the Cantor-Bendixson rank of $SG$ is $n+1$, so the injective dimension of the algebraic model is $n$.