Residue fields for a class of rational $\mathbf{E}_\infty$-rings and applications

TL;DR

This paper constructs residue fields for certain rational $ ext{E}_ty$-rings, proving a nilpotence theorem analog, and uses this to describe the Galois theory, thick subcategories, and Picard group of these rings.

Contribution

It introduces a method to realize residue fields for noetherian rational $ ext{E}_ty$-rings and applies this to analyze their Galois theory and subcategory structure.

Findings

Constructed residue fields for rational $ ext{E}_ty$-rings.

Proved an analog of the nilpotence theorem for these residue fields.

Provided a complete algebraic description of Galois groups and thick subcategories.

Abstract

Let $A$ be an $\mathbf{E}_{\infty}$-ring spectrum over the rational numbers. If $A$ satisfies a noetherian condition on its homotopy groups $\pi_*(A)$, we construct a collection of $\mathbf{E}_{\infty}$-$A$-algebras that realize on homotopy the residue fields of $\pi_*(A)$. We prove an analog of the nilpotence theorem for these residue fields. As a result, we are able to give a complete algebraic description of the Galois theory of $A$ and of the thick subcategories of perfect $A$-modules. We also obtain partial information on the Picard group of $A$.