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

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 $A$.