Residue fields for a class of rational E∞-rings and applications

Let A   be an E∞E∞-ring over the rational numbers. If A   satisfies a noetherian condition on its homotopy groups π⁎(A)π⁎(A), we construct a collection of E∞E∞-A  -algebras that realize on homotopy the residue fields of π⁎(A)π⁎(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.