The author develops various results about the homotopy theory of “rational \(E_{\infty}\)-rings” which are noetherian in a sense explained below. The rational \(E_{\infty}\)-rings considered here are equivalent to unbounded commutative differential algebras over the rationals or to commutative structured ring spectra over \(H\mathbb Q\), the Eilenberg-Mac Lane spectrum of the rationals.
A rational \(E_{\infty}\)-ring \(A\) is defined to be noetherian if the ring of even degree homotopy groups \(\pi_{\mathrm{even}}(A)\) is noetherian and if the odd degree homotopy groups \(\pi_{\mathrm{odd}}(A)\) form a finitely generated module over \(\pi_{\mathrm{even}}(A)\). The first main result is that every rational noetherian \(E_{\infty}\)-ring \(A\) with a unit in degree 2 admits residue fields. That is, for every prime ideal \(\mathfrak p\) in \(\pi_0(A)\) there exists a unique \(E_{\infty}\)-algebra \(\kappa(\mathfrak p)\) under \(A\) which is even periodic and which induces the reduction \(\pi_0(A) \to \pi_0(A)_{\mathfrak p}/{\mathfrak p}\pi_0(A)_{\mathfrak p}\) on \(\pi_0\).
One application of the existence of fraction fields is a nilpotence theorem that is analogous to (but easier to prove than) the important Hopkins-Smith nilpotence theorem. It states that if \(A\) is as above and \(B\) is an \(A\)-algebra, an element in \(\pi_*(B)\) is nilpotent if and only if for every prime ideal \(\mathfrak p \subset \pi_0(A)\) its image in \(\pi_*(B \otimes_A \kappa(\mathfrak p))\) is nilpotent. Another application is a thick subcategory theorem that provides a classification of thick subcategories of the category of perfect \(A\)-modules in terms of specialization-closed subsets of the collection of homogeneous prime ideals of \(\pi_{\mathrm{even}}(A)\).
In earlier work [Adv. Math. 291, 403–541 (2016; Zbl 1338.55009)], the author defined the Galois group of a stable homotopy theory (or, more precisely, of a suitable symmetric monoidal stable \(\infty\)-category). Another result proved in the present paper is that the Galois group of the category of \(A\)-modules admits a purely algebraic description if \(A\) is a noetherian rational \(E_{\infty}\)-ring. The author also shows that for such \(A\), the cokernel of the map of Picard groups \(\mathrm{Pic}(\pi_*(A)) \to \mathrm{Pic}(A)\) is torsion-free.
In the last section, the author discusses examples of non-noetherian rational \(E_{\infty}\)-rings that illustrate the failure of various of the above results in the non-noetherian case. The examples include a Galois extension (in the sense of J. Rognes [Mem. Am. Math. Soc. 898, 137 p. (2008; Zbl 1166.55001)]) of rational ring spectra which is not of algebraic origin.