GALOIS EXTENSIONS OF STRUCTURED RING SPECTRA

By John Rognes. 2005

150 citations recorded by the source corpus.

We introduce the notion of a Galois extension of commutative S-algebras (E1 ring spectra), often localized with respect to a flxed homology theory. There are numerous examples, including some involving Eilenberg{MacLane spectra of commu- tative rings, real and complex topological K-theory, Lubin{Tate spectra and cochain S-algebras. We establish the main theorem of Galois theory in this generality. Its proof involves the notions of separable (andextensions of commutative S- algebras, and the Goerss{Hopkins{Miller theory for E1 mapping spaces. We show that the global sphere spectrum S is separably closed (using Minkowski's discrimi- nant theorem), and we estimate the separable closure of its localization with respect to each of the Morava K-theories. We also deflne Hopf{Galois extensions of com- mutative S-algebras, and study the complex cobordism spectrum MU as a common integral model for all of the local Lubin{Tate Galois extensions.

Subfields: Geometry and Topology, Mathematical Physics, Pathology and Forensic Medicine

View Paper: https://www.semanticscholar.org/paper/122fec5604a1ca4378c20fa482d3d6f26ca62924

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

In Mathematical Physics

In Pathology and Forensic Medicine

Citations in the corpus

Homotopy completion and topological Quillen homology of structured ring spectra (2011, background): “The topological analog for commutative ring spectra, called topological André-Quillen homology, was originally studied by Basterra [6]; see also BakerGilmour-Reinhard [4], Baker-Richter [5], Basterra-Mandell [7, 8], Goerss-Hopkins [25], Lazarev [46], Mandell [52], Richter [62], Rognes [63, 64] and ”
The localization sequence for the algebraic K-theory of topological K-theory (2006, methodology): “The analysis is based on descent conjectures coming from Rognes’ Galois theory of S -algebras [8], which relate the layers of the tower to the K-theory of the Morava E -theory ring spectra En and to the K-theory of the p-completed Johnson–Wilson ring spectra E(n)p .”
Nilpotence and descent in equivariant stable homotopy theory (2015, methodology): “We then develop an analogous Z/2-Galois descent picture from equivariant complex to real K-theory (which is due to Rognes [65] for G = 1).”
The Picard group of topological modular forms via descent theory (2014, background): “Theorem E Let G be a finite group, and let A!B be a faithful G–Galois extension of E1–rings in the sense of Rognes [59].”
Separability and triangulated categories (2011, background): “For instance, Rognes [23] offers a thorough investigation of (commutative) separable and étale algebras in stable homotopy theory and provides many examples.”
The Galois group of a stable homotopy theory (2014, methodology): “This is the primary example (together with E-localized versions) considered in [71].”
A modular description of the K(2)-local sphere at the prime 3 (2005, background): “ion (such a subgroup exists since G is compact p-adic analytic). Let Y be the spectrum EhU∧M. The spectrum Y is dualizable since the spectrum EhU is dualizable. In fact, the spectrum EhU is self-dual [42]. 41 Proof of (1). It suﬃces to show that for any spectrum X, if Y ∧Xis null, then Xmust be null”
The C2–spectrum Tmf1(3) and its invertible modules (2015, background): “The extensions TMF0(3)→ TMF1(3) (tmf1(3)[ā −1 1 ]) hC2 → tmf1(3)[ā−11 ] (tmf1(3)[ā −1 3 ]) hC2 → tmf1(3)[ā−13 ] (tmf1(3)[ā −1 1 ā −1 3 ]) hC2 → tmf1(3)[ā−11 ā−13 ] are faithful C2-Galois extensions in the sense of Rognes.”