|Location:||MSRI: Baker Board Room|
In the 1960's Auslander and Goldman, followed shortly by Chase, Harrison, and Rosenberg, generalized the notion of Galois extension from fields to commutative rings and extended the classical Galois correspondence to this new framework. Inspired by their work, Rognes defined a notion of (homotopic) Galois extension for commutative ring spectra and proved a Galois correspondence result for such extensions.
Generalizing from group actions to coactions of Hopf algebras, one obtains the notion of Hopf-Galois extensions of rings, first studied by Chase and Sweedler and by Kreimer and Takeuchi. Rognes showed that the unit map from the sphere spectrum S to MU is not a homotopic Galois extension for any group G, but that it is homotopic Hopf-Galois, in an appropriate sense, with respect to the obvious coaction of S[BU] on MU. Though there is a formal framework in which to study homotopic Hopf-Galois extensions, not much is known about possible homotopic Hopf-Galois correspondences. I will begin by recalling the classical theory of Galois extensions of commutative rings and by sketching Rognes's homotopical generalizations. I will then briefly describe the general notion of homotopic Hopf-Galois extensions and explain recent joint work with my student Varvara Karpova on one direction of a homotopic Hopf-Galois correspondence.