Homotopic Hopf-Galois extensions of commutative differential graded algebras

This thesis is concerned with the definition and the study of properties of homotopic Hopf-Galois extensions in the category Ch 0 k of chain complexes over a field k, equipped with its projective model structure. Given a differential graded k-Hopf algebra H of finite type, we define a homotopic H-Hopf-Galois extension to be a morphism ' : B ! A of augmented H-comodule dg-k-algebras, where B is equipped with the trivial H-coaction, for which the associated Galois functor ( ')⇤ : M W can ' A ! M W⇢ A and the comparison functor (i') : ModAhcoH !ModB are Quillen equivalences. Here AhcoH denotes the object of homotopy H-coinvariants of the dg-algebra A, and ModAhcoH denotes the category of right modules over AhcoH in Ch 0 k , endowed with the model category structure right-induced by the forgetful functor from Ch 0 k (and similarly for B). The categories M W can ' A and M W⇢ A denote, respectively, the categories of right A⌦B Aand A⌦H-comodules in the category ModA, and they are equipped with the model category structures left-induced from ModA by the forgetful functor. We investigate the behavior of homotopic Hopf-Galois extensions of commutative dg-k-algebras under base change. First, we study their preservation under base change. Given a homotopic H-Hopf-Galois extension ' : B ! A, with B, A commutative, and a morphism f : B ! B0 of commutative dgk-algebras, we determine conditions on ' and f , under which the induced morphism ' : B0 ! B⌦B A is also a homotopic H-Hopf-Galois extension. Secondly, we examine the reflection of such extensions under base change. We suppose that the induced morphism ' : B0 ! B⌦B A is a homotopic HHopf-Galois extension, and we specify conditions on ' and f that guarantee that ' : B ! A was a homotopic H-Hopf-Galois extension. The main result of this thesis establishes one direction of a Hopf-Galois correspondence for homotopic Hopf-Galois extensions over co-commutative dg-k-Hopf-algebras of finite type. We show that if ' : B ! A is a homotopic H-Hopf-Galois extension, and g : H ! K is an inclusion of co-commutative dg-k-Hopf-algebras of finite type, then AhcoK ! A is always a homotopic KHopf-Galois extension, and B ! AhcoK is a homotopic HhcoK-Hopf-Galois extension, provided that A is semi-free as a B-module. We end with an example, derived from the context of simplicial sets, which offers interesting possibilities of application of our main result to principal fibrations of simplicial sets.