Communicated by E.M. Friedlander MSC: 55P91; 55Q70; 14F35 a b s t r a c tFor a profinite group, we construct a model structure on profinite spaces and profinite spectra with a continuous action. This yields descent spectral sequences for the homotopy groups of homotopy fixed point spaces and for stable homotopy groups of homotopy orbit spaces. Our main example is the Galois action on profinite étale topological types of varieties over a field. One motivation is to understand Grothendieck's section conjecture in terms of homotopy fixed points.

We show that there is a stable homotopy theory of profinite spaces and use it for two main applications. On the one hand we construct ań etale topological realization of the stable A 1 -homotopy theory of smooth schemes over a base field of arbitrary characteristic in analogy to the complex realization functor for fields of characteristic zero. On the other hand we get a natural setting forétale cohomology theories. In particular, we define and discuss anétale topological cobordism theory for schemes. It is equipped with an Atiyah-Hirzebruch spectral sequence starting frométale cohomology. Finally, we construct maps from algebraic toétale cobordism and discuss algebraic cobordism with finite coefficients over an algebraically closed field after inverting a Bott element. * et (X; Z/ℓ ν ) defined above. Profiniteétale Morava K-Theory

We provide a new and conceptually simplified construction of continuous homotopy fixed point spectra for Lubin-Tate spectra under the action of the extended Morava stabilizer group. Moreover, our new construction of a homotopy fixed point spectral sequence converging to the homotopy groups of the homotopy fixed points of Lubin-Tate spectra is isomorphic to an Adams spectral sequence converging to the homotopy groups of the spectra constructed by Devinatz and Hopkins. The new idea is built on the theory of profinite spectra with a continuous action by a profinite group.

We study profinite completion of spaces in the model category of profinite spaces and construct a rigidification of the completion functors of Artin-Mazur and Sullivan which extends also to non-connected spaces. Another new aspect is an equivariant profinite completion functor and equivariant fibrant replacement functor for a profinite group acting on a space. This is crucial for applications where, for example, Galois groups are involved, or for profinite Teichmüller theory where equivariant completions are applied. Along the way we collect and survey the most important known results about profinite completion of spaces.

We construct a stable model structure on profinite spectra with a continuous action of an arbitrary profinite group. The motivation is to provide a natural framework in a subsequent paper for a new and conceptually simplified construction of continuous homotopy fixed point spectra and of continuous homotopy fixed point spectral sequences for Lubin-Tate spectra under the action of the extended Morava stabilizer group.