arXiv subjects
Publications and source records attributed to Gereon Quick.
We prove a Verdier Hypercovering Theorem for cohomology theories arising from motivic spectra. This allows us to construct for smooth quasi-projective complex varieties a natural morphism from etale algebraic to Hodge filtered complex cobordism, which extends the map from etale motivic to Deligne-Beilinson cohomology.
We construct Hodge filtered cohomology groups for complex manifolds that combine the topological information of generalized cohomology theories with geometric data of Hodge filtered holomorphic forms. This theory provides a natural generalization of Deligne cohomology. For smooth complex algebraic varieties, we show that the theory satisfies a projective bundle formula and $\A^1$-homotopy invariance. Moreover, we obtain transfer maps along projective morphisms.
We show that the existence of rational points on smooth varieties over a field can be detected using homotopy fixed points of etale topological types under the Galois action. As our main example we show that the surjectivity statement in Grothendieck's Section Conjecture would follow from the surjectivity of the map from fixed points to continuous homotopy fixed points on the level of connected components. Along the way we define a new model for the continuous etale homotopy fixed point space of a smooth variety over a field under the Galois action.
In this short note we show that the homotopy category of smooth compactifications of smooth algebraic varieties is equivalent to the homotopy category of smooth varieties over a field of characteristic zero. As an application we show that the functor sending a variety to the p-th step of the Hodge filtration of its complex cohomology is representable in the homotopy category of simplicial presheaves on smooth complex varieties. The main motivation are recent applications of homotopy theory to Deligne-Beilinson cohomology theories.
We construct a stable model structure on profinite symmetric spectra with a continuous action of an arbitrary profinite group. This provides a natural framework for a new construction of homotopy fixed point spectra and of homotopy fixed point spectral sequences for the action of the extended Morava stabilizer group on Lubin-Tate spectra. These continuous homotopy fixed points are canonically equivalent to the homotopy fixed points of Devinatz and Hopkins but have a drastically simplified construction.
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 Teichmueller theory where equivariant completions are applied. Along the way we collect and survey the most important known results about profinite completion of spaces.
For 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 space 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.
Following an idea of Totaro, we prove that the classical integral cycle class map from algebraic cycles to étale cohomology factors through a quotient of $\ell$-adic étale cobordism over an algebraically closed field of positive characteristic. This shows that there is a strong topological obstruction for cohomology classes to be algebraic and that examples of Atiyah, Hirzebruch and Totaro also work in positive characteristic.
We construct a model structure on simplicial profinite sets such that the homotopy groups carry a natural profinite structure. This yields a rigid profinite completion functor for spaces and pro-spaces. One motivation is the étale homotopy theory of schemes in which higher profinite étale homotopy groups fit well with the étale fundamental group which is always profinite. We show that the profinite étale topological realization functor is a good object in several respects.
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 an étale topological realization of the stable motivic 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 for étale cohomology theories. In particular, we define and discuss an étale topological cobordism theory for schemes. It is equipped with an Atiyah-Hirzebruch spectral sequence starting from étale cohomology. Finally, we construct maps from algebraic to étale cobordism and discuss algebraic cobordism with finite coefficients over an algebraically closed field after inverting a Bott element.
Profinite etale cobordism is a cohomology theory for smooth schemes of finite type over a field. Using an idea of Friedlander, it is constructed as an etale topological analog of the algebraic cobordism theories of Voevodsky and Levine-Morel. With finite coefficients, profinite etale cobordism has a converging Atiyah-Hirzebruch spectral sequence starting from etale cohomology. There are interesting comparison maps from algebraic to profinite etale cobordism. We conjecture that algebraic and etale cobordism with finite coefficients are isomorphic after inverting a Bott element. For the development of the theory, we construct a stable category of profinite spectra and an etale realization of the category of stable motivic spectra over a field.