Rational and p-local motivic homotopy theory
Let F be a perfect field and k an arbitrary field. The main goal of this thesis is to investigate algebraic models for the Morel-Voevodsky unstable motivic homotopy category H(F) after A1-k localization. More specifically, we extend results of Goerss to the A1-algebraic topology setting: we study the homotopy theory of the category of presheaves of simplicial coalgebras over a field k and their Nisnevich (étale) and A1-localizations. For k algebraically closed, we determine the A1-k homotopy type. Furthermore, for rational coefficients, we introduce the notion of A1-nilpotent space and we provide an explicit formula for the rational-A1 localization functor for such spaces. For this goal, we study the category of G-discrete motivic spaces, with G a profinite group. The category of presheaves with Voevodsky transfers with the monoidal structure given by a Day convolution product is locally presentable.