$p$-completions in motivic homotopy theory

Speaker: Klaus Mattis – University of Mainz

Abstract: In homotopy theory, one uses the $p$-completion functors to simplify questions about spectra or anima to the much easier case of $p$-complete spectra or anima. A similar thing can be done in motivic homotopy theory: I will define the unstable $p$-completion functor of a general $\infty$-topos and of the category of motivic spaces. Then I will sketch how one can translate classical results about unstable $p$-completion from homotopy theory to the motivic setting. In particular, I will explain how one can obtain for every nilpotent motivic space $X$ a short exact sequence $$0 \longrightarrow L_0 \pi_n(X) \longrightarrow \pi_n^p(X_p^\wedge) \longrightarrow L_1 \pi_{n-1}(X) \longrightarrow 0,$$ which is analogous to the classical situation. Here, the $L_i$ are versions of the derived $p$-completion functors, and $\pi_n^p$ is a certain “$p$-completed homotopy sheaf”.