Unstable $p$-completion in motivic homotopy theory

Abstract

We define unstable $p$-completion in general $\infty$-topoi and the unstable motivic homotopy category, and prove that the $p$-completion of a nilpotent sheaf or motivic space can be computed on its Postnikov tower. We then show that the ($p$-completed) homotopy groups of the $p$-completion of a nilpotent motivic space $X$ fit into short exact sequences $0\to {\mathbb{L}}_0{\pi }_n(X)\to {\pi }_n^p(X_p^{\wedge })\to {\mathbb{L}}_1{\pi }_{n-1}(X)\to 0$, where the ${\mathbb{L}}_i$ are (versions of) the derived $p$-completion functors, analogous to the classical situation.

arXiv comment98 pages. Comments welcome!