Unipotent homotopy theory of schemes

Building on Toën’s work on affine stacks, we develop a certain homotopy theory for schemes, which we call “unipotent homotopy theory.” Over a field of characteristic $p>0$, we prove that the unipotent homotopy group schemes ${\pi }_i^{\mathrm{U}}(\cdot )$ introduced in our paper recover the unipotent Nori fundamental group scheme, the $p$-adic étale homotopy groups, as well as certain formal groups introduced by Artin and Mazur. We prove a version of the classical Freudenthal suspension theorem as well as a profiniteness theorem for unipotent homotopy group schemes. We also introduce the notion of a formal sphere and use it to show that for Calabi-Yau varieties of dimension $n$, the group schemes ${\pi }_i^{\mathrm{U}}(\cdot )$ are derived invariants for all $i\ge 0$; the case $i=n$ is related to recent work of Antieau and Bragg involving topological Hochschild homology. Using the unipotent homotopy group schemes, we establish a correspondence between formal Lie groups and certain higher algebraic structures.