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 p>0, we prove that the unipotent homotopy group schemes π i U (⋅) ᵢ^ {U} (\, \, ) introduced in our paper recover the unipotent Nori fundamental group scheme (see M. V. Nori Compositio Math. 33 (1976), pp. 29–41), the p p -adic étale homotopy groups (see M. Artin and B. Mazur Etale homotopy, Springer-Verlag, Berlin-New York, 1969), as well as certain formal groups introduced by Artin and Mazur Ann. Sci. École Norm. Sup. (4) 10 (1977), pp. 87–131. 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 n, the group schemes π i U (⋅) ᵢ^ {U} (\, \, ) are derived invariants for all i ≥ 0 i 0 ; the case i = n i=n is related to recent work of Antieau and Bragg Algebr. Geom. 9 (2022), pp. 364–399 involving topological Hochschild homology. Using the unipotent homotopy group schemes, we establish a correspondence between formal Lie groups and certain higher algebraic structures.