Unipotent homotopy theory of schemes

TL;DR

This paper develops a unipotent homotopy theory for schemes over fields of characteristic p, connecting it to classical invariants like the Nori fundamental group and formal groups, and establishing new invariance and structural results.

Contribution

It introduces unipotent homotopy group schemes for schemes, linking them to existing invariants and proving foundational theorems like Freudenthal suspension and profiniteness.

Findings

Unipotent homotopy group schemes recover classical invariants.

Proved a version of Freudenthal suspension theorem.

Established derived invariance for Calabi-Yau varieties.

Abstract

Building on To\"en'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 \'etale 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.