Unipotent homotopy theory of schemes
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866918126665334784 |
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| author | Mondal, Shubhodip Reinecke, Emanuel |
| author_facet | Mondal, Shubhodip Reinecke, Emanuel |
| contents | 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 $π_i^{\mathrm{U}}(\,\cdot\,)$ 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 $π_i^{\mathrm{U}}(\,\cdot\,)$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_10703 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Unipotent homotopy theory of schemes Mondal, Shubhodip Reinecke, Emanuel Algebraic Geometry Algebraic Topology 14F35, 14L05, 55P99, 18F20 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 $π_i^{\mathrm{U}}(\,\cdot\,)$ 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 $π_i^{\mathrm{U}}(\,\cdot\,)$ 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. |
| title | Unipotent homotopy theory of schemes |
| topic | Algebraic Geometry Algebraic Topology 14F35, 14L05, 55P99, 18F20 |
| url | https://arxiv.org/abs/2302.10703 |