Motivic homotopy theory for perfect schemes
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914071358472192 |
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| author | Dahlhausen, Christian Hekking, Jeroen Wolters, Storm |
| author_facet | Dahlhausen, Christian Hekking, Jeroen Wolters, Storm |
| contents | We construct a perfect version of Morel--Voevodsky's motivic homotopy category over a perfect base scheme in positive characteristic. By checking the axioms of a coefficient system, we establish a six-functor formalism. We show that multiplication by $p$ is already invertible in the perfect motivic homotopy catgory. By work of Elmanto--Khan the functor sending an $\mathbb{F}_p$-scheme $S$ to the category $\mathrm{S}\mathcal{H}(S)[1/p]$ is invariant under universal homeomorphisms, hence under perfections. Our result gives an explicit model for the localization of $\mathrm{S}\mathcal{H}$ at the universal homeomorphisms, which we conclude is the same as $\mathrm{S}\mathcal{H}[1/p]$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_01390 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Motivic homotopy theory for perfect schemes Dahlhausen, Christian Hekking, Jeroen Wolters, Storm Algebraic Geometry K-Theory and Homology 14F42, 19E08 We construct a perfect version of Morel--Voevodsky's motivic homotopy category over a perfect base scheme in positive characteristic. By checking the axioms of a coefficient system, we establish a six-functor formalism. We show that multiplication by $p$ is already invertible in the perfect motivic homotopy catgory. By work of Elmanto--Khan the functor sending an $\mathbb{F}_p$-scheme $S$ to the category $\mathrm{S}\mathcal{H}(S)[1/p]$ is invariant under universal homeomorphisms, hence under perfections. Our result gives an explicit model for the localization of $\mathrm{S}\mathcal{H}$ at the universal homeomorphisms, which we conclude is the same as $\mathrm{S}\mathcal{H}[1/p]$. |
| title | Motivic homotopy theory for perfect schemes |
| topic | Algebraic Geometry K-Theory and Homology 14F42, 19E08 |
| url | https://arxiv.org/abs/2510.01390 |