Logarithmic motivic homotopy theory
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866912460562235392 |
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| author | Binda, Federico Park, Doosung Østvær, Paul Arne |
| author_facet | Binda, Federico Park, Doosung Østvær, Paul Arne |
| contents | This work is dedicated to the construction of a new motivic homotopy theory for (log) schemes, generalizing Morel-Voevodsky's (un)stable $\mathbb{A}^1$-homotopy category. Our framework can be used to represent log topological Hochschild and cyclic homology, as well as algebraic $K$-theory of regular schemes. Additionally, we can realize the cyclotomic trace as a morphism between motivic spectra. Among our applications, we provide a generalized framework of oriented cohomology theories that enables us to produce new residue sequences for (topological) Hochschild, periodic, and cyclic homology of classical schemes. We also compute $THH$ and its variants for Grassmannians, and we define a new version of algebraic cobordism. Finally, we give a construction of a log étale stable realization functor, as well as a Kato-Nakayama realization functor, which is of independent interest for applications in log geometry. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_02729 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Logarithmic motivic homotopy theory Binda, Federico Park, Doosung Østvær, Paul Arne Algebraic Geometry K-Theory and Homology This work is dedicated to the construction of a new motivic homotopy theory for (log) schemes, generalizing Morel-Voevodsky's (un)stable $\mathbb{A}^1$-homotopy category. Our framework can be used to represent log topological Hochschild and cyclic homology, as well as algebraic $K$-theory of regular schemes. Additionally, we can realize the cyclotomic trace as a morphism between motivic spectra. Among our applications, we provide a generalized framework of oriented cohomology theories that enables us to produce new residue sequences for (topological) Hochschild, periodic, and cyclic homology of classical schemes. We also compute $THH$ and its variants for Grassmannians, and we define a new version of algebraic cobordism. Finally, we give a construction of a log étale stable realization functor, as well as a Kato-Nakayama realization functor, which is of independent interest for applications in log geometry. |
| title | Logarithmic motivic homotopy theory |
| topic | Algebraic Geometry K-Theory and Homology |
| url | https://arxiv.org/abs/2303.02729 |