Trace methods for stable categories I: The linear approximation of algebraic K-theory
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866911476978024448 |
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| author | Harpaz, Yonatan Nikolaus, Thomas Saunier, Victor |
| author_facet | Harpaz, Yonatan Nikolaus, Thomas Saunier, Victor |
| contents | We study algebraic K-theory and topological Hochschild homology in the setting of bimodules over a stable category, a datum we refer to as a laced category. We show that in this setting both K-theory and THH carry universal properties, the former defined in terms of additivity and the latter via trace properties. We then use these universal properties in order to construct a trace map from laced K-theory to THH, and show that it exhibits THH as the first Goodwillie derivative of laced K-theory in the bimodule direction, generalizing the celebrated identification of stable K-theory by Dundas-McCarthy, a result which is the entryway to trace methods. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_04743 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Trace methods for stable categories I: The linear approximation of algebraic K-theory Harpaz, Yonatan Nikolaus, Thomas Saunier, Victor Algebraic Topology K-Theory and Homology 18N60, 19D55, 16E40 We study algebraic K-theory and topological Hochschild homology in the setting of bimodules over a stable category, a datum we refer to as a laced category. We show that in this setting both K-theory and THH carry universal properties, the former defined in terms of additivity and the latter via trace properties. We then use these universal properties in order to construct a trace map from laced K-theory to THH, and show that it exhibits THH as the first Goodwillie derivative of laced K-theory in the bimodule direction, generalizing the celebrated identification of stable K-theory by Dundas-McCarthy, a result which is the entryway to trace methods. |
| title | Trace methods for stable categories I: The linear approximation of algebraic K-theory |
| topic | Algebraic Topology K-Theory and Homology 18N60, 19D55, 16E40 |
| url | https://arxiv.org/abs/2411.04743 |