The spectrum of units of algebraic $K$-theory
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912084965457920 |
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| author | Carmeli, Shachar Luecke, Kiran |
| author_facet | Carmeli, Shachar Luecke, Kiran |
| contents | It is well known that the $[0,1]$ and $[0,2]$ Postnikov truncations of the units of the topological $K$-theories $\glone \KO$ and $\glone \KU$, respectively, are split, and that the splitting is provided by the ($\Z/2$-graded) line bundles. In this paper we give a similar splitting for the $[0,1]$-truncation of the units of algebraic $K$-theory, considered as a sheaf on affine schemes. A crucial step is to produce the splitting for $\glone K(\Z)$. Along the way we also give a complete calculation of the connective spectrum of strict units of $K(\Z)$ and $K(\F_\ell)$ for a prime $\ell$. Finally, we show that the units of algebraic $K$-theory do not split as a presheaf. In fact we show they do not even split pointwise. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_10126 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The spectrum of units of algebraic $K$-theory Carmeli, Shachar Luecke, Kiran K-Theory and Homology Algebraic Topology 19D55, 55P60 It is well known that the $[0,1]$ and $[0,2]$ Postnikov truncations of the units of the topological $K$-theories $\glone \KO$ and $\glone \KU$, respectively, are split, and that the splitting is provided by the ($\Z/2$-graded) line bundles. In this paper we give a similar splitting for the $[0,1]$-truncation of the units of algebraic $K$-theory, considered as a sheaf on affine schemes. A crucial step is to produce the splitting for $\glone K(\Z)$. Along the way we also give a complete calculation of the connective spectrum of strict units of $K(\Z)$ and $K(\F_\ell)$ for a prime $\ell$. Finally, we show that the units of algebraic $K$-theory do not split as a presheaf. In fact we show they do not even split pointwise. |
| title | The spectrum of units of algebraic $K$-theory |
| topic | K-Theory and Homology Algebraic Topology 19D55, 55P60 |
| url | https://arxiv.org/abs/2410.10126 |