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arXiv · 2410.10126

The spectrum of units of algebraic $K$-theory

Shachar Carmeli·Kiran Luecke
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Abstract

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.

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Shachar Carmeli, Kiran Luecke. 2024-10-25. The spectrum of units of algebraic $K$-theory. https://arxiv.org/abs/2410.10126

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