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The spectrum of units of algebraic $K$-theory

Shachar Carmeli, Kiran Luecke

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.

The spectrum of units of algebraic $K$-theory

Abstract

It is well known that the and Postnikov truncations of the units of the topological -theories and , respectively, are split, and that the splitting is provided by the (-graded) line bundles. In this paper we give a similar splitting for the -truncation of the units of algebraic -theory, considered as a sheaf on affine schemes. A crucial step is to produce the splitting for . Along the way we also give a complete calculation of the connective spectrum of strict units of and for a prime . Finally, we show that the units of algebraic -theory do not split as a presheaf. In fact we show they do not even split pointwise.

Paper Structure

This paper contains 32 sections, 51 theorems, 112 equations.

Table of Contents

  1. Introduction and outline
  2. Background and motivation
  3. Main results
  4. Methods
  5. Acknowledgements
  6. Preliminaries for the splitting of $\mathfrak{gl}_1 R$
  7. $p$-completion
  8. $T(1)$-localization
  9. The homotopy type of $\mathfrak{gl}_1^{\pm 1} R[0,1]$
  10. Splitting $\mathfrak{sl}_1 R$
  11. Strict units and the logarithmic fiber
  12. Preliminaries
  13. Main results
  14. Strict units in $L_{T(1)} K(\mathbb{F}_\ell)$, $p\neq \ell$
  15. Strict units in $L_{T(1)} K(\mathbb{Z})$
  16. ...and 17 more sections

Key Result

Theorem A

The Zariski sheaf $\mathfrak{gl}_1 K$ is split at 1. That is, in the category of Zariski sheaves of connective spectra on affine schemes there is an isomorphism

Theorems & Definitions (97)

  • Theorem A: \ref{['sheafsplit_main']}
  • Theorem B: \ref{['pointwisecounterexample']}
  • Theorem C: \ref{['glintsplittingpcomplete']}
  • Theorem D: \ref{['low_K_correct_splitting']}
  • Theorem E: \ref{['strictunitsKZ']}, \ref{['strictunitsKFl']}
  • Definition 2.0.1
  • Definition 2.0.2
  • Lemma 2.1.1
  • proof
  • Corollary 2.1.2
  • ...and 87 more