The spectrum of units of algebraic K-theory

Abstract

It is well known that the [ 0 , 1 ] and [ 0 , 2 ] Postnikov truncations of the units of the topological K -theories 𝔤 𝔩 1 ⁢ K O and 𝔤 𝔩 1 ⁢ K U , 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 𝔤 𝔩 1 ⁢ 𝐾 ⁡ ( ℤ ) . Along the way we also give a complete calculation of the connective spectrum of strict units of 𝐾 ⁡ ( ℤ ) and 𝐾 ⁡ ( 𝔽 ℓ ) for a prime ℓ . 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.