On very effective hermitian K-theory

Abstract

We argue that the very effective cover of hermitian K-theory in the sense of motivic homotopy theory is a convenient algebro-geometric generalization of the connective real topological K-theory spectrum. This means the very effective cover acquires the correct Betti realization, its motivic cohomology has the desired structure as a module over the motivic Steenrod algebra, and that its motivic Adams and slice spectral sequences are amenable to calculations.

On equivariant and motivic slices

“…where the last equivalence follows from the computation of the slices of the sphere spectrum, cf [33, Proposition 2.9]. By Lemma 3.8, Cell s q (1) ≃ s q (1), and so we conclude that s Cell q (1) ≃ s q (1).…”

“…In the situation of Lemma 2.12, we have that Condition (2) and Condition (3) hold automatically -the former by assumption, and the latter by an easy adjunction argument. However, there seems to be no reason for Condition (1) to hold in general. The extra condition appears to arise because the assumptions in Lemma 2.12 are stronger -if…”

“…We now give a full computation of the cellular slices of the motivic sphere spectrum. This is done using precisely the techniques of Voevodsky, Levine, and others, in computing s q (1). We again let E s,q 2 (MU) denote the E 2 term of the Adams-Novikov spectral sequence, i.e., E s,q 2 (MU) ∼ = Ext s,q MU * MU (MU * , MU * ).…”

Remarks on motivic Moore spectra

“…The effect of the canonical map kq nh → cKW nh on slices is readily obtained, except for the occurrence of Sq 1 . The latter follows from the determination of the first slice differential for cKW nh = cKW ∨ Σ 1,0 cKW, compared with possible first slice differentials for kq nh compatible with the first slice differential for kq, as described in [1,Theorem 3.5]. Determining the first slice differential for kq nh is then essentially straightforward.…”

“…Proof. The determination of the relevant part of the slice spectral sequence for kq given in [1,Proposition 27] implies that the K MW -module π 1−(⋆) f 1 kq is an extension of the K MW -module h ⋆,1+⋆ = π 1−(⋆) s 1 kq (on which η operates trivially) and the K MW -module h 1+⋆,2+⋆ /Sq 2 h ⋆−1,1+⋆ (on which η operates trivially as well). The K M -module h ⋆,1+⋆ = π 1−(⋆) s 1 kq is generated by the image of η top ∈ π 1,0 1, and the K M -module h 1+⋆,2+⋆ /Sq 2 h ⋆−1,1+⋆ is generated by the image of ηη top ∈ π 2,1 1.…”