On equivariant and motivic slices

TL;DR

This paper compares motivic and equivariant slice filtrations over fields with real embeddings, establishing conditions for their equivalence and computing related spectral sequences.

Contribution

It provides conditions under which motivic and equivariant slice towers are equivalent, especially for spectra related to $MGL$, $MR$, and Landweber spectra, linking motivic and equivariant homotopy theories.

Findings

The towers agree for spectra from localized quotients of $MGL$ and $MR$.

Equivariant spectra from localized quotients of $MR$ are even in the Hill--Meier sense.

Computed the slice spectral sequence for $ ext{BP} ext{⟨}n angle/2$ for $1 \,\le\, n \le \infty$.

Abstract

Let $k$ be a field with a real embedding. We compare the motivic slice filtration of a motivic spectrum over $Spec(k)$ with the $C_2$-equivariant slice filtration of its equivariant Betti realization, giving conditions under which realization induces an equivalence between the associated slice towers. In particular, we show that, up to reindexing, the towers agree for all spectra obtained from localized quotients of $MGL$ and $MR$, and for motivic Landweber exact spectra and their realizations. As a consequence, we deduce that equivariant spectra obtained from localized quotients of $MR$ are even in the sense of Hill--Meier, and give a computation of the slice spectral sequence converging to $\pi_{*,*}BP\langle n \rangle/2$ for $1 \le n \le \infty$.