On the Conservativity of the Functor Assigning to a Motivic Spectrum its Motive

TL;DR

This paper investigates the conservativity of the functor assigning motives to motivic spectra, showing it preserves information under certain field conditions and extending known results to broader contexts.

Contribution

It establishes the conservativity of the motive functor for compact spectra over fields with finite virtual 2-étale cohomological dimension, introducing 'real motives' for this extension.

Findings

The functor M is conservative on compact spectra over certain fields.

M induces an injection on Picard groups under specified conditions.

Reproves a variant of R"ondings-{\

Abstract

Given a 0-connective motivic spectrum $E \in SH(k)$ over a perfect field k, we determine $h_0$ of the associated motive $M E \in DM(k)$ in terms of $\pi_0 (E)$. Using this we show that if k has finite 2-\'etale cohomological dimension, then the functor M is conservative when restricted to the subcategory of compact spectra, and induces an injection on Picard groups. We extend the conservativity result to fields of finite virtual 2-\'etale cohomological dimension by considering what we call "real motives". As a by-product we reprove a variant of a rigidity Theorem of R\"ondings-{\O}stv{\ae}r.