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

Abstract

Given a 0-connective motivic spectrum $E\in SH(k)$ over a perfect field k, we determine $h_0$ of the associated motive $ME\in DM(k)$ in terms of ${\pi }_0(E)$. Using this we show that if k has finite 2-étale 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-étale cohomological dimension by considering what we call "real motives". As a by-product we reprove a variant of a rigidity Theorem of Röndings-Østvær.

Minor corrections. Accepted for publication in Duke

References in corpus (5)

• From algebraic cobordism to motivic cohomology
• Étale motives
• Motivic and Real Etale Stable Homotopy Theory
• The Witt group of real algebraic varieties
• Witt sheaves and the $\eta$-inverted sphere spectrum

Cited by in corpus (8)

• Motivic and Real Etale Stable Homotopy Theory
• Perfection in motivic homotopy theory
• On the effectivity of spectra representing motivic cohomology theories
• Localizations and completions in motivic homotopy theory
• Rigidity in etale motivic stable homotopy theory
• Motivic Tambara Functors
• Topological models for stable motivic invariants of regular number rings
• Towards conservativity of ${\mathbb{G}}_m$-stabilization