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arXiv · 1506.07375

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

Tom Bachmann
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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.

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Tom Bachmann. 2015-06-24. On the Conservativity of the Functor Assigning to a Motivic Spectrum its Motive. https://doi.org/10.1215/00127094-2018-0002

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