## Duke Mathematical Journal

### On the conservativity of the functor assigning to a motivic spectrum its motive

Tom Bachmann

#### Abstract

Given a $0$-connective motivic spectrum $E\in\mathbf{SH}(k)$ over a perfect field $k$, we determine $\underline{h}_{0}$ of the associated motive $ME\in\mathbf{DM}(k)$ in terms of $\underline{\pi}_{0}(E)$. Using this, we show that if $k$ has finite $2$-étale cohomological dimension, then the functor $M:\mathbf{SH}(k)\to\mathbf{DM}(k)$ 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.

#### Article information

Source
Duke Math. J., Volume 167, Number 8 (2018), 1525-1571.

Dates
Revised: 21 December 2017
First available in Project Euclid: 28 March 2018

https://projecteuclid.org/euclid.dmj/1522224100

Digital Object Identifier
doi:10.1215/00127094-2018-0002

Mathematical Reviews number (MathSciNet)
MR3807316

Zentralblatt MATH identifier
06896952

#### Citation

Bachmann, Tom. On the conservativity of the functor assigning to a motivic spectrum its motive. Duke Math. J. 167 (2018), no. 8, 1525--1571. doi:10.1215/00127094-2018-0002. https://projecteuclid.org/euclid.dmj/1522224100

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