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Register a new userMotivic Brown-Peterson invariants of the rationals
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Fix the base field Q of rational numbers and let BP<n> denote the family of motivic truncated Brown-Peterson spectra over Q. We employ a local-to-global philosophy in order to compute the motivic Adams spectral sequence converging to the bi-graded homotopy groups of BP<n>. Along the way, we provide a new computation of the homotopy groups of BP<n> over the 2-adic rationals, prove a motivic Hasse principle for the spectra BP<n>, and deduce several classical and recent theorems about the K-theory of particular fields.
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We compute topological Hochschild homology of sufficiently structured forms of truncated Brown--Peterson spectra with coefficients. In particular, we compute $operatorname{THH}_*(operatorname{taf}^D;M)$ for $Min { Hmathbb{Z}_{(3)},k(1),k(2)}$ where $operatorname{taf}^D$ is the $E_{infty}$ form of $BPlangle 2rangle$ constructed by Hill--Lawson. We compute $operatorname{THH}_*(operatorname{tmf}_1(3);M)$ when $Min { Hmathbb{Z}_{(2)},k(2)}$ where $operatorname{tmf}_1(3)$ is the $E_{infty}$ form of $BPlangle 2rangle$ constructed by Lawson--Naumann. We also compute $operatorname{THH}_*(Blangle nrangle;M)$ for $M=Hmathbb{Z}_{(p)}$ and certain $E_3$ forms $Blangle nrangle$ of $BPlangle nrangle$. For example at $p=2$, this result applies to the $E_3$ forms of $BPlangle nrangle$ constructed by Hahn--Wilson.
We equip $mathrm{BP} langle n rangle$ with an $mathbb{E}_3$-$mathrm{BP}$-algebra structure, for each prime $p$ and height $n$. The algebraic $K$-theory of this $mathbb{E}_3$-ring is of chromatic height exactly $n+1$. Specifically, it is an fp-spectrum of fp-type $n+1$, which can be viewed as a higher height version of the Lichtenbaum-Quillen conjecture.
We show that the analogue of the Peterson conjecture on the action of Steenrod squares does not hold in motivic cohomology.
We provide a complete analysis of the motivic Adams spectral sequences converging to the bigraded coefficients of the 2-complete algebraic Johnson-Wilson spectra BPGL<n> over p-adic fields. These spectra interpolate between integral motivic cohomology (n=0), a connective version of algebraic K-theory (n=1), and the algebraic Brown-Peterson spectrum. We deduce that, over p-adic fields, the 2-complete BPGL<n> split over 2-complete BPGL<0>, implying that the slice spectral sequence for BPGL collapses. This is the first in a series of two papers investigating motivic invariants of p-adic fields, and it lays the groundwork for an understanding of the motivic Adams-Novikov spectral sequence over such base fields.
We construct a period regulator for motivic cohomology of an algebraic scheme over a subfield of the complex numbers. For the field of algebraic numbers we formulate a period conjecture for motivic cohomology by saying that this period regulator is surjective. Showing that a suitable Betti--de Rham realization of 1-motives is fully faithful we can verify this period conjecture in several cases. The divisibility properties of motivic cohomology imply that our conjecture is a neat generalization of the classical Grothendieck period conjecture for algebraic cycles on smooth and proper schemes. These divisibility properties are treated in an appendix by B. Kahn (extending previous work of Bloch and Colliot-Thel`ene--Raskind).
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