Motivic Brown-Peterson invariants of the rationals

Fix the base field Q of rational numbers and let BP 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. Along the way, we provide a new computation of the homotopy groups of BP over the 2-adic rationals, prove a motivic Hasse principle for the spectra BP, and deduce several classical and recent theorems about the K-theory of particular fields.Comment: 32 pages, 6 figures; Introduction and exposition improved, typos corrected, now published in Geometry & Topolog

Framed transfers and motivic fundamental classes

We relate the recognition principle for infinite P1-loop spaces to the theory of motivic fundamental classes of Deglise, Jin and Khan. We first compare two kinds of transfers that are naturally defined on cohomology theories represented by motivic spectra: the framed transfers given by the recognition principle, which arise from Voevodsky's computation of the Nisnevish sheaf associated with An/(An-0), and the Gysin transfers defined via Verdier's deformation to the normal cone. We then introduce the category of finite R-correspondences for R a motivic ring spectrum, generalizing Voevodsky's category of finite correspondences and Calmes and Fasel's category of finite Milnor-Witt correspondences. Using the formalism of fundamental classes, we show that the natural functor from the category of framed correspondences to the category of R-module spectra factors through the category of finite R-correspondences