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Motivic Brown-Peterson invariants of the rationals. (English) Zbl 1276.55023
Let \(BP\langle n\rangle\) be the the family of motivic truncated Brown-Peterson spectra at the prime 2 over a characteristic 0 base field \(k\). The authors perform some calculations for different motivic spectra related to \(BP\). The main tool is the motivic Adams-Novikov spectral sequence. One of the main results is the calculation of the bigraded homotopy groups of the 2-completed \(BP\langle n\rangle\). Another example is the following. Theorem. Let \(k\) be a field with finite virtual cohomological dimension at 2. Let \(x_i, i\geq 1\) denote the standard Lazard ring polynomial generators in degree \(i(1+\alpha)\). Let \(BP=BP_k\) and \(MGL=MGL_k\) denote the 2-complete Brown-Peterson and algebraic cobordism spectra over \(k\). Then there is an equivalence \[ \bigvee_{x_I}\sum^{|x_I|}BP\to MGL \] for a suitable collection of indices \(\{ I\}\). This is the algebraic analogue of a theorem of Milnor and Novikov for the usual cobordism spectrum \(MGL\).
The authors compute the rational homotopy group of \(MGL\) and analyze the motivic Adams spectral sequence for \(BP\) over the rationals and conditions for its convergence.

MSC:
55T15 Adams spectral sequences
19D50 Computations of higher \(K\)-theory of rings
19E15 Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects)
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