Let p be an odd prime and BP*(pt) ≅ $¥mathbb Z$(p)[v1,v2,…] the coefficient ring of the Brown-Peterson cohomology theory BP*(−) with |vi| = −2pi + 2. We study ABP*,*'(−) theory, which is the counter part in algebraic geometry of the BP*(−) theory. Let k be a field with k ⊂ $¥mathbb C$ and K*M(k) the Milnor K-theory. For a nonzero symbol a ∈ Kn+1M(k)/p, a norm variety Va is a smooth variety such that a|k(Va) = 0 ∈ Kn+1M(k(Va))/p and V a($¥mathbb C$) = vn. In particular, we compute ABP*,*'(Ma) for the Rost motive Ma which is a direct summand of the motive M(Va) of some norm variety Va.
"Algebraic BP-theory and norm varieties." Hokkaido Math. J. 41 (2) 275 - 316, June 2012. https://doi.org/10.14492/hokmj/1340714416