Hokkaido Mathematical Journal

Hokkaido Mathematical Journal, 41 (2012) pp.275-316

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Abstract

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.

MSC(Primary) 14C15 57T25, 55R35, 57T05 algebraic cobordism, BP-theory, norm variety