Algebraic BP-theory and norm varieties

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 |v_i| = −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 ∈ K_n+1^M(k)/p, a norm variety V_a is a smooth variety such that a|_k(Va) = 0 ∈ K_n+1^M(k(V_a))/p and V _a($¥mathbb C$) = v_n. In particular, we compute ABP^*,*'(M_a) for the Rost motive M_a which is a direct summand of the motive M(V_a) of some norm variety V_a.