## Hokkaido Mathematical Journal

- Hokkaido Math. J.
- Volume 41, Number 2 (2012), 275-316.

### Algebraic *BP*-theory and norm varieties

#### Abstract

Let *p* be an odd prime and *BP*^{*}(*pt*) ≅ $¥mathbb Z$_{(p)}[*v*_{1},*v*_{2},…] the coefficient ring of the Brown-Peterson cohomology theory *BP*^{*}(−) with |*v*_{i}| = −2*p*^{i} + 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}.

#### Article information

**Source**

Hokkaido Math. J., Volume 41, Number 2 (2012), 275-316.

**Dates**

First available in Project Euclid: 26 June 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.hokmj/1340714416

**Digital Object Identifier**

doi:10.14492/hokmj/1340714416

**Mathematical Reviews number (MathSciNet)**

MR2977048

**Zentralblatt MATH identifier**

1321.14005

**Subjects**

Primary: 14C15: (Equivariant) Chow groups and rings; motives 57T25: Homology and cohomology of H-spaces

Secondary: 55R35: Classifying spaces of groups and $H$-spaces 57T05: Hopf algebras [See also 16T05]

**Keywords**

algebraic cobordism BP-theory norm variety

#### Citation

YAGITA, Nobuaki. Algebraic BP -theory and norm varieties. Hokkaido Math. J. 41 (2012), no. 2, 275--316. doi:10.14492/hokmj/1340714416. https://projecteuclid.org/euclid.hokmj/1340714416