motivic cohomology and classifying spaces

点击翻译按钮获取中文摘要

A Suslin and V. Voevodsky constructed and developed the motivic cohomology theory H^{*, *''} (X; Z/p) for algebraic sets X over the base field k. This theory is the counter part in algebraic geometry of the usual mod p singular cohomology in algebraic topology. Let ch (k)=0 and fix an embedding that k is subset C. As the counter part of the complex cobordism theory MU^* (X), Voevodsky defined the algebraic cobordism theory MGL^{*, *''} (X) and used it in the first proof of the Milnor Conjecture. Given a nonzero symbol a in K_{n+1}^M (k)/p, the norm variety V_a is a variety such that a=0 in K_{n+1}^M (k (V_a))/p and V_a c=v_n. Here v_n is the 2 (p^n-1) complex manifold generating the coefficient ring of the BP-theory in algebraic topology. In these papers we write down the properties of ABP^{*, *''} (X) which is the algebraic counter part of BP-theory. For example we give a construction of the Atiyah-Hirzebruch spectral sequence for ABP^{*, *} (X; \bZ/p); its existence (of$MGL$-version) was announced by Hopkins and Morel more that several years before, however any proof (or even statement), does not appear yet. We study the cohomology operations, products and Gysin maps explicitly in $ABP^{*, *''}$-theory.