THE BOREL CHARACTER

The main purpose of this article is to define a quadratic analogue of the Chern character, the so-called Borel character, that identifies rational higher Grothendieck-Witt groups with a sum of rational Milnor-Witt (MW)-motivic cohomologies and rational motivic cohomologies. We also discuss the notion of ternary laws due to Walter, a quadratic analogue of formal group laws, and compute what we call the additive ternary laws, associated with MW-motivic cohomology. Finally, we provide an application of the Borel character by showing that the Milnor-Witt K-theory of a field F embeds into suitable higher Grothendieck-Witt groups of F modulo explicit torsion.

On formal group laws over the quotients of Lazard’s ring

We present a formal power series {\sum A_{ij}x^{i}y^{j}} over the Lazard ring Λ and the formal group laws {F_{n}} , {n\geq 2} , over the quotient rings of Λ. For each {F_{n}} , we construct a complex cobordism theory with singularities with the coefficient ring {\mathbb{Q}[p_{1},\dots,p_{2n}]} , with parameters {p_{i}} , {|p_{i}|=2i} .

Kernels in the Category of Formal Group Laws

Fontaine described the category of formal groups over the ring of Witt vectors over a finite field of characteristic p with the aid of triples consisting of the module of logarithms, the Dieudonné module, and the morphism from the former to the latter. We propose an explicit construction for the kernels in this category in term of Fontaine's triples. The construction is applied to the formal norm homomorphism in the case of an unramified extension of ℚp and of a totally ramiûed extension of degree less or equal than p. A similar consideration applied to a global extension allows us to establish the existence of a strict isomorphism between the formal norm torus and a formal group law coming from L-series.

Some explicit expressions concerning BP

This paper provides some explicit expressions concerning the formal group laws of the following cohomology theories: BP, the Brown–Peterson cohomology,