Algebra/Topology Seminar by Dustin Clausen
Algebra/Topology Seminar by Dustin Clausen (Copenhagen)
Title: A general approach to Artin maps
Abstract: If F is a global field, then the Artin map for F is a certain homomorphism from (A_F)^*/F^* to the abelianized absolute Galois group of F. Its very existence implies the Artin reciprocity law, a generalization of the quadratic reciprocity law. But there are other fields of arithmetic significance than the global fields, and many of them also have Artin maps. For example, if F is a local field, then the Artin map has source F^* instead. We will describe a way to produce these Artin maps which is uniform in the field, and works even in much more general situations, e.g. for an arbitrary non-commutative ring. A consequence is a new proof of the Artin reciprocity law. The methods are homotopy theoretic, based on a simple topological construction with tori and two new kinds of K-theory.