Higher-dimensional Contou-Carrère symbol, II

The talk is based on a joint work with Sergey Gorchinskiy. The $n$-dimensional Contou-Carrère symbol is a universal deformation of the $n$-dimensional tame symbol such that it satisfies the Steinberg property from algebraic $K$-theory and it is possible to obtain the $n$-dimensional residue from this symbol. I will give various equivalent definitions of the $n$-dimensional Contou-Carrère symbol: 1) by an explicit "analytic’’ formula over ${\mathbb Q}$-algebras, 2) by means of the action of the group of continuous automorphisms of iterated Laurent series over a ring, 3) by means of algebraic $K$-theory. I will explain also the universal property for the $n$-dimensional Contou-Carrère symbol, the proof of which is based on the statement that the tangent map to the map given by the $n$-dimensional Contou-Carrère symbol is the $n$-dimensional residue.