Polymultiplicative maps associated with the algebra of Iterated Laurent series and the higher-dimensional Contou-Carrere Symbol

TL;DR

This paper investigates functorial polymultiplicative maps on iterated Laurent series algebras, showing they align with the higher-dimensional Contou-Carrère symbol under invariance conditions, revealing a deep algebraic structure.

Contribution

It establishes a characterization of polymultiplicative maps as powers of the higher-dimensional Contou-Carrère symbol, extending understanding of algebraic invariants in iterated Laurent series.

Findings

Polymultiplicative maps invariant under automorphisms are powers of the Contou-Carrère symbol.

The result provides a classification of such maps in terms of known algebraic symbols.

The work connects automorphism invariance with the structure of algebraic symbols in higher dimensions.

Abstract

We study functorial polymultiplicative maps from the multiplicative group of the algebra of $n$-times iterated Laurent series over a commutative ring in $n+1$ variables into the multiplicative group of the ring. It is proven that if such a map is invariant under continuous automorphisms of this algebra, then it coincides, up to a sign, with an integer power of the $n$-dimensional Contou-Carr\`ere symbol.