Higher-dimensional Contou-Carr\`ere symbol and continuous automorphisms

TL;DR

This paper proves the invariance of the higher-dimensional Contou-Carr extbackslash'ere symbol under continuous automorphisms and provides a new explicit formula applicable over arbitrary rings without using algebraic K-theory.

Contribution

It establishes invariance of the higher-dimensional Contou-Carr extbackslash'ere symbol under continuous automorphisms and derives a new explicit, ring-agnostic formula.

Findings

Invariance of the symbol under continuous automorphisms.

New explicit formula valid over arbitrary rings.

Elimination of algebraic K-theory from the formula.

Abstract

We prove that the higher-dimensional Contou-Carr\`ere symbol is invariant under continuous automorphisms of algebras of iterated Laurent series over a ring. Applying this property, we obtain a new explicit formula for the higher-dimensional Contou-Carr\`ere symbol. Unlike previously known formulas, this formula is given over an arbitrary ring, not necessarily a $\mathbb Q$-algebra, and does not involve algebraic $K$-theory.