Explicit formula for the higher-dimensional Contou-Carrère symbol

Continuous homomorphisms between algebras of iterated Laurent series over a ring

“…The results of this paper will be applied in a forthcoming paper [9] to further investigations of the n-dimensional Contou-Carrère symbol, which was studied by Contou-Carrère himself in [1,2] for the case n = 1, by the second named author and Zhu in [14] for the case n = 2, and by the authors in [6,7,8] for arbitrary n.…”

Higher-dimensional Contou-Carrère symbol and continuous automorphisms

“…It follows that the map sgn is invariant under automorphisms of Z n . One has (equivalent) explicit formulas for the map sgn, see the proof of [ [5]), for any Q-algebra A, there is a unique multilinear antisymmetric map CC n : L n (A) * ×(n+1) −→ A * that satisfies the following properties:…”

A higher-dimensional Contou-Carrère symbol: local theory

“…0.8]. For the induction step, we consider loop functors of all functors in formula (19) with n being replaced by n−1. Then we use the isomorphism Z ≃ LZ induced by morphisms (13) and proved by the second named author and Zhu [37, Lem.…”

“…This gives a new sense to formula (8) as the result of direct calculations of certain canonical maps. (Note that we announced the equality between formulas (7) and (8) in the short note [19], which did not contain proofs.) Also, we show that for any natural number N, the extension of any morphism of functors (L n G m ) ×N Q → (G m ) Q from Q-algebras to all rings is unique, see Theorem 6.12.…”