The two-dimensional Contou-Carrère symbol and reciprocity laws

Abstract

Abstract. We define a two-dimensional Contou-Carrère symbol, which is a deformation of the two-dimensional tame symbol and is a natural generalization of the (usual) one-dimensional Contou-Carrère symbol. We give several constructions of this symbol and investigate its properties. Using higher categorical methods, we prove reciprocity laws on algebraic surfaces for this symbol. We also relate the two-dimensional Contou-Carrère symbol to the two-dimensional class field theory.

Explicit formula for the higher-dimensional Contou-Carrère symbol

“…We note that in view of Proposition 1 the explicit formulae in items (i)-(iii) of Theorem 1 uniquely determine an antisymmetric and (n + 1)-multiplicative map CCn for any Q-algebra A. The cases n = 1 and n = 2 of Theorem 1 were proved in [3].…”

“…((tn)) be the ring of iterated Laurent series over A. From [1], [2] ( § 2.1), and [3] ( § 7) we have that for any natural number m the functorial (with respect to A) canonical map ∂m is defined from the group Km(A((t))) to the group Km−1(A). For any natural number n we define a map γn+1 : Kn+1(L n (A)) → A * as the composition det ·∂2 · · · ∂n+1, where the natural map det from K1(A) to A * is induced by the determinant.…”

“…This terminology is justified by the facts (proved in [3], § 7) that CC1 coincides with the classical Contou-Carrère symbol defined in [4] ( § 2.9) and [5], and CC2 coincides with the two-dimensional Contou-Carrère symbol defined and studied in detail in [3].…”

Continuous homomorphisms between algebras of iterated Laurent series over a ring

“…It is not clear how to apply methods from [11] and [12] in the case n > 1 for an arbitrary commutative ring A (cf. [14,Rem. 5.2] for n = 2).…”

Higher-dimensional Contou-Carrère symbol and continuous automorphisms

“…which is functorial with respect to a ring A, and where for a ring R we denote by R * the group of its invertible elements. The one-dimensional Contou-Carrère symbol (or, was obtained in [11] only over such rings, see [11,Def. 3.5,Prop.…”