A generalized Contou-Carrère symbol and its reciprocity laws in higher dimensions

Oliver Braunling,Michael Groechenig,Jesse Wolfson

doi:10.1090/btran/81

Abstract

We generalize Contou-Carrère symbols to higher dimensions. To an ( n + 1 ) (n+1) -tuple f 0 , … , f n ∈ A ( ( t 1 ) ) ⋯ ( ( t n ) ) × f_0,\dots ,f_n \in A((t_1))\cdots ((t_n))^{\times } , where A A denotes an algebra over a field k k , we associate an element ( f 0 , … , f n ) ∈ A × (f_0,\dots ,f_n) \in A^{\times } , extending the higher tame symbol for k = A k = A , and earlier constructions for n = 1 n = 1 by Contou-Carrère, and n = 2 n = 2 by Osipov–Zhu. It is based on the concept of higher commutators for central extensions by spectra. Using these tools, we describe the higher Contou-Carrère symbol as a composition of boundary maps in algebraic K K -theory, and prove a version of Parshin–Kato reciprocity for higher Contou-Carrère symbols.

Highlights

This article concerns a higher-dimensional generalization of the Contou-Carrere symbol [CC94]
The original symbol plays a key role in the local theory of generalized Jacobians for a relative curve, as developed by Contou-Carrere [CC79], [CC90]
If the relative curve is just a plain curve over a field, the symbol specializes to the tame symbol

Summary

Introduction

This article concerns a higher-dimensional generalization of the Contou-Carrere symbol [CC94]. Iterated use of the boundary maps ∂ corresponds to an iterated use of Tate categories and a straightforward generalization of Diagram (24) All these constructions (including Equation (25), [BGW18b, Theorem 1.4 (2)]) work for arbitrary rings A and for A((T )) and not just k((t)), making it possible to use it in a relative setting. In the setting of higher dimensional reciprocity laws, we can morally interpret the ring AX,ζ of Theorem 1.4 as the ring of A-valued rational functions of an exotic “curve” Xζ associated to the almost saturated flag {Zj}j=i In principle, this “curve” should be obtained by iteratively completing X at the Zj and removing the special point Zj. at present, the theories of Berkovich or rigid analytic spaces are insufficient to handle such constructions. It is a relatively straightforward matter to show that this composition is zero when restricted to tuples of invertible elements of AX,ζ

K-theory

The CC symbol via boundary maps

Spectral extensions and higher commutators

The CC symbol via Tate categories

Comparison of both definitions

Reciprocity