On the local residue symbol in the style of Tate and Beilinson

Abstract

Tate gave a famous construction of the residue symbol on curves by using some non-commutative operator algebra in the context of algebraic geometry. We explain Beilinson's multidimensional generalization, which is not so well-documented in the literature. We provide a new approach using Hochschild homology.Suppose X/k is a smooth proper algebraic curve over a field. One can define the residue of a rational 1-form ω at a closed point x as (0.1)in terms of a local coordinate t, i.e. by picking an isomorphism Frac O X,x ≃ κ(x)((t)). This works, but is unwieldy since it depends on the choice of the isomorphism and one needs to prove that it is well-defined, cf. Serre [Ser97, Ch. II]. One could ask for a bit more:Aim: Construct the local residue symbol without ever needing to choose coordinates.J. Tate [Tat68] has pioneered an approach which circumvents choices of coordinates at all times by employing ideas in the style of functional analysis: The local fieldThis work has been partially supported by the DFG SFB/TR45 "Periods, moduli spaces, and arithmetic of algebraic varieties" and the Alexander von Humboldt Stiftung.

Local Beilinson–Tate operators

“…The purpose of this paper is to present an analytic variant of the Beilinson-Tate construction for topological local fields, and to study its properties. Presumably our analytic construction agrees with the geometric construction of [Be1,Br2], and the resulting residue functional is the same as residue functional from [Ye1] -see Conjectures 0.9 and 0.12 below.…”

“…For a long time assertions (4) and (5) were essentially neglected and remained cryptic. Very recently we heard about renewed interest in the work of Beilinson, mainly by Braunling, Groechenig and Wolfson [Br1,Br2,BGW]. The papers [Br1,Br2] discuss item (4) above.…”

“…Very recently we heard about renewed interest in the work of Beilinson, mainly by Braunling, Groechenig and Wolfson [Br1,Br2,BGW]. The papers [Br1,Br2] discuss item (4) above. A long term goal of this team is to understand and make precise the global aspect (5) of Beilinson's construction, and then to apply this construction in various directions.…”

“…Here is a definition from [Br2], that distills Beilinson's definition from [Be1]. The notation we use is closer to the original notation of Tate.…”

“…For n = 1 this is Tate's original abstract residue from [Ta]. For n ≥ 2 this was worked out in the recent paper [Br2], using Lie algebra homology and Hochschild homology.…”