The local symbol complex of a Reciprocity Functor

TL;DR

This paper investigates the local symbol complex associated with reciprocity functors on smooth curves over algebraically closed fields, establishing an isomorphism between its homology and a specific K-group of reciprocity functors.

Contribution

It introduces a new connection between the homology of the local symbol complex and K-groups of reciprocity functors, under certain conditions.

Findings

Homology of the local symbol complex is isomorphic to a K-group of reciprocity functors.

Provides conditions under which this isomorphism holds.

Enhances understanding of reciprocity functors in algebraic geometry.

Abstract

For a reciprocity functor $\mathcal{M}$ we consider the local symbol complex $\mathcal{M}\otimes^{M}\mathbb{G}_{m}(\eta_{C})\to\oplus_{P\in C}\mathcal{M}(k)\to\mathcal{M}(k)$, where $C$ is a smooth complete curve over an algebraically closed field $k$ with generic point $\eta_{C}$ and $\otimes^{M}$ is the product of Mackey functors. We prove that if $\mathcal{M}$ satisfies certain conditions, then the homology of the above complex is isomorphic to the $K$-group of reciprocity functors $T(\mathcal{M},\underline{CH}_{0}(C)^{0})(Spec k)$.