The local symbol complex of a reciprocity functor

Abstract

For a reciprocity functor $\mathcal{ℳ}$ we consider the local symbol complex

$$\left(\mathcal{ℳ}{\otimes }^M{\mathbb{G}}_m\right)\left({\eta }_C\right)\to \underset{P\in C}{\oplus }\mathcal{ℳ}\left(k\right)\to \mathcal{ℳ}\left(k\right),$$

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 assumptions, then the homology of this complex is isomorphic to the $K$-group of reciprocity functors $T\left(\mathcal{ℳ},{\underset{¯}{CH}}_0{\left(C\right)}^0\right)\left(Speck\right)$.