#### Vol. 1, No. 3, 2016

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The local symbol complex of a reciprocity functor

### Evangelia Gazaki

Vol. 1 (2016), No. 3, 317–338
##### Abstract

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

$\left(\mathsc{ℳ}{\otimes }^{M}{\mathbb{G}}_{m}\right)\left({\eta }_{C}\right)\to \underset{P\in C}{\oplus }\mathsc{ℳ}\left(k\right)\to \mathsc{ℳ}\left(k\right),$

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 $\mathsc{ℳ}$ satisfies certain assumptions, then the homology of this complex is isomorphic to the $K$-group of reciprocity functors $T\left(\mathsc{ℳ},{\underset{¯}{CH}}_{0}{\left(C\right)}^{0}\right)\left(Spec\phantom{\rule{0.3em}{0ex}}k\right)$.

##### Keywords
reciprocity functor, Milnor $K$-group, local symbol
Primary: 14C25