Reciprocity laws and $K\mkern-2mu$-theory

Abstract

We associate to a full flag $\mathcal{ℱ}$ in an $n$-dimensional variety $X$ over a field $k$, a “symbol map” ${\mu }_{\mathcal{ℱ}}:K\left(F_X\right)\to {\Sigma }^{n}K\left(k\right)$. Here, $F_X$ is the field of rational functions on $X$, and $K\left(\cdot \right)$ is the $K$-theory spectrum. We prove a “reciprocity law” for these symbols: given a partial flag, the sum of all symbols of full flags refining it is $0$. Examining this result on the level of $K$-groups, we derive the following known reciprocity laws: the degree of a principal divisor is zero, the Weil reciprocity law, the residue theorem, the Contou-Carrère reciprocity law (when $X$ is a smooth complete curve), as well as the Parshin reciprocity law and the higher residue reciprocity law (when $X$ is higher-dimensional).