Reciprocity laws and $K$-theory

We associate to a full flag $\mathcal{F}$ in an $n$-dimensional variety $X$ over a field $k$, a “symbol map” ${\mu }_{\mathcal{F}}:K(F_X)\to {\mathrm{\Sigma }}^{n}K(k)$. Here, $F_X$ is the field of rational functions on $X$, and $K(\cdot )$ 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 re-obtain various “reciprocity laws”. Namely, when $X$ is a smooth complete curve, we obtain degree of a principal divisor is zero, Weil reciprocity, Residue theorem, Contou-Carr`{e}re reciprocity. When $X$ is higher-dimensional, we obtain Parshin reciprocity.