Reciprocity laws and K-theory

We associate to a full flag F in an n-dimensional variety X over a field k, a “symbol map” μ_F: K(F_X)→∑ⁿK(k). Here, F_X is the field of rational functions on X, and K (∙) 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).