Abstract
We associate to a full flag in an -dimensional variety over a field , a “symbol map” . Here, is the field of rational functions on , and is the -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 . Examining this result on the level of -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 is a smooth complete curve), as well as the Parshin reciprocity law and the higher residue reciprocity law (when is higher-dimensional).
Citation
Evgeny Musicantov. Alexander Yom Din. "Reciprocity laws and $K\mkern-2mu$-theory." Ann. K-Theory 2 (1) 27 - 46, 2017. https://doi.org/10.2140/akt.2017.2.27
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