We associate to a full flag F in an n-dimensional variety X over a field k, a “symbol map” μF: K(FX)→∑nK(k). Here, FX 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).
Bibliographical noteFunding Information:
This research was partially supported by the ERC grant 291612 and by the ISF grant 533/14. MSC2010: 19F15. Keywords: reciprocity laws, K-theory, symbols in arithmetic, Parshin symbol, Parshin reciprocity, Contou-Carrère symbol, Tate vector spaces.
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- Contou-Carrère symbol
- Parshin reciprocity
- Parshin symbol
- Reciprocity laws
- Symbols in arithmetic
- Tate vector spaces