#### Vol. 2, No. 1, 2017

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

### Evgeny Musicantov and Alexander Yom Din

Vol. 2 (2017), No. 1, 27–46
DOI: 10.2140/akt.2017.2.27
##### Abstract

We associate to a full flag $\mathsc{ℱ}$ in an $n$-dimensional variety $X$ over a field $k$, a “symbol map” ${\mu }_{\mathsc{ℱ}}:K\left({F}_{X}\right)\to {\Sigma }^{n}K\left(k\right)$. 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).

##### Keywords
reciprocity laws, $K\mkern-2mu$-theory, symbols in arithmetic, Parshin symbol, Parshin reciprocity, Contou-Carrère symbol, Tate vector spaces
Primary: 19F15