Beilinson regulator

{{Short description|Concept in mathematics}}

{{One source|date=April 2023}}

In mathematics, especially in algebraic geometry, the Beilinson regulator is the Chern class map from algebraic K-theory to Deligne cohomology:

:$K\_n\; (X)\; \backslash rightarrow\; \backslash oplus\_\{p\; \backslash geq\; 0\}\; H\_D^\{2p-n\}\; (X,\; \backslash mathbf\; Q(p)).$

Here, X is a complex smooth projective variety, for example. It is named after Alexander Beilinson. The Beilinson regulator features in Beilinson's conjecture on special values of L-functions.

The Dirichlet regulator map (used in the proof of Dirichlet's unit theorem) for the ring of integers $\backslash mathcal\; O\_F$ of a number field F

:$\backslash mathcal\; O\_F^\backslash times\; \backslash rightarrow\; \backslash mathbf\; R^\{r\_1\; +\; r\_2\},\; \backslash \; \backslash \; x\; \backslash mapsto\; (\backslash log\; |\backslash sigma\; (x)|)\_\backslash sigma$

is a particular case of the Beilinson regulator. (As usual, $\backslash sigma:\; F\; \backslash subset\; \backslash mathbf\; C$ runs over all complex embeddings of F, where conjugate embeddings are considered equivalent.) Up to a factor 2, the Beilinson regulator is also generalization of the Borel regulator.