Beilinson regulator
# Beilinson regulator

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

    $K_{n}(X)\rightarrow \oplus _{p\geq 0}H_{D}^{2p-n}(X,\mathbf {Q} (p)).$

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 ${\mathcal {O}}_{F}$ of a number field F

    ${\mathcal {O}}_{F}^{\times }\rightarrow \mathbf {R} ^{r_{1}+r_{2}},\ \ x\mapsto (\log |\sigma (x)|)_{\sigma }$

is a particular case of the Beilinson regulator. (As usual, $\sigma :F\subset \mathbf {C}$ 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.

