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.