Beilinson regulator

In mathematics, especially in algebraic geometry, the Beilinson regulator is the Chern class map from algebraic K-theory to Deligne cohomology:
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 formula_2 of a number field "F"
is a particular case of the Beilinson regulator. (As usual, formula_4 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.