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

:<math>K_n (X) \rightarrow \oplus_{p \geq 0} H_D^{2p-n} (X, \mathbf Q(p)).</math>

Here, ''X'' is a complex smooth [[projective variety]], for example. It is named after [[Alexander Beilinson]]. The Beilinson regulator features in [[Beilinson conjecture|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]] <math>\mathcal O_F</math> of a [[number field]] ''F''

:<math>\mathcal O_F^\times \rightarrow \mathbf R^{r_1 + r_2}, \ \ x \mapsto (\log |\sigma (x)|)_\sigma </math>

is a particular case of the Beilinson regulator. (As usual, <math>\sigma: F \subset \mathbf C</math> runs over all [[complex embedding]]s of ''F'', where conjugate embeddings are considered equivalent.) Up to a factor 2, the Beilinson regulator is also generalization of the [[Borel regulator]].

== References ==
* {{cite book|title=Beilinson's conjectures on special values of L-functions|year=1988|publisher=Academic Press|isbn=0-12-581120-9|editor=M. Rapoport, N. Schappacher and P. Schneider}}

[[Category:Algebraic geometry]]
[[Category:Algebraic K-theory]]

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