Beilinson regulator

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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:

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

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

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

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